Cosmological Constraints on Unstable ParticlesNumerical Bounds and Analytic Approximations

Keith R Dienes12lowast Jason Kumar3dagger Patrick Stengel45Dagger Brooks Thomas6sect1Department of Physics University of Arizona Tucson AZ 85721 USA

2Department of Physics University of Maryland College Park MD 20742 USA3Department of Physics and Astronomy University of Hawaii Honolulu HI 96822 USA

4Leinweber Center for Theoretical Physics Department of PhysicsUniversity of Michigan Ann Arbor MI 48109 USA

5Oskar Klein Centre for Cosmoparticle Physics Department of PhysicsStockholm University Alba Nova 10691 Stockholm Sweden

6Department of Physics Lafayette College Easton PA 18042 USA

Many extensions of the Standard Model predict large numbers of additional unstable particleswhose decays in the early universe are tightly constrained by observational data For example thedecays of such particles can alter the ratios of light-element abundances give rise to distortions inthe cosmic microwave background alter the ionization history of the universe and contribute to thediffuse photon flux Constraints on new physics from such considerations are typically derived fora single unstable particle species with a single well-defined mass and characteristic lifetime In thispaper by contrast we investigate the cosmological constraints on theories involving entire ensemblesof decaying particles mdash ensembles which span potentially broad ranges of masses and lifetimes Inaddition to providing a detailed numerical analysis of these constraints we also formulate a set ofsimple analytic approximations for these constraints which may be applied to generic ensembles ofunstable particles which decay into electromagnetically-interacting final states We then illustratehow these analytic approximations can be used to constrain a variety of toy scenarios for physicsbeyond the Standard Model For ease of reference we also compile our results in the form of a tablewhich can be consulted independently of the rest of the paper It is thus our hope that this workmight serve as a useful reference for future model-builders concerned with cosmological constraintson decaying particles regardless of the particular model under study

Contents

I Introduction 1

II Electromagnetic Injection Overview andClassification of Relevant Processes 3

III Impact on Light-Element Abundances 4A Reprocessed Injection Spectrum 4B Light-Element ProductionDestruction 6C Constraints on Primordial Light-Element

Abundances 7D Towards an Analytic Approximation

Linearization and Decoupling 9E Analytic Approximation Contribution from

Primary Processes 12F Analytic Approximation Contribution from

Secondary Processes 15G The Fruits of Linearization Light-Element

Constraints on Ensembles of UnstableParticles 18

IV Spectral Distortions in the CMB 23

lowastE-mail address dienesemailarizonaedudaggerE-mail address jkumarhawaiieduDaggerE-mail address patrickstengelfysiksusesectE-mail address thomasbdlafayetteedu

A The micro Parameter 23B The Compton y Parameter 25C The Fruits of Linearization

CMB-Distortion Constraints on Ensemblesof Unstable Particles 25

V Modifications to the Ionization History ofthe Universe 27

VI Contributions to the Diffuse PhotonBackground 28

VII Applications Two Examples 30A Results for Uniform Mass Splittings 31B Results for Exponentially Rising Mass

Splittings 35

VIII Conclusions Discussion and Summary ofResults 38

Acknowledgments 39

References 39

I INTRODUCTION

Many proposals for physics beyond the StandardModel (SM) predict the existence of additional unstableparticles The decays of such particles can have a variety

arX

iv1

810

1058

7v2

[he

p-ph

] 1

4 Fe

b 20

19

2

of observable consequences mdash especially if the final statesinto which these particles decay involve visible-sectorparticles Indeed electromagnetic or hadronic showersprecipitated by unstable-particle decays within the recentcosmological past can alter the primordial abundances oflight nuclei both during and after Big-Bang nucleosyn-thesis (BBN) [1ndash5] give rise to spectral distortions inthe cosmic microwave background (CMB) [6 7] alter theionization history of the universe [8ndash12] and give rise tocharacteristic features in the diffuse photon backgroundThese considerations therefore place stringent constraintson models for new physics involving unstable particles

Much previous work has focused on examining thecosmological consequences of a single particle speciesdecaying in isolation and the corresponding limits onthe properties of such a particle species are now wellestablished Indeed simple analytic approximations canbe derived which accurately model the effects that thedecays of such a particle can have on many of therelevant observables [3 6 13] However many theoriesfor new physics involve not merely one or a few unstableparticles but rather a large mdash and potentially vastmdash number of such particles with a broad spectrum ofmasses lifetimes and cosmological abundances Forexample theories with additional spacetime dimensionsgive rise to infinite towers of Kaluza-Klein (KK) exci-tations for any field which propagates in the higher-dimensional bulk Likewise string theories generallypredict large numbers of light moduli [14ndash16] or axion-like particles [17ndash19] Collections of similar light fieldsalso arise in supergravity theories [20] as well as inother scenarios for new physics [21 22] There evenexist approaches to the dark-matter problem such asthe Dynamical Dark Matter (DDM) framework [23 24]which posit the existence of potentially vast ensemblesof unstable dark-sector particles It is therefore crucialto understand the cosmological consequences of entireensembles of decaying particles in the early universe andif possible to formulate a corresponding set of analyticapproximations which model the effects of these decays

For a variety of reasons assessing the effects of anentire ensemble of decaying particles with a broad rangeof masses lifetimes and cosmological abundances isnot merely a matter of trivially generalizing the resultsobtained in the single-particle case The decay of a givenunstable particle amounts to an injection of additionalelectromagnetic radiation andor other energetic parti-cles into the evolution of the universe and injection atdifferent characteristic timescales during this evolutioncan have markedly different effects on the same observ-able Moreover since many of these observables evolvein time according to a complicated system of coupledequations the effects of injection at any particular timetinj depend in a non-trivial way on the entire injectionhistory prior to tinj through feedback effects

In principle the cosmological constraints on ensem-bles of decaying particles in the early universe canbe evaluated numerically Indeed there are several

publicly available codes [12 25 26] which can readilybe modified in order to assess the effects of an arbitraryadditional injection history on the relevant observablesComputational methods can certainly yield useful resultsin any particular individual case However anothercomplementary approach which can provide additionalphysical insight into the underlying dynamics involvesthe formulation of approximate analytic expressions forthe relevant observables mdash expressions analogous tothose which already exist for a single particle speciesdecaying in isolation Our aim in this paper is to derivesuch a set of analytic expressions mdash expressions whichare applicable to generic theories involving large numbersof unstable particles but which nevertheless provideaccurate approximations for the relevant cosmologicalobservables Thus our results can serve as a usefulreference for future model-builders concerned with cos-mological constraints on decaying particles regardless ofthe particular model under study

In this paper we shall focus primarily on the case inwhich electromagnetic injection dominates mdash ie thecase in which the energy liberated by the decays of theseparticles is released primarily in the form of photonselectrons and positrons rather than hadrons We alsoemphasize that the approximations we shall derive in thispaper are not ad hoc in nature in particular they are notthe results of empirical fits Rather as we shall see theyemerge organically from the underlying physics and thuscarry direct information about the underlying processesinvolved

This paper is organized as follows In Sect II webegin by establishing the notation and conventions thatwe shall use throughout this paper We also reviewthe various scattering processes through which energeticphotons injected by particle decay interact with otherparticles present in the radiation bath In subsequentsections we then turn our attention to entire ensemblesof unstable particles focusing on the electromagneticinjections arising from decays occurring after the BBNepoch Each section is devoted to a different cosmologicalconsideration arising from such injection and in eachcase we ultimately obtain a simple analytic approxi-mation for the corresponding constraint For examplein Sect III we consider the constraints associated withthe modification of the abundances of light nuclei afterBBN and in Sect IV we consider limits on distortionsof the CMB-photon spectrum Likewise in Sect V weconsider the constraints associated with the ionizationhistory of the universe and its impact on the CMB andin Sect VI we consider the constraints associated withadditional contributions from unstable-particle decays tothe diffuse photon background Ultimately the resultsfrom these sections furnish us with the tools needed toconstrain decaying ensembles of various types This isthen illustrated in Sect VII where we consider howour results may be applied to two classes of ensembleswhose constituents exhibit different representative massspectra Finally in Sect VIII we conclude with a

3

discussion of our main results and avenues for futurework For future reference we also provide (in Table IV)a summarycompilation of our main results

II ELECTROMAGNETIC INJECTIONOVERVIEW AND CLASSIFICATION OF

RELEVANT PROCESSES

Our aim in this paper is to assess the cosmologicalconstraints on an ensemble consisting of a potentiallylarge number N of unstable particle species χi withmasses mi and decay widths Γi (or equivalently lifetimesτi equiv Γminus1

i ) where the index i = 0 1 N minus 1 labelsthese particle species in order of increasing mass miWe shall characterize the cosmological abundance ofeach of the χi in terms of a quantity Ωi which we callthe ldquoextrapolated abundancerdquo This quantity representsthe abundance that the species χi would have had atpresent time had it been absolutely stable We shallassume that the total abundance Ωtot equiv

sumi Ωi of the

ensemble is sufficiently small that the universe remainsradiation-dominated until the time of matter-radiationequality tMRE asymp 1012 s Moreover we shall focus onthe regime in which mi amp 1 GeV for all χi and allof the ensemble constituents are non-relativistic by endof the BBN epoch Within this regime as we shalldiscuss in further detail below the spectrum of energeticphotons produced by electromagnetic injection takes acharacteristic form which to a very good approximationdepends only on the overall energy density injected [1]By contrast for much lighter decaying particles theform of the resulting photon spectrum can differ fromthis characteristic form as a result of the immediatedecay products lacking sufficient energy to induce theproduction of e+eminus pairs by scattering off backgroundphotons [27 28] The cosmological constraints on asingle electromagnetically-decaying particle species witha mass below 1 GeV mdash investigated earlier to constrainneutrinos (see eg Ref [29]) mdash have recently beeninvestigated in Refs [30 31]

The considerations which place the most stringent con-straints on the ensemble depend on the values of Ωi and τifor the individual ensemble constituents For ensemblesof particles with lifetimes in the range 1 s τi tnowwhere tnow denotes the present age of the universe thedominant constraints are those related to the abundancesof light elements to spectral distortions of the CMBand to the ionization history of the universe Theeffect of electromagnetic injection on the correspondingobservables is sensitive to the overall energy densityinjected and to the timescales over which that energyis injected but not to the details of the decay kinematicsor the particular channels through which the χi decayThus in order to retain as much generality as possiblein our analysis we shall focus on ensembles for which allconstituents with non-negligible Ωi have lifetimes withinthis range moreover we shall refrain from specifying any

particular decay channel for the χi when assessing thebounds on these ensembles due to these considerationsBy contrast the constraints on decaying ensembles thatfollow from limits on features in the diffuse photonbackground do depend on the particulars of the decaykinematics Thus when analyzing these constraints inSect VI we not only present a general expression for therelevant observable mdash namely the contribution to thedifferential photon flux from the decaying ensemble mdashbut also apply this result to a concrete example involvinga particular decay topology

Many of the constraints on electromagnetic injec-tion are insensitive to the details of the decay kine-matics because the injection of photons and otherelectromagnetically-interacting particles prior to CMBdecoupling sets into motion a complicated chain of inter-actions which serve to redistribute the energies of theseparticles In particular the effects of electromagneticinjection on cosmological observables ultimately dependon the interplay between three broad classes of processesthrough which these photons interact with other particlesin the background plasma These are

bull Class I Cascade and cooling processes whichrapidly redistribute the energy of the injectedphotons Processes in this class include γ+ γBG rarre+ + eminus and γ + γBG rarr γ + γ where γBG

denotes a background photon as well as inverse-Compton scattering and e+eminus pair production offnuclei These processes occur on timescales farshorter than the timescales associated with otherrelevant processes and thus may be considered tobe effectively instantaneous As we shall see theseprocesses serve to establish a non-thermal popula-tion of photons with a characteristic spectrum

bull Class II Processes through which the non-thermalpopulation of photons established by Class-I pro-cesses can have a direct effect on cosmologicalobservables These include the photoproductionand photodisintegration of light elements during orafter BBN as well as the photoionization of neutralhydrogen and helium after recombination

bull Class III Processes which serve to bring thenon-thermal population of photons established byClass-I processes into kinetic andor thermal equi-librium with the radiation bath Processes in thisclass include Compton scattering bremsstrahlungand e+eminus pair production off nuclei

We emphasize that these classes are not necessarily mu-tually exclusive and that certain processes play differentroles during different cosmological epochs

Any energy injected in the form of photons prior tolast scattering is rapidly redistributed to lower energiesdue to the Class-I processes discussed above The resultis a non-thermal contribution to the photon spectrumat high energies with a normalization that depends onthe total injected power and a generic shape which

4

is essentially independent of the shape of the initialinjection spectrum directly produced by χi decays Thisldquoreprocessedrdquo photon spectrum serves as a source offor Class-II processes mdash processes which include forexample reactions that alter the abundances of lightnuclei and scattering processes which contribute to theionization of neutral hydrogen and helium after recombi-nation Since all information about the detailed shape ofthe initial injection spectrum from decays is effectivelywashed out by Class-I processes in establishing thisreprocessed photon spectrum the results of our analysisare largely independent of the kinematics of χi decayThis is ultimately why many of our results mdash includingthose pertaining to the alteration of light abundancesafter BBN distortions in the CMB and the ionizationhistory of the universe mdash are likewise largely insensitiveto the decay kinematics of the χi

The timescale over which injected photons can causethese alterations is controlled by the Class-III processesPrior to CMB decoupling these processes serve toldquodegraderdquo the reprocessed photon spectrum establishedby Class-I processes by bringing this non-thermal pop-ulation of photons into kinetic or thermal equilibriumwith the photons in the radiation bath As this occursthese Class-III interactions reduce the energies of thephotons below the threshold for Class-II processes whilealso potentially altering the shape of the CMB-photonspectrum These Class-III processes eventually freezeout as well after which point any photons injected byparticle decays simply contribute to the diffuse extra-galactic photon background

III IMPACT ON LIGHT-ELEMENTABUNDANCES

We begin our analysis of the cosmological constraintson ensembles of unstable particles by considering theeffect that the decays of these particles have on the abun-dances of light nuclei generated during BBN We shallassume that these decays occur after BBN has concludedie after initial abundances for these nuclei are alreadyestablished We begin by reviewing the properties of thenon-thermal photon spectrum which is established by therapid reprocessing of injected photons from these decaysby Class-I processes We then review the correspondingconstraints on a single unstable particle species [3] mdashconstraints derived from a numerical analysis of thecoupled system of Boltzmann equations which govern theevolution of these abundances We then set the stage forour eventual analysis by deriving a set of analytic approx-imations for the above constraints and demonstratingthat the results obtained from these approximationsare in excellent agreement with the results of a fullnumerical computation within our regime of interestFinally we apply our analytic approximations in orderto constrain scenarios involving an entire ensemble ofmultiple decaying particles exhibiting a range of masses

and lifetimes

A Reprocessed Injection Spectrum

As discussed in Sect II the initial spectrum of photonsinjected at time tinj is redistributed effectively instan-taneously by Class-I processes A detailed treatmentof the Boltzmann equations governing these processesin a radiation-dominated epoch can be found eg inRef [1] For injection at times tinj 1012 s theresulting reprocessed photon spectrum turns out to take acharacteristic form which we may parametrize as follows

dnγ(E tinj)

dEdtinj=

dρ(tinj)

dtinjK(E tinj) (31)

The quantity dρ(tinj)dtinj appearing in this expressionwhich specifies the overall normalization of the contri-bution to the reprocessed photon spectrum representsthe energy density injected by particle decays during theinfinitesimal time interval from tinj to tinj + dtinj ThefunctionK(E tinj) on the other hand specifies the shapeof the spectrum as a function of the photon energy EThis function is normalized such thatint infin

0

K(E tinj)EdE = 1 (32)

It can be shown that for any process that injectsenergy primarily through electromagnetic (rather thanhadronic) channels the function K(E tinj) takes theuniversal form [1 32 33]

K(E tinj) =

K0(EXE)32 E lt EXK0(EXE)2 EX lt E lt EC0 E gt EC

(33)where K0 is an overall normalization constant and whereEC and EX are energy scales associated with specificClass-I processes whose interplay determines the shapeof the reprocessed photon spectrum The normalizationconvention in Eq (32) implies that K0 is given by

K0 =1

E2X

[2 + ln(ECEX)

] (34)

Physically the energy scales EC and EX appearing inEq (33) can be understood as follows The scale ECrepresents the energy above which the photon spectrumis effectively extinguished by the pair-production processγ + γBG rarr e+ + eminus in conjunction with interactionsbetween the resulting electron and positron and otherparticles in the thermal bath The energy scale EXrepresents the threshold above which γ + γBG rarr γ + γis the dominant process through which photons loseenergy By contrast below this energy threshold thedominant processes are Compton scattering and e+eminus

5

pair-production off nuclei Note that while the nor-malization of the reprocessed photon spectrum is setby dρ(tinj)dtinj the shape of this spectrum is entirelycontrolled by the temperature Tinj at injection This

temperature behaves like Tinj prop tminus12inj in a radiation-

dominated epoch This implies that as tinj increases thevalue of EC also increases This reflects the fact thatthe thermal bath is colder at later injection times andthus an injected photon must be more energetic in orderfor the pair-production process γ + γBG rarr e+ + eminus tobe effective Numerically the values of EC and EX at agiven injection time tinj are estimated to be [1 3]

EC =m2e

22Tinjasymp(103 MeV

)times(tinj

108 s

)12

EX =m2e

80Tinjasymp(28 MeV

)times(tinj

108 s

)12

(35)

where me is the electron mass and Tinj is the temperatureof the thermal bath at tinj

The reprocessed photon spectrum in Eq (31) is thespectrum which effectively contributes to the photo-production andor photodisintegration of light elementsafter the BBN epoch In order to illustrate how theshape of this spectrum depends on the injection timetinj we plot in Fig 1 the function K(E tinj) whichdetermines the shape of this spectrum as a function ofE for several different values of the injection time tinjSince the ultraviolet cutoff EC in the photon spectrumincreases with tinj injection at later times can initiatephotoproduction and photodisintegration reactions withhigher energy thresholds The dashed vertical line whichwe include for reference represents the lowest thresholdenergy associated with any such reaction which can havea significant impact on the primordial abundance of anylight nucleus which is tightly constrained by observationAs discussed in Sect III B this reaction turns outto be the deuterium-photodisintegration reaction D +γ rarr n + p which has a threshold energy of roughly22 MeV Thus the portion of the photon spectrumwhich lies within the gray shaded region in Fig 1 has noeffect on the abundance of any relevant nucleus Since EClies below this threshold for tinj asymp 104 s electromagneticinjection between the end of BBN and this timescale hasessentially no effect on the abundances of light nuclei

One additional complication that we must take intoaccount in assessing the effect of injection on the abun-dances of light nuclei is that the reprocessed photonspectrum established by Class-I processes immediatelyafter injection at tinj is subsequently degraded by Class-III processes which slowly act to bring this reprocessedspectrum into thermal equilibrium with the backgroundphotons in the radiation bath The timescale δtth(E tinj)on which these processes act on a photon of energy E is

01 1 10 100 1000 10410-7

10-5

0001

0100

10

E [MeV]

K[MeV

-2]

tinj = 104 s

tinj = 106 s

tinj = 108 s

tinj = 1010 s

tinj = 1012 s

FIG 1 The function K(E tinj) plotted as a function ofphoton energy E for several different values of injection timetinj As discussed in the text this function determines theshape of the reprocessed photon spectrum for an instanta-neous injection of photons of energy E at time tinj Thedashed vertical line at E asymp 22 MeV indicates the lowestthreshold energy associated with any photoproduction orphotodisintegration reaction which can significantly alter theabundance of any light nucleus whose primordial abundanceis tightly constrained by observation

roughly

δtth(E tinj) asymp [nB(tinj)σth(E)]minus1

(36)

where nB(tinj) is the number density of baryons at thetime of injection and where σth(E) is the characteristiccross-section for the relevant scattering processes whichinclude Compton scattering and e+eminus pair productionoff nuclei The cross-sections for all relevant individualcontributing processes can be found in Ref [1] Note thatwhile at low energies σth(E) is well approximated by theThomson cross-section this approximation breaks downat higher energies as other processes become relevant

Given these observations the spectrum of the resultingnon-thermal population of photons at time t not onlyrepresents the sum of all contributions from injection atall times tinj lt t but also reflects the subsequent degra-dation of these contributions by the Class-III processeswhich serve to thermalize this population of photons withthe radiation bath This overall non-thermal photonspectrum takes the form

dnγ(E t)

dE=

int t

0

dnγ(E tinj)

dEdtinjG(t tinj)dtinj (37)

where

G(t tinj) equiv eminus(tminustinj)δtth(Etinj)Θ(tminus tinj) (38)

6

is the Greenrsquos function which solves the differentialequation

dG(t tinj)

dt+

G(t tinj)

δtth(E tinj)= δ(tminus tinj) (39)

The population of non-thermal photons described byEq (37) serves as the source for the initial photo-production and photodisintegration reactions ultimatelyresponsible for the modification of light-element abun-dances after BBN We shall therefore henceforth refer tophotons in this population as ldquoprimaryrdquo photons

In what follows we will find it useful to employwhat we shall call the ldquouniform-decay approximationrdquoSpecifically we shall approximate the full exponentialdecay of each dark-sector species χi as if the entirepopulation of such particles throughout the universe wereto decay precisely at the same time τi As we shall seethis will prove critical in allowing us to formulate ourultimate analytic approximations We shall neverthelessfind that the results of our approximations are generallyin excellent agreement with the results of a full numericalanalysis

Within the uniform-decay approximation the contri-bution to dρdt(tinj) from the decay of a single unstableparticle species χ takes the form of a Dirac δ-function

dρ(tinj)

dtinj= ρχ(tinj)εχδ(tinj minus τχ) (310)

where ρχ(tinj) is the energy density of χ at time tinj andwhere εχ is the fraction of the energy density releasedby χ decays which is transferred to photons It thereforefollows that in this approximation the primary-photonspectrum in Eq (37) reduces to

dnγ(E t)

dE= ρχ(τχ)εχK(E τχ)

times eminus(tminusτχ)δtth(Etinj)Θ(tminus τχ) (311)

B Light-Element ProductionDestruction

Generally speaking the overall rate of change of thenumber density na of a nuclear species Na due tothe injection of electromagnetic energy at late times isgoverned by a Boltzmann equation of the form

dnadt

+ 3Hna = C(p)a + C(s)

a (312)

where H is the Hubble parameter and where C(p)a and

C(s)a are the collision terms associated with two different

classes of scattering processes which contribute to thisoverall rate of change We shall describe these individualcollision terms in detail below Since na is affected byHubble expansion it is more convenient to work with thecorresponding comoving number density Ya equiv nanB

where nB denotes the total number density of baryonsThe Boltzmann equation for Ya then takes the form

dYadt

=1

nB

[C(p)a + C(s)

a

] (313)

The collision term C(p)a represents the collective con-

tribution from Class-II processes directly involving thepopulation of primary photons described by Eq (311)

The principal processes which contribute to C(p)a are

photoproduction processes of the form Nb+γ rarr Na+Ncand photodisintegration processes of the form Na + γ rarrNb + Nc where Nb and Nc are other nuclei in thethermal plasma The collision term associated with theseprocesses takes the form

C(p)a =

sumbc

[YbnB

int EC

E(ac)b

dnγ(E t)

dEσ

(ac)b (E)dE

minus YanB

int EC

E(bc)a

dnγ(E t)

dEσ(bc)a (E)dE

] (314)

where the indices b and c run over the nuclei present

in the plasma where σ(ac)b (E) and σ

(bc)a (E) respectively

denote the cross-sections for the corresponding photo-production and photodisintegration processes discussed

above and where E(ac)b and E

(bc)a are the respective

energy thresholds for these processes Expressions forthese cross-sections and values for the correspondingenergy thresholds can be found eg in Ref [3]

By contrast C(s)a represents the collective contribution

from additional secondary processes which involve notthe primary photons themselves but rather a non-thermalpopulation of energetic nuclei produced by interactionsinvolving those primary photons In principle thesesecondary processes include both reactions that producenuclei of species Na and reactions which destroy themIn practice however because the non-thermal populationof any given species Nb generated by processes involvingprimary photons is comparatively small the effect ofsecondary processes on the populations of most nuclearspecies is likewise small As we shall discuss in moredetail in Sect III C the only exception is 6Li whichis not produced in any significant amount during theBBN epoch but which can potentially be produced bysecondary processes initiated by photon injection atsubsequent times Since these processes involve theproduction rather than the destruction of 6Li we focus onthe effect of secondary processes on nuclei which appearin the final state rather than the initial state in whatfollows

The energetic nuclei which participate in secondaryprocesses are the products of the same kinds of reactionswhich lead to the collision term in Eq (314) Thusthe kinetic-energy spectrum dnb(Eb t)dEb of the non-thermal population of a nuclear species Nb produced inthis manner is in large part determined by the energyspectrum of the primary photons In calculating this

7

spectrum one must in principle account for the fact thata photon of energy Eγ can give rise to a range of possibleEb values due to the range of possible scattering anglesbetween the three-momentum vectors of the incomingphoton and the excited nucleus in the center-of-massframe However it can be shown [34] that a reasonable

approximation for the collision term C(s)a for 6Li is

nevertheless obtained by taking Eb to be a one-to-onefunction of Eγ of the form [35]

Eb = E(Eγ) equiv 1

4

[Eγ minus E(bd)

c

] (315)

where E(bd)c is the energy threshold for the primary

process Nc + γ rarr Nb + Nd In this approximationdnb(Eb t)dEb takes the form [3]

dnb(Eb t)

dEb=sumcd

ncb(Eb t)

int EC

Emin(Eb)

[dnγ(Eprimeγ t)

dEprimeγ

times σ(bd)c (Eprimeγ)eminusβb(E

primeγ t)

]dEprimeγ (316)

where b(Eb t) is the energy-loss rate for Nb due toCoulomb scattering with particles in the thermal back-ground plasma The exponential factor βb(E

primeγ t) ac-

counts for the collective effect of additional processeswhich act to reduce the number of nuclei of species NbThe lower limit of integration in Eq (316) is given by

Emin(Eb) = max[Eminus1(Eb) E(bd)c ] where Eminus1(Eb) is the

inverse of the function defined in Eq (315) In otherwords Eminus1(Eb) is the photon energy which correspondsto a kinetic energy Eb for the excited nucleus

In principle the processes which contribute toβa(Eprimeγ t) include both decay processes (in the case inwhich Nb is unstable) and photodisintegration processesof the form Nb+γ rarr Nc+Nd involving a primary photonIn practice the photodisintegration rate due to theseprocesses is much slower that the energy-loss rate due toCoulomb scattering for any species of interest Moreoveras we shall see in Sect III C the only nuclear specieswhose non-thermal population has a significant effect onthe 6Li abundance are tritium (T) and the helium isotope3He Because these two species are mirror nuclei thesecondary processes in which they participate affect the6Li abundance in the same way and have almost identicalcross-sections and energy thresholds Thus in terms oftheir effect on the production of 6Li the populations ofT and 3He may effectively be treated together as if theywere the population of a single nuclear species Althoughtritium is unstable and decays via beta decay to 3Hewith a lifetime of τT asymp 56times 108 s these decays have noimpact on the combined population of T and 3He Wemay therefore safely approximate βb(E

primeγ t) asymp 0 for this

combined population of excited nuclei in what followsThe most relevant processes through which an en-

ergetic nucleus Nb in this non-thermal population canalter the abundance of another nuclear species Nb are

scattering processes of the form Nb + Nf rarr Na + Xin which an energetic nucleus Nb from the non-thermalpopulation generated by primary processes scatters witha background nucleus Nf resulting in the production of anucleus of species Na and some other particle X (whichcould be either an additional nucleus or a photon) Inprinciple processes of the form Nb +Na rarr Nf +X canalso act to reduce the abundance of Na However asdiscussed above this reduction has a negligible impact onYa for any nuclear species Na which already has a sizablecomoving number density at the end of BBN Thuswe focus here on production rather than destructionwhen assessing the impact of secondary processes on theprimordial abundances of light nuclei

With this simplification the collision term associatedwith secondary production processes takes the form

C(s)a =

sumbf

YfnB

int E(EC)

E(aX)bf

[dnb(Eb t)

dEb

times σ(aX)bf (Eb)|v(Eb)|

]dEb (317)

where v(Eb) is the (non-relativistic) relative velocityof nuclei Nb and Nf in the background frame wherednb(Eb t)dEb is the differential energy spectrum of the

non-thermal population of Nb where σ(aX)bf is the cross-

section for the scattering process Nb+Nf rarr Na+X with

corresponding threshold energy E(aX)bf and where E(EC)

is the cutoff in dnb(Eb t)dEb produced by primaryprocesses

C Constraints on Primordial Light-ElementAbundances

The nuclear species whose primordial abundances arethe most tightly constrained by observation mdash and whichare therefore relevant for constraining the late decays ofunstable particles mdash are D 4He 6Li and 7Li The abun-dance of 3He during the present cosmological epoch hasalso been constrained by observation [36 37] Howeveruncertainties in the contribution to this abundance fromstellar sources make it difficult to translate the results ofthese measurements into bounds on the primordial 3Heabundance [38 39] The effect of these uncertainties canbe mitigated in part if we consider the ratio (D+ 3He)Hrather than 3HeH as the former is expected to be largelyunaffected by stellar processing [40ndash42] In this paperwe focus our attention on D 4He 6Li and 7Li as therelationship between the measured abundances of thesenuclei and their corresponding primordial abundances ismore transparent

The observational constraints on the primordial abun-dances of these nuclei can be summarized as followsBounds on the primordial 4He abundance are typicallyphrased in terms of the primordial helium mass fraction

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

2

of observable consequences mdash especially if the final statesinto which these particles decay involve visible-sectorparticles Indeed electromagnetic or hadronic showersprecipitated by unstable-particle decays within the recentcosmological past can alter the primordial abundances oflight nuclei both during and after Big-Bang nucleosyn-thesis (BBN) [1ndash5] give rise to spectral distortions inthe cosmic microwave background (CMB) [6 7] alter theionization history of the universe [8ndash12] and give rise tocharacteristic features in the diffuse photon backgroundThese considerations therefore place stringent constraintson models for new physics involving unstable particles

Much previous work has focused on examining thecosmological consequences of a single particle speciesdecaying in isolation and the corresponding limits onthe properties of such a particle species are now wellestablished Indeed simple analytic approximations canbe derived which accurately model the effects that thedecays of such a particle can have on many of therelevant observables [3 6 13] However many theoriesfor new physics involve not merely one or a few unstableparticles but rather a large mdash and potentially vastmdash number of such particles with a broad spectrum ofmasses lifetimes and cosmological abundances Forexample theories with additional spacetime dimensionsgive rise to infinite towers of Kaluza-Klein (KK) exci-tations for any field which propagates in the higher-dimensional bulk Likewise string theories generallypredict large numbers of light moduli [14ndash16] or axion-like particles [17ndash19] Collections of similar light fieldsalso arise in supergravity theories [20] as well as inother scenarios for new physics [21 22] There evenexist approaches to the dark-matter problem such asthe Dynamical Dark Matter (DDM) framework [23 24]which posit the existence of potentially vast ensemblesof unstable dark-sector particles It is therefore crucialto understand the cosmological consequences of entireensembles of decaying particles in the early universe andif possible to formulate a corresponding set of analyticapproximations which model the effects of these decays

For a variety of reasons assessing the effects of anentire ensemble of decaying particles with a broad rangeof masses lifetimes and cosmological abundances isnot merely a matter of trivially generalizing the resultsobtained in the single-particle case The decay of a givenunstable particle amounts to an injection of additionalelectromagnetic radiation andor other energetic parti-cles into the evolution of the universe and injection atdifferent characteristic timescales during this evolutioncan have markedly different effects on the same observ-able Moreover since many of these observables evolvein time according to a complicated system of coupledequations the effects of injection at any particular timetinj depend in a non-trivial way on the entire injectionhistory prior to tinj through feedback effects

In principle the cosmological constraints on ensem-bles of decaying particles in the early universe canbe evaluated numerically Indeed there are several

publicly available codes [12 25 26] which can readilybe modified in order to assess the effects of an arbitraryadditional injection history on the relevant observablesComputational methods can certainly yield useful resultsin any particular individual case However anothercomplementary approach which can provide additionalphysical insight into the underlying dynamics involvesthe formulation of approximate analytic expressions forthe relevant observables mdash expressions analogous tothose which already exist for a single particle speciesdecaying in isolation Our aim in this paper is to derivesuch a set of analytic expressions mdash expressions whichare applicable to generic theories involving large numbersof unstable particles but which nevertheless provideaccurate approximations for the relevant cosmologicalobservables Thus our results can serve as a usefulreference for future model-builders concerned with cos-mological constraints on decaying particles regardless ofthe particular model under study

In this paper we shall focus primarily on the case inwhich electromagnetic injection dominates mdash ie thecase in which the energy liberated by the decays of theseparticles is released primarily in the form of photonselectrons and positrons rather than hadrons We alsoemphasize that the approximations we shall derive in thispaper are not ad hoc in nature in particular they are notthe results of empirical fits Rather as we shall see theyemerge organically from the underlying physics and thuscarry direct information about the underlying processesinvolved

This paper is organized as follows In Sect II webegin by establishing the notation and conventions thatwe shall use throughout this paper We also reviewthe various scattering processes through which energeticphotons injected by particle decay interact with otherparticles present in the radiation bath In subsequentsections we then turn our attention to entire ensemblesof unstable particles focusing on the electromagneticinjections arising from decays occurring after the BBNepoch Each section is devoted to a different cosmologicalconsideration arising from such injection and in eachcase we ultimately obtain a simple analytic approxi-mation for the corresponding constraint For examplein Sect III we consider the constraints associated withthe modification of the abundances of light nuclei afterBBN and in Sect IV we consider limits on distortionsof the CMB-photon spectrum Likewise in Sect V weconsider the constraints associated with the ionizationhistory of the universe and its impact on the CMB andin Sect VI we consider the constraints associated withadditional contributions from unstable-particle decays tothe diffuse photon background Ultimately the resultsfrom these sections furnish us with the tools needed toconstrain decaying ensembles of various types This isthen illustrated in Sect VII where we consider howour results may be applied to two classes of ensembleswhose constituents exhibit different representative massspectra Finally in Sect VIII we conclude with a

3

discussion of our main results and avenues for futurework For future reference we also provide (in Table IV)a summarycompilation of our main results

II ELECTROMAGNETIC INJECTIONOVERVIEW AND CLASSIFICATION OF

RELEVANT PROCESSES

Our aim in this paper is to assess the cosmologicalconstraints on an ensemble consisting of a potentiallylarge number N of unstable particle species χi withmasses mi and decay widths Γi (or equivalently lifetimesτi equiv Γminus1

i ) where the index i = 0 1 N minus 1 labelsthese particle species in order of increasing mass miWe shall characterize the cosmological abundance ofeach of the χi in terms of a quantity Ωi which we callthe ldquoextrapolated abundancerdquo This quantity representsthe abundance that the species χi would have had atpresent time had it been absolutely stable We shallassume that the total abundance Ωtot equiv

sumi Ωi of the

ensemble is sufficiently small that the universe remainsradiation-dominated until the time of matter-radiationequality tMRE asymp 1012 s Moreover we shall focus onthe regime in which mi amp 1 GeV for all χi and allof the ensemble constituents are non-relativistic by endof the BBN epoch Within this regime as we shalldiscuss in further detail below the spectrum of energeticphotons produced by electromagnetic injection takes acharacteristic form which to a very good approximationdepends only on the overall energy density injected [1]By contrast for much lighter decaying particles theform of the resulting photon spectrum can differ fromthis characteristic form as a result of the immediatedecay products lacking sufficient energy to induce theproduction of e+eminus pairs by scattering off backgroundphotons [27 28] The cosmological constraints on asingle electromagnetically-decaying particle species witha mass below 1 GeV mdash investigated earlier to constrainneutrinos (see eg Ref [29]) mdash have recently beeninvestigated in Refs [30 31]

The considerations which place the most stringent con-straints on the ensemble depend on the values of Ωi and τifor the individual ensemble constituents For ensemblesof particles with lifetimes in the range 1 s τi tnowwhere tnow denotes the present age of the universe thedominant constraints are those related to the abundancesof light elements to spectral distortions of the CMBand to the ionization history of the universe Theeffect of electromagnetic injection on the correspondingobservables is sensitive to the overall energy densityinjected and to the timescales over which that energyis injected but not to the details of the decay kinematicsor the particular channels through which the χi decayThus in order to retain as much generality as possiblein our analysis we shall focus on ensembles for which allconstituents with non-negligible Ωi have lifetimes withinthis range moreover we shall refrain from specifying any

particular decay channel for the χi when assessing thebounds on these ensembles due to these considerationsBy contrast the constraints on decaying ensembles thatfollow from limits on features in the diffuse photonbackground do depend on the particulars of the decaykinematics Thus when analyzing these constraints inSect VI we not only present a general expression for therelevant observable mdash namely the contribution to thedifferential photon flux from the decaying ensemble mdashbut also apply this result to a concrete example involvinga particular decay topology

Many of the constraints on electromagnetic injec-tion are insensitive to the details of the decay kine-matics because the injection of photons and otherelectromagnetically-interacting particles prior to CMBdecoupling sets into motion a complicated chain of inter-actions which serve to redistribute the energies of theseparticles In particular the effects of electromagneticinjection on cosmological observables ultimately dependon the interplay between three broad classes of processesthrough which these photons interact with other particlesin the background plasma These are

bull Class I Cascade and cooling processes whichrapidly redistribute the energy of the injectedphotons Processes in this class include γ+ γBG rarre+ + eminus and γ + γBG rarr γ + γ where γBG

denotes a background photon as well as inverse-Compton scattering and e+eminus pair production offnuclei These processes occur on timescales farshorter than the timescales associated with otherrelevant processes and thus may be considered tobe effectively instantaneous As we shall see theseprocesses serve to establish a non-thermal popula-tion of photons with a characteristic spectrum

bull Class II Processes through which the non-thermalpopulation of photons established by Class-I pro-cesses can have a direct effect on cosmologicalobservables These include the photoproductionand photodisintegration of light elements during orafter BBN as well as the photoionization of neutralhydrogen and helium after recombination

bull Class III Processes which serve to bring thenon-thermal population of photons established byClass-I processes into kinetic andor thermal equi-librium with the radiation bath Processes in thisclass include Compton scattering bremsstrahlungand e+eminus pair production off nuclei

We emphasize that these classes are not necessarily mu-tually exclusive and that certain processes play differentroles during different cosmological epochs

Any energy injected in the form of photons prior tolast scattering is rapidly redistributed to lower energiesdue to the Class-I processes discussed above The resultis a non-thermal contribution to the photon spectrumat high energies with a normalization that depends onthe total injected power and a generic shape which

4

is essentially independent of the shape of the initialinjection spectrum directly produced by χi decays Thisldquoreprocessedrdquo photon spectrum serves as a source offor Class-II processes mdash processes which include forexample reactions that alter the abundances of lightnuclei and scattering processes which contribute to theionization of neutral hydrogen and helium after recombi-nation Since all information about the detailed shape ofthe initial injection spectrum from decays is effectivelywashed out by Class-I processes in establishing thisreprocessed photon spectrum the results of our analysisare largely independent of the kinematics of χi decayThis is ultimately why many of our results mdash includingthose pertaining to the alteration of light abundancesafter BBN distortions in the CMB and the ionizationhistory of the universe mdash are likewise largely insensitiveto the decay kinematics of the χi

The timescale over which injected photons can causethese alterations is controlled by the Class-III processesPrior to CMB decoupling these processes serve toldquodegraderdquo the reprocessed photon spectrum establishedby Class-I processes by bringing this non-thermal pop-ulation of photons into kinetic or thermal equilibriumwith the photons in the radiation bath As this occursthese Class-III interactions reduce the energies of thephotons below the threshold for Class-II processes whilealso potentially altering the shape of the CMB-photonspectrum These Class-III processes eventually freezeout as well after which point any photons injected byparticle decays simply contribute to the diffuse extra-galactic photon background

III IMPACT ON LIGHT-ELEMENTABUNDANCES

We begin our analysis of the cosmological constraintson ensembles of unstable particles by considering theeffect that the decays of these particles have on the abun-dances of light nuclei generated during BBN We shallassume that these decays occur after BBN has concludedie after initial abundances for these nuclei are alreadyestablished We begin by reviewing the properties of thenon-thermal photon spectrum which is established by therapid reprocessing of injected photons from these decaysby Class-I processes We then review the correspondingconstraints on a single unstable particle species [3] mdashconstraints derived from a numerical analysis of thecoupled system of Boltzmann equations which govern theevolution of these abundances We then set the stage forour eventual analysis by deriving a set of analytic approx-imations for the above constraints and demonstratingthat the results obtained from these approximationsare in excellent agreement with the results of a fullnumerical computation within our regime of interestFinally we apply our analytic approximations in orderto constrain scenarios involving an entire ensemble ofmultiple decaying particles exhibiting a range of masses

and lifetimes

A Reprocessed Injection Spectrum

As discussed in Sect II the initial spectrum of photonsinjected at time tinj is redistributed effectively instan-taneously by Class-I processes A detailed treatmentof the Boltzmann equations governing these processesin a radiation-dominated epoch can be found eg inRef [1] For injection at times tinj 1012 s theresulting reprocessed photon spectrum turns out to take acharacteristic form which we may parametrize as follows

dnγ(E tinj)

dEdtinj=

dρ(tinj)

dtinjK(E tinj) (31)

The quantity dρ(tinj)dtinj appearing in this expressionwhich specifies the overall normalization of the contri-bution to the reprocessed photon spectrum representsthe energy density injected by particle decays during theinfinitesimal time interval from tinj to tinj + dtinj ThefunctionK(E tinj) on the other hand specifies the shapeof the spectrum as a function of the photon energy EThis function is normalized such thatint infin

0

K(E tinj)EdE = 1 (32)

It can be shown that for any process that injectsenergy primarily through electromagnetic (rather thanhadronic) channels the function K(E tinj) takes theuniversal form [1 32 33]

K(E tinj) =

K0(EXE)32 E lt EXK0(EXE)2 EX lt E lt EC0 E gt EC

(33)where K0 is an overall normalization constant and whereEC and EX are energy scales associated with specificClass-I processes whose interplay determines the shapeof the reprocessed photon spectrum The normalizationconvention in Eq (32) implies that K0 is given by

K0 =1

E2X

[2 + ln(ECEX)

] (34)

Physically the energy scales EC and EX appearing inEq (33) can be understood as follows The scale ECrepresents the energy above which the photon spectrumis effectively extinguished by the pair-production processγ + γBG rarr e+ + eminus in conjunction with interactionsbetween the resulting electron and positron and otherparticles in the thermal bath The energy scale EXrepresents the threshold above which γ + γBG rarr γ + γis the dominant process through which photons loseenergy By contrast below this energy threshold thedominant processes are Compton scattering and e+eminus

5

pair-production off nuclei Note that while the nor-malization of the reprocessed photon spectrum is setby dρ(tinj)dtinj the shape of this spectrum is entirelycontrolled by the temperature Tinj at injection This

temperature behaves like Tinj prop tminus12inj in a radiation-

dominated epoch This implies that as tinj increases thevalue of EC also increases This reflects the fact thatthe thermal bath is colder at later injection times andthus an injected photon must be more energetic in orderfor the pair-production process γ + γBG rarr e+ + eminus tobe effective Numerically the values of EC and EX at agiven injection time tinj are estimated to be [1 3]

EC =m2e

22Tinjasymp(103 MeV

)times(tinj

108 s

)12

EX =m2e

80Tinjasymp(28 MeV

)times(tinj

108 s

)12

(35)

where me is the electron mass and Tinj is the temperatureof the thermal bath at tinj

The reprocessed photon spectrum in Eq (31) is thespectrum which effectively contributes to the photo-production andor photodisintegration of light elementsafter the BBN epoch In order to illustrate how theshape of this spectrum depends on the injection timetinj we plot in Fig 1 the function K(E tinj) whichdetermines the shape of this spectrum as a function ofE for several different values of the injection time tinjSince the ultraviolet cutoff EC in the photon spectrumincreases with tinj injection at later times can initiatephotoproduction and photodisintegration reactions withhigher energy thresholds The dashed vertical line whichwe include for reference represents the lowest thresholdenergy associated with any such reaction which can havea significant impact on the primordial abundance of anylight nucleus which is tightly constrained by observationAs discussed in Sect III B this reaction turns outto be the deuterium-photodisintegration reaction D +γ rarr n + p which has a threshold energy of roughly22 MeV Thus the portion of the photon spectrumwhich lies within the gray shaded region in Fig 1 has noeffect on the abundance of any relevant nucleus Since EClies below this threshold for tinj asymp 104 s electromagneticinjection between the end of BBN and this timescale hasessentially no effect on the abundances of light nuclei

One additional complication that we must take intoaccount in assessing the effect of injection on the abun-dances of light nuclei is that the reprocessed photonspectrum established by Class-I processes immediatelyafter injection at tinj is subsequently degraded by Class-III processes which slowly act to bring this reprocessedspectrum into thermal equilibrium with the backgroundphotons in the radiation bath The timescale δtth(E tinj)on which these processes act on a photon of energy E is

01 1 10 100 1000 10410-7

10-5

0001

0100

10

E [MeV]

K[MeV

-2]

tinj = 104 s

tinj = 106 s

tinj = 108 s

tinj = 1010 s

tinj = 1012 s

FIG 1 The function K(E tinj) plotted as a function ofphoton energy E for several different values of injection timetinj As discussed in the text this function determines theshape of the reprocessed photon spectrum for an instanta-neous injection of photons of energy E at time tinj Thedashed vertical line at E asymp 22 MeV indicates the lowestthreshold energy associated with any photoproduction orphotodisintegration reaction which can significantly alter theabundance of any light nucleus whose primordial abundanceis tightly constrained by observation

roughly

δtth(E tinj) asymp [nB(tinj)σth(E)]minus1

(36)

where nB(tinj) is the number density of baryons at thetime of injection and where σth(E) is the characteristiccross-section for the relevant scattering processes whichinclude Compton scattering and e+eminus pair productionoff nuclei The cross-sections for all relevant individualcontributing processes can be found in Ref [1] Note thatwhile at low energies σth(E) is well approximated by theThomson cross-section this approximation breaks downat higher energies as other processes become relevant

Given these observations the spectrum of the resultingnon-thermal population of photons at time t not onlyrepresents the sum of all contributions from injection atall times tinj lt t but also reflects the subsequent degra-dation of these contributions by the Class-III processeswhich serve to thermalize this population of photons withthe radiation bath This overall non-thermal photonspectrum takes the form

dnγ(E t)

dE=

int t

0

dnγ(E tinj)

dEdtinjG(t tinj)dtinj (37)

where

G(t tinj) equiv eminus(tminustinj)δtth(Etinj)Θ(tminus tinj) (38)

6

is the Greenrsquos function which solves the differentialequation

dG(t tinj)

dt+

G(t tinj)

δtth(E tinj)= δ(tminus tinj) (39)

The population of non-thermal photons described byEq (37) serves as the source for the initial photo-production and photodisintegration reactions ultimatelyresponsible for the modification of light-element abun-dances after BBN We shall therefore henceforth refer tophotons in this population as ldquoprimaryrdquo photons

In what follows we will find it useful to employwhat we shall call the ldquouniform-decay approximationrdquoSpecifically we shall approximate the full exponentialdecay of each dark-sector species χi as if the entirepopulation of such particles throughout the universe wereto decay precisely at the same time τi As we shall seethis will prove critical in allowing us to formulate ourultimate analytic approximations We shall neverthelessfind that the results of our approximations are generallyin excellent agreement with the results of a full numericalanalysis

Within the uniform-decay approximation the contri-bution to dρdt(tinj) from the decay of a single unstableparticle species χ takes the form of a Dirac δ-function

dρ(tinj)

dtinj= ρχ(tinj)εχδ(tinj minus τχ) (310)

where ρχ(tinj) is the energy density of χ at time tinj andwhere εχ is the fraction of the energy density releasedby χ decays which is transferred to photons It thereforefollows that in this approximation the primary-photonspectrum in Eq (37) reduces to

dnγ(E t)

dE= ρχ(τχ)εχK(E τχ)

times eminus(tminusτχ)δtth(Etinj)Θ(tminus τχ) (311)

B Light-Element ProductionDestruction

Generally speaking the overall rate of change of thenumber density na of a nuclear species Na due tothe injection of electromagnetic energy at late times isgoverned by a Boltzmann equation of the form

dnadt

+ 3Hna = C(p)a + C(s)

a (312)

where H is the Hubble parameter and where C(p)a and

C(s)a are the collision terms associated with two different

classes of scattering processes which contribute to thisoverall rate of change We shall describe these individualcollision terms in detail below Since na is affected byHubble expansion it is more convenient to work with thecorresponding comoving number density Ya equiv nanB

where nB denotes the total number density of baryonsThe Boltzmann equation for Ya then takes the form

dYadt

=1

nB

[C(p)a + C(s)

a

] (313)

The collision term C(p)a represents the collective con-

tribution from Class-II processes directly involving thepopulation of primary photons described by Eq (311)

The principal processes which contribute to C(p)a are

photoproduction processes of the form Nb+γ rarr Na+Ncand photodisintegration processes of the form Na + γ rarrNb + Nc where Nb and Nc are other nuclei in thethermal plasma The collision term associated with theseprocesses takes the form

C(p)a =

sumbc

[YbnB

int EC

E(ac)b

dnγ(E t)

dEσ

(ac)b (E)dE

minus YanB

int EC

E(bc)a

dnγ(E t)

dEσ(bc)a (E)dE

] (314)

where the indices b and c run over the nuclei present

in the plasma where σ(ac)b (E) and σ

(bc)a (E) respectively

denote the cross-sections for the corresponding photo-production and photodisintegration processes discussed

above and where E(ac)b and E

(bc)a are the respective

energy thresholds for these processes Expressions forthese cross-sections and values for the correspondingenergy thresholds can be found eg in Ref [3]

By contrast C(s)a represents the collective contribution

from additional secondary processes which involve notthe primary photons themselves but rather a non-thermalpopulation of energetic nuclei produced by interactionsinvolving those primary photons In principle thesesecondary processes include both reactions that producenuclei of species Na and reactions which destroy themIn practice however because the non-thermal populationof any given species Nb generated by processes involvingprimary photons is comparatively small the effect ofsecondary processes on the populations of most nuclearspecies is likewise small As we shall discuss in moredetail in Sect III C the only exception is 6Li whichis not produced in any significant amount during theBBN epoch but which can potentially be produced bysecondary processes initiated by photon injection atsubsequent times Since these processes involve theproduction rather than the destruction of 6Li we focus onthe effect of secondary processes on nuclei which appearin the final state rather than the initial state in whatfollows

The energetic nuclei which participate in secondaryprocesses are the products of the same kinds of reactionswhich lead to the collision term in Eq (314) Thusthe kinetic-energy spectrum dnb(Eb t)dEb of the non-thermal population of a nuclear species Nb produced inthis manner is in large part determined by the energyspectrum of the primary photons In calculating this

7

spectrum one must in principle account for the fact thata photon of energy Eγ can give rise to a range of possibleEb values due to the range of possible scattering anglesbetween the three-momentum vectors of the incomingphoton and the excited nucleus in the center-of-massframe However it can be shown [34] that a reasonable

approximation for the collision term C(s)a for 6Li is

nevertheless obtained by taking Eb to be a one-to-onefunction of Eγ of the form [35]

Eb = E(Eγ) equiv 1

4

[Eγ minus E(bd)

c

] (315)

where E(bd)c is the energy threshold for the primary

process Nc + γ rarr Nb + Nd In this approximationdnb(Eb t)dEb takes the form [3]

dnb(Eb t)

dEb=sumcd

ncb(Eb t)

int EC

Emin(Eb)

[dnγ(Eprimeγ t)

dEprimeγ

times σ(bd)c (Eprimeγ)eminusβb(E

primeγ t)

]dEprimeγ (316)

where b(Eb t) is the energy-loss rate for Nb due toCoulomb scattering with particles in the thermal back-ground plasma The exponential factor βb(E

primeγ t) ac-

counts for the collective effect of additional processeswhich act to reduce the number of nuclei of species NbThe lower limit of integration in Eq (316) is given by

Emin(Eb) = max[Eminus1(Eb) E(bd)c ] where Eminus1(Eb) is the

inverse of the function defined in Eq (315) In otherwords Eminus1(Eb) is the photon energy which correspondsto a kinetic energy Eb for the excited nucleus

In principle the processes which contribute toβa(Eprimeγ t) include both decay processes (in the case inwhich Nb is unstable) and photodisintegration processesof the form Nb+γ rarr Nc+Nd involving a primary photonIn practice the photodisintegration rate due to theseprocesses is much slower that the energy-loss rate due toCoulomb scattering for any species of interest Moreoveras we shall see in Sect III C the only nuclear specieswhose non-thermal population has a significant effect onthe 6Li abundance are tritium (T) and the helium isotope3He Because these two species are mirror nuclei thesecondary processes in which they participate affect the6Li abundance in the same way and have almost identicalcross-sections and energy thresholds Thus in terms oftheir effect on the production of 6Li the populations ofT and 3He may effectively be treated together as if theywere the population of a single nuclear species Althoughtritium is unstable and decays via beta decay to 3Hewith a lifetime of τT asymp 56times 108 s these decays have noimpact on the combined population of T and 3He Wemay therefore safely approximate βb(E

primeγ t) asymp 0 for this

combined population of excited nuclei in what followsThe most relevant processes through which an en-

ergetic nucleus Nb in this non-thermal population canalter the abundance of another nuclear species Nb are

scattering processes of the form Nb + Nf rarr Na + Xin which an energetic nucleus Nb from the non-thermalpopulation generated by primary processes scatters witha background nucleus Nf resulting in the production of anucleus of species Na and some other particle X (whichcould be either an additional nucleus or a photon) Inprinciple processes of the form Nb +Na rarr Nf +X canalso act to reduce the abundance of Na However asdiscussed above this reduction has a negligible impact onYa for any nuclear species Na which already has a sizablecomoving number density at the end of BBN Thuswe focus here on production rather than destructionwhen assessing the impact of secondary processes on theprimordial abundances of light nuclei

With this simplification the collision term associatedwith secondary production processes takes the form

C(s)a =

sumbf

YfnB

int E(EC)

E(aX)bf

[dnb(Eb t)

dEb

times σ(aX)bf (Eb)|v(Eb)|

]dEb (317)

where v(Eb) is the (non-relativistic) relative velocityof nuclei Nb and Nf in the background frame wherednb(Eb t)dEb is the differential energy spectrum of the

non-thermal population of Nb where σ(aX)bf is the cross-

section for the scattering process Nb+Nf rarr Na+X with

corresponding threshold energy E(aX)bf and where E(EC)

is the cutoff in dnb(Eb t)dEb produced by primaryprocesses

C Constraints on Primordial Light-ElementAbundances

The nuclear species whose primordial abundances arethe most tightly constrained by observation mdash and whichare therefore relevant for constraining the late decays ofunstable particles mdash are D 4He 6Li and 7Li The abun-dance of 3He during the present cosmological epoch hasalso been constrained by observation [36 37] Howeveruncertainties in the contribution to this abundance fromstellar sources make it difficult to translate the results ofthese measurements into bounds on the primordial 3Heabundance [38 39] The effect of these uncertainties canbe mitigated in part if we consider the ratio (D+ 3He)Hrather than 3HeH as the former is expected to be largelyunaffected by stellar processing [40ndash42] In this paperwe focus our attention on D 4He 6Li and 7Li as therelationship between the measured abundances of thesenuclei and their corresponding primordial abundances ismore transparent

The observational constraints on the primordial abun-dances of these nuclei can be summarized as followsBounds on the primordial 4He abundance are typicallyphrased in terms of the primordial helium mass fraction

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

3

discussion of our main results and avenues for futurework For future reference we also provide (in Table IV)a summarycompilation of our main results

II ELECTROMAGNETIC INJECTIONOVERVIEW AND CLASSIFICATION OF

RELEVANT PROCESSES

Our aim in this paper is to assess the cosmologicalconstraints on an ensemble consisting of a potentiallylarge number N of unstable particle species χi withmasses mi and decay widths Γi (or equivalently lifetimesτi equiv Γminus1

i ) where the index i = 0 1 N minus 1 labelsthese particle species in order of increasing mass miWe shall characterize the cosmological abundance ofeach of the χi in terms of a quantity Ωi which we callthe ldquoextrapolated abundancerdquo This quantity representsthe abundance that the species χi would have had atpresent time had it been absolutely stable We shallassume that the total abundance Ωtot equiv

sumi Ωi of the

ensemble is sufficiently small that the universe remainsradiation-dominated until the time of matter-radiationequality tMRE asymp 1012 s Moreover we shall focus onthe regime in which mi amp 1 GeV for all χi and allof the ensemble constituents are non-relativistic by endof the BBN epoch Within this regime as we shalldiscuss in further detail below the spectrum of energeticphotons produced by electromagnetic injection takes acharacteristic form which to a very good approximationdepends only on the overall energy density injected [1]By contrast for much lighter decaying particles theform of the resulting photon spectrum can differ fromthis characteristic form as a result of the immediatedecay products lacking sufficient energy to induce theproduction of e+eminus pairs by scattering off backgroundphotons [27 28] The cosmological constraints on asingle electromagnetically-decaying particle species witha mass below 1 GeV mdash investigated earlier to constrainneutrinos (see eg Ref [29]) mdash have recently beeninvestigated in Refs [30 31]

The considerations which place the most stringent con-straints on the ensemble depend on the values of Ωi and τifor the individual ensemble constituents For ensemblesof particles with lifetimes in the range 1 s τi tnowwhere tnow denotes the present age of the universe thedominant constraints are those related to the abundancesof light elements to spectral distortions of the CMBand to the ionization history of the universe Theeffect of electromagnetic injection on the correspondingobservables is sensitive to the overall energy densityinjected and to the timescales over which that energyis injected but not to the details of the decay kinematicsor the particular channels through which the χi decayThus in order to retain as much generality as possiblein our analysis we shall focus on ensembles for which allconstituents with non-negligible Ωi have lifetimes withinthis range moreover we shall refrain from specifying any

particular decay channel for the χi when assessing thebounds on these ensembles due to these considerationsBy contrast the constraints on decaying ensembles thatfollow from limits on features in the diffuse photonbackground do depend on the particulars of the decaykinematics Thus when analyzing these constraints inSect VI we not only present a general expression for therelevant observable mdash namely the contribution to thedifferential photon flux from the decaying ensemble mdashbut also apply this result to a concrete example involvinga particular decay topology

Many of the constraints on electromagnetic injec-tion are insensitive to the details of the decay kine-matics because the injection of photons and otherelectromagnetically-interacting particles prior to CMBdecoupling sets into motion a complicated chain of inter-actions which serve to redistribute the energies of theseparticles In particular the effects of electromagneticinjection on cosmological observables ultimately dependon the interplay between three broad classes of processesthrough which these photons interact with other particlesin the background plasma These are

bull Class I Cascade and cooling processes whichrapidly redistribute the energy of the injectedphotons Processes in this class include γ+ γBG rarre+ + eminus and γ + γBG rarr γ + γ where γBG

denotes a background photon as well as inverse-Compton scattering and e+eminus pair production offnuclei These processes occur on timescales farshorter than the timescales associated with otherrelevant processes and thus may be considered tobe effectively instantaneous As we shall see theseprocesses serve to establish a non-thermal popula-tion of photons with a characteristic spectrum

bull Class II Processes through which the non-thermalpopulation of photons established by Class-I pro-cesses can have a direct effect on cosmologicalobservables These include the photoproductionand photodisintegration of light elements during orafter BBN as well as the photoionization of neutralhydrogen and helium after recombination

bull Class III Processes which serve to bring thenon-thermal population of photons established byClass-I processes into kinetic andor thermal equi-librium with the radiation bath Processes in thisclass include Compton scattering bremsstrahlungand e+eminus pair production off nuclei

We emphasize that these classes are not necessarily mu-tually exclusive and that certain processes play differentroles during different cosmological epochs

Any energy injected in the form of photons prior tolast scattering is rapidly redistributed to lower energiesdue to the Class-I processes discussed above The resultis a non-thermal contribution to the photon spectrumat high energies with a normalization that depends onthe total injected power and a generic shape which

4

is essentially independent of the shape of the initialinjection spectrum directly produced by χi decays Thisldquoreprocessedrdquo photon spectrum serves as a source offor Class-II processes mdash processes which include forexample reactions that alter the abundances of lightnuclei and scattering processes which contribute to theionization of neutral hydrogen and helium after recombi-nation Since all information about the detailed shape ofthe initial injection spectrum from decays is effectivelywashed out by Class-I processes in establishing thisreprocessed photon spectrum the results of our analysisare largely independent of the kinematics of χi decayThis is ultimately why many of our results mdash includingthose pertaining to the alteration of light abundancesafter BBN distortions in the CMB and the ionizationhistory of the universe mdash are likewise largely insensitiveto the decay kinematics of the χi

The timescale over which injected photons can causethese alterations is controlled by the Class-III processesPrior to CMB decoupling these processes serve toldquodegraderdquo the reprocessed photon spectrum establishedby Class-I processes by bringing this non-thermal pop-ulation of photons into kinetic or thermal equilibriumwith the photons in the radiation bath As this occursthese Class-III interactions reduce the energies of thephotons below the threshold for Class-II processes whilealso potentially altering the shape of the CMB-photonspectrum These Class-III processes eventually freezeout as well after which point any photons injected byparticle decays simply contribute to the diffuse extra-galactic photon background

III IMPACT ON LIGHT-ELEMENTABUNDANCES

We begin our analysis of the cosmological constraintson ensembles of unstable particles by considering theeffect that the decays of these particles have on the abun-dances of light nuclei generated during BBN We shallassume that these decays occur after BBN has concludedie after initial abundances for these nuclei are alreadyestablished We begin by reviewing the properties of thenon-thermal photon spectrum which is established by therapid reprocessing of injected photons from these decaysby Class-I processes We then review the correspondingconstraints on a single unstable particle species [3] mdashconstraints derived from a numerical analysis of thecoupled system of Boltzmann equations which govern theevolution of these abundances We then set the stage forour eventual analysis by deriving a set of analytic approx-imations for the above constraints and demonstratingthat the results obtained from these approximationsare in excellent agreement with the results of a fullnumerical computation within our regime of interestFinally we apply our analytic approximations in orderto constrain scenarios involving an entire ensemble ofmultiple decaying particles exhibiting a range of masses

and lifetimes

A Reprocessed Injection Spectrum

As discussed in Sect II the initial spectrum of photonsinjected at time tinj is redistributed effectively instan-taneously by Class-I processes A detailed treatmentof the Boltzmann equations governing these processesin a radiation-dominated epoch can be found eg inRef [1] For injection at times tinj 1012 s theresulting reprocessed photon spectrum turns out to take acharacteristic form which we may parametrize as follows

dnγ(E tinj)

dEdtinj=

dρ(tinj)

dtinjK(E tinj) (31)

The quantity dρ(tinj)dtinj appearing in this expressionwhich specifies the overall normalization of the contri-bution to the reprocessed photon spectrum representsthe energy density injected by particle decays during theinfinitesimal time interval from tinj to tinj + dtinj ThefunctionK(E tinj) on the other hand specifies the shapeof the spectrum as a function of the photon energy EThis function is normalized such thatint infin

0

K(E tinj)EdE = 1 (32)

It can be shown that for any process that injectsenergy primarily through electromagnetic (rather thanhadronic) channels the function K(E tinj) takes theuniversal form [1 32 33]

K(E tinj) =

K0(EXE)32 E lt EXK0(EXE)2 EX lt E lt EC0 E gt EC

(33)where K0 is an overall normalization constant and whereEC and EX are energy scales associated with specificClass-I processes whose interplay determines the shapeof the reprocessed photon spectrum The normalizationconvention in Eq (32) implies that K0 is given by

K0 =1

E2X

[2 + ln(ECEX)

] (34)

Physically the energy scales EC and EX appearing inEq (33) can be understood as follows The scale ECrepresents the energy above which the photon spectrumis effectively extinguished by the pair-production processγ + γBG rarr e+ + eminus in conjunction with interactionsbetween the resulting electron and positron and otherparticles in the thermal bath The energy scale EXrepresents the threshold above which γ + γBG rarr γ + γis the dominant process through which photons loseenergy By contrast below this energy threshold thedominant processes are Compton scattering and e+eminus

5

pair-production off nuclei Note that while the nor-malization of the reprocessed photon spectrum is setby dρ(tinj)dtinj the shape of this spectrum is entirelycontrolled by the temperature Tinj at injection This

temperature behaves like Tinj prop tminus12inj in a radiation-

dominated epoch This implies that as tinj increases thevalue of EC also increases This reflects the fact thatthe thermal bath is colder at later injection times andthus an injected photon must be more energetic in orderfor the pair-production process γ + γBG rarr e+ + eminus tobe effective Numerically the values of EC and EX at agiven injection time tinj are estimated to be [1 3]

EC =m2e

22Tinjasymp(103 MeV

)times(tinj

108 s

)12

EX =m2e

80Tinjasymp(28 MeV

)times(tinj

108 s

)12

(35)

where me is the electron mass and Tinj is the temperatureof the thermal bath at tinj

The reprocessed photon spectrum in Eq (31) is thespectrum which effectively contributes to the photo-production andor photodisintegration of light elementsafter the BBN epoch In order to illustrate how theshape of this spectrum depends on the injection timetinj we plot in Fig 1 the function K(E tinj) whichdetermines the shape of this spectrum as a function ofE for several different values of the injection time tinjSince the ultraviolet cutoff EC in the photon spectrumincreases with tinj injection at later times can initiatephotoproduction and photodisintegration reactions withhigher energy thresholds The dashed vertical line whichwe include for reference represents the lowest thresholdenergy associated with any such reaction which can havea significant impact on the primordial abundance of anylight nucleus which is tightly constrained by observationAs discussed in Sect III B this reaction turns outto be the deuterium-photodisintegration reaction D +γ rarr n + p which has a threshold energy of roughly22 MeV Thus the portion of the photon spectrumwhich lies within the gray shaded region in Fig 1 has noeffect on the abundance of any relevant nucleus Since EClies below this threshold for tinj asymp 104 s electromagneticinjection between the end of BBN and this timescale hasessentially no effect on the abundances of light nuclei

One additional complication that we must take intoaccount in assessing the effect of injection on the abun-dances of light nuclei is that the reprocessed photonspectrum established by Class-I processes immediatelyafter injection at tinj is subsequently degraded by Class-III processes which slowly act to bring this reprocessedspectrum into thermal equilibrium with the backgroundphotons in the radiation bath The timescale δtth(E tinj)on which these processes act on a photon of energy E is

01 1 10 100 1000 10410-7

10-5

0001

0100

10

E [MeV]

K[MeV

-2]

tinj = 104 s

tinj = 106 s

tinj = 108 s

tinj = 1010 s

tinj = 1012 s

FIG 1 The function K(E tinj) plotted as a function ofphoton energy E for several different values of injection timetinj As discussed in the text this function determines theshape of the reprocessed photon spectrum for an instanta-neous injection of photons of energy E at time tinj Thedashed vertical line at E asymp 22 MeV indicates the lowestthreshold energy associated with any photoproduction orphotodisintegration reaction which can significantly alter theabundance of any light nucleus whose primordial abundanceis tightly constrained by observation

roughly

δtth(E tinj) asymp [nB(tinj)σth(E)]minus1

(36)

where nB(tinj) is the number density of baryons at thetime of injection and where σth(E) is the characteristiccross-section for the relevant scattering processes whichinclude Compton scattering and e+eminus pair productionoff nuclei The cross-sections for all relevant individualcontributing processes can be found in Ref [1] Note thatwhile at low energies σth(E) is well approximated by theThomson cross-section this approximation breaks downat higher energies as other processes become relevant

Given these observations the spectrum of the resultingnon-thermal population of photons at time t not onlyrepresents the sum of all contributions from injection atall times tinj lt t but also reflects the subsequent degra-dation of these contributions by the Class-III processeswhich serve to thermalize this population of photons withthe radiation bath This overall non-thermal photonspectrum takes the form

dnγ(E t)

dE=

int t

0

dnγ(E tinj)

dEdtinjG(t tinj)dtinj (37)

where

G(t tinj) equiv eminus(tminustinj)δtth(Etinj)Θ(tminus tinj) (38)

6

is the Greenrsquos function which solves the differentialequation

dG(t tinj)

dt+

G(t tinj)

δtth(E tinj)= δ(tminus tinj) (39)

The population of non-thermal photons described byEq (37) serves as the source for the initial photo-production and photodisintegration reactions ultimatelyresponsible for the modification of light-element abun-dances after BBN We shall therefore henceforth refer tophotons in this population as ldquoprimaryrdquo photons

In what follows we will find it useful to employwhat we shall call the ldquouniform-decay approximationrdquoSpecifically we shall approximate the full exponentialdecay of each dark-sector species χi as if the entirepopulation of such particles throughout the universe wereto decay precisely at the same time τi As we shall seethis will prove critical in allowing us to formulate ourultimate analytic approximations We shall neverthelessfind that the results of our approximations are generallyin excellent agreement with the results of a full numericalanalysis

Within the uniform-decay approximation the contri-bution to dρdt(tinj) from the decay of a single unstableparticle species χ takes the form of a Dirac δ-function

dρ(tinj)

dtinj= ρχ(tinj)εχδ(tinj minus τχ) (310)

where ρχ(tinj) is the energy density of χ at time tinj andwhere εχ is the fraction of the energy density releasedby χ decays which is transferred to photons It thereforefollows that in this approximation the primary-photonspectrum in Eq (37) reduces to

dnγ(E t)

dE= ρχ(τχ)εχK(E τχ)

times eminus(tminusτχ)δtth(Etinj)Θ(tminus τχ) (311)

B Light-Element ProductionDestruction

Generally speaking the overall rate of change of thenumber density na of a nuclear species Na due tothe injection of electromagnetic energy at late times isgoverned by a Boltzmann equation of the form

dnadt

+ 3Hna = C(p)a + C(s)

a (312)

where H is the Hubble parameter and where C(p)a and

C(s)a are the collision terms associated with two different

classes of scattering processes which contribute to thisoverall rate of change We shall describe these individualcollision terms in detail below Since na is affected byHubble expansion it is more convenient to work with thecorresponding comoving number density Ya equiv nanB

where nB denotes the total number density of baryonsThe Boltzmann equation for Ya then takes the form

dYadt

=1

nB

[C(p)a + C(s)

a

] (313)

The collision term C(p)a represents the collective con-

tribution from Class-II processes directly involving thepopulation of primary photons described by Eq (311)

The principal processes which contribute to C(p)a are

photoproduction processes of the form Nb+γ rarr Na+Ncand photodisintegration processes of the form Na + γ rarrNb + Nc where Nb and Nc are other nuclei in thethermal plasma The collision term associated with theseprocesses takes the form

C(p)a =

sumbc

[YbnB

int EC

E(ac)b

dnγ(E t)

dEσ

(ac)b (E)dE

minus YanB

int EC

E(bc)a

dnγ(E t)

dEσ(bc)a (E)dE

] (314)

where the indices b and c run over the nuclei present

in the plasma where σ(ac)b (E) and σ

(bc)a (E) respectively

denote the cross-sections for the corresponding photo-production and photodisintegration processes discussed

above and where E(ac)b and E

(bc)a are the respective

energy thresholds for these processes Expressions forthese cross-sections and values for the correspondingenergy thresholds can be found eg in Ref [3]

By contrast C(s)a represents the collective contribution

from additional secondary processes which involve notthe primary photons themselves but rather a non-thermalpopulation of energetic nuclei produced by interactionsinvolving those primary photons In principle thesesecondary processes include both reactions that producenuclei of species Na and reactions which destroy themIn practice however because the non-thermal populationof any given species Nb generated by processes involvingprimary photons is comparatively small the effect ofsecondary processes on the populations of most nuclearspecies is likewise small As we shall discuss in moredetail in Sect III C the only exception is 6Li whichis not produced in any significant amount during theBBN epoch but which can potentially be produced bysecondary processes initiated by photon injection atsubsequent times Since these processes involve theproduction rather than the destruction of 6Li we focus onthe effect of secondary processes on nuclei which appearin the final state rather than the initial state in whatfollows

The energetic nuclei which participate in secondaryprocesses are the products of the same kinds of reactionswhich lead to the collision term in Eq (314) Thusthe kinetic-energy spectrum dnb(Eb t)dEb of the non-thermal population of a nuclear species Nb produced inthis manner is in large part determined by the energyspectrum of the primary photons In calculating this

7

spectrum one must in principle account for the fact thata photon of energy Eγ can give rise to a range of possibleEb values due to the range of possible scattering anglesbetween the three-momentum vectors of the incomingphoton and the excited nucleus in the center-of-massframe However it can be shown [34] that a reasonable

approximation for the collision term C(s)a for 6Li is

nevertheless obtained by taking Eb to be a one-to-onefunction of Eγ of the form [35]

Eb = E(Eγ) equiv 1

4

[Eγ minus E(bd)

c

] (315)

where E(bd)c is the energy threshold for the primary

process Nc + γ rarr Nb + Nd In this approximationdnb(Eb t)dEb takes the form [3]

dnb(Eb t)

dEb=sumcd

ncb(Eb t)

int EC

Emin(Eb)

[dnγ(Eprimeγ t)

dEprimeγ

times σ(bd)c (Eprimeγ)eminusβb(E

primeγ t)

]dEprimeγ (316)

where b(Eb t) is the energy-loss rate for Nb due toCoulomb scattering with particles in the thermal back-ground plasma The exponential factor βb(E

primeγ t) ac-

counts for the collective effect of additional processeswhich act to reduce the number of nuclei of species NbThe lower limit of integration in Eq (316) is given by

Emin(Eb) = max[Eminus1(Eb) E(bd)c ] where Eminus1(Eb) is the

inverse of the function defined in Eq (315) In otherwords Eminus1(Eb) is the photon energy which correspondsto a kinetic energy Eb for the excited nucleus

In principle the processes which contribute toβa(Eprimeγ t) include both decay processes (in the case inwhich Nb is unstable) and photodisintegration processesof the form Nb+γ rarr Nc+Nd involving a primary photonIn practice the photodisintegration rate due to theseprocesses is much slower that the energy-loss rate due toCoulomb scattering for any species of interest Moreoveras we shall see in Sect III C the only nuclear specieswhose non-thermal population has a significant effect onthe 6Li abundance are tritium (T) and the helium isotope3He Because these two species are mirror nuclei thesecondary processes in which they participate affect the6Li abundance in the same way and have almost identicalcross-sections and energy thresholds Thus in terms oftheir effect on the production of 6Li the populations ofT and 3He may effectively be treated together as if theywere the population of a single nuclear species Althoughtritium is unstable and decays via beta decay to 3Hewith a lifetime of τT asymp 56times 108 s these decays have noimpact on the combined population of T and 3He Wemay therefore safely approximate βb(E

primeγ t) asymp 0 for this

combined population of excited nuclei in what followsThe most relevant processes through which an en-

ergetic nucleus Nb in this non-thermal population canalter the abundance of another nuclear species Nb are

scattering processes of the form Nb + Nf rarr Na + Xin which an energetic nucleus Nb from the non-thermalpopulation generated by primary processes scatters witha background nucleus Nf resulting in the production of anucleus of species Na and some other particle X (whichcould be either an additional nucleus or a photon) Inprinciple processes of the form Nb +Na rarr Nf +X canalso act to reduce the abundance of Na However asdiscussed above this reduction has a negligible impact onYa for any nuclear species Na which already has a sizablecomoving number density at the end of BBN Thuswe focus here on production rather than destructionwhen assessing the impact of secondary processes on theprimordial abundances of light nuclei

With this simplification the collision term associatedwith secondary production processes takes the form

C(s)a =

sumbf

YfnB

int E(EC)

E(aX)bf

[dnb(Eb t)

dEb

times σ(aX)bf (Eb)|v(Eb)|

]dEb (317)

where v(Eb) is the (non-relativistic) relative velocityof nuclei Nb and Nf in the background frame wherednb(Eb t)dEb is the differential energy spectrum of the

non-thermal population of Nb where σ(aX)bf is the cross-

section for the scattering process Nb+Nf rarr Na+X with

corresponding threshold energy E(aX)bf and where E(EC)

is the cutoff in dnb(Eb t)dEb produced by primaryprocesses

C Constraints on Primordial Light-ElementAbundances

The nuclear species whose primordial abundances arethe most tightly constrained by observation mdash and whichare therefore relevant for constraining the late decays ofunstable particles mdash are D 4He 6Li and 7Li The abun-dance of 3He during the present cosmological epoch hasalso been constrained by observation [36 37] Howeveruncertainties in the contribution to this abundance fromstellar sources make it difficult to translate the results ofthese measurements into bounds on the primordial 3Heabundance [38 39] The effect of these uncertainties canbe mitigated in part if we consider the ratio (D+ 3He)Hrather than 3HeH as the former is expected to be largelyunaffected by stellar processing [40ndash42] In this paperwe focus our attention on D 4He 6Li and 7Li as therelationship between the measured abundances of thesenuclei and their corresponding primordial abundances ismore transparent

The observational constraints on the primordial abun-dances of these nuclei can be summarized as followsBounds on the primordial 4He abundance are typicallyphrased in terms of the primordial helium mass fraction

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

4

is essentially independent of the shape of the initialinjection spectrum directly produced by χi decays Thisldquoreprocessedrdquo photon spectrum serves as a source offor Class-II processes mdash processes which include forexample reactions that alter the abundances of lightnuclei and scattering processes which contribute to theionization of neutral hydrogen and helium after recombi-nation Since all information about the detailed shape ofthe initial injection spectrum from decays is effectivelywashed out by Class-I processes in establishing thisreprocessed photon spectrum the results of our analysisare largely independent of the kinematics of χi decayThis is ultimately why many of our results mdash includingthose pertaining to the alteration of light abundancesafter BBN distortions in the CMB and the ionizationhistory of the universe mdash are likewise largely insensitiveto the decay kinematics of the χi

The timescale over which injected photons can causethese alterations is controlled by the Class-III processesPrior to CMB decoupling these processes serve toldquodegraderdquo the reprocessed photon spectrum establishedby Class-I processes by bringing this non-thermal pop-ulation of photons into kinetic or thermal equilibriumwith the photons in the radiation bath As this occursthese Class-III interactions reduce the energies of thephotons below the threshold for Class-II processes whilealso potentially altering the shape of the CMB-photonspectrum These Class-III processes eventually freezeout as well after which point any photons injected byparticle decays simply contribute to the diffuse extra-galactic photon background

III IMPACT ON LIGHT-ELEMENTABUNDANCES

We begin our analysis of the cosmological constraintson ensembles of unstable particles by considering theeffect that the decays of these particles have on the abun-dances of light nuclei generated during BBN We shallassume that these decays occur after BBN has concludedie after initial abundances for these nuclei are alreadyestablished We begin by reviewing the properties of thenon-thermal photon spectrum which is established by therapid reprocessing of injected photons from these decaysby Class-I processes We then review the correspondingconstraints on a single unstable particle species [3] mdashconstraints derived from a numerical analysis of thecoupled system of Boltzmann equations which govern theevolution of these abundances We then set the stage forour eventual analysis by deriving a set of analytic approx-imations for the above constraints and demonstratingthat the results obtained from these approximationsare in excellent agreement with the results of a fullnumerical computation within our regime of interestFinally we apply our analytic approximations in orderto constrain scenarios involving an entire ensemble ofmultiple decaying particles exhibiting a range of masses

and lifetimes

A Reprocessed Injection Spectrum

As discussed in Sect II the initial spectrum of photonsinjected at time tinj is redistributed effectively instan-taneously by Class-I processes A detailed treatmentof the Boltzmann equations governing these processesin a radiation-dominated epoch can be found eg inRef [1] For injection at times tinj 1012 s theresulting reprocessed photon spectrum turns out to take acharacteristic form which we may parametrize as follows

dnγ(E tinj)

dEdtinj=

dρ(tinj)

dtinjK(E tinj) (31)

The quantity dρ(tinj)dtinj appearing in this expressionwhich specifies the overall normalization of the contri-bution to the reprocessed photon spectrum representsthe energy density injected by particle decays during theinfinitesimal time interval from tinj to tinj + dtinj ThefunctionK(E tinj) on the other hand specifies the shapeof the spectrum as a function of the photon energy EThis function is normalized such thatint infin

0

K(E tinj)EdE = 1 (32)

It can be shown that for any process that injectsenergy primarily through electromagnetic (rather thanhadronic) channels the function K(E tinj) takes theuniversal form [1 32 33]

K(E tinj) =

K0(EXE)32 E lt EXK0(EXE)2 EX lt E lt EC0 E gt EC

(33)where K0 is an overall normalization constant and whereEC and EX are energy scales associated with specificClass-I processes whose interplay determines the shapeof the reprocessed photon spectrum The normalizationconvention in Eq (32) implies that K0 is given by

K0 =1

E2X

[2 + ln(ECEX)

] (34)

Physically the energy scales EC and EX appearing inEq (33) can be understood as follows The scale ECrepresents the energy above which the photon spectrumis effectively extinguished by the pair-production processγ + γBG rarr e+ + eminus in conjunction with interactionsbetween the resulting electron and positron and otherparticles in the thermal bath The energy scale EXrepresents the threshold above which γ + γBG rarr γ + γis the dominant process through which photons loseenergy By contrast below this energy threshold thedominant processes are Compton scattering and e+eminus

5

pair-production off nuclei Note that while the nor-malization of the reprocessed photon spectrum is setby dρ(tinj)dtinj the shape of this spectrum is entirelycontrolled by the temperature Tinj at injection This

temperature behaves like Tinj prop tminus12inj in a radiation-

dominated epoch This implies that as tinj increases thevalue of EC also increases This reflects the fact thatthe thermal bath is colder at later injection times andthus an injected photon must be more energetic in orderfor the pair-production process γ + γBG rarr e+ + eminus tobe effective Numerically the values of EC and EX at agiven injection time tinj are estimated to be [1 3]

EC =m2e

22Tinjasymp(103 MeV

)times(tinj

108 s

)12

EX =m2e

80Tinjasymp(28 MeV

)times(tinj

108 s

)12

(35)

where me is the electron mass and Tinj is the temperatureof the thermal bath at tinj

The reprocessed photon spectrum in Eq (31) is thespectrum which effectively contributes to the photo-production andor photodisintegration of light elementsafter the BBN epoch In order to illustrate how theshape of this spectrum depends on the injection timetinj we plot in Fig 1 the function K(E tinj) whichdetermines the shape of this spectrum as a function ofE for several different values of the injection time tinjSince the ultraviolet cutoff EC in the photon spectrumincreases with tinj injection at later times can initiatephotoproduction and photodisintegration reactions withhigher energy thresholds The dashed vertical line whichwe include for reference represents the lowest thresholdenergy associated with any such reaction which can havea significant impact on the primordial abundance of anylight nucleus which is tightly constrained by observationAs discussed in Sect III B this reaction turns outto be the deuterium-photodisintegration reaction D +γ rarr n + p which has a threshold energy of roughly22 MeV Thus the portion of the photon spectrumwhich lies within the gray shaded region in Fig 1 has noeffect on the abundance of any relevant nucleus Since EClies below this threshold for tinj asymp 104 s electromagneticinjection between the end of BBN and this timescale hasessentially no effect on the abundances of light nuclei

One additional complication that we must take intoaccount in assessing the effect of injection on the abun-dances of light nuclei is that the reprocessed photonspectrum established by Class-I processes immediatelyafter injection at tinj is subsequently degraded by Class-III processes which slowly act to bring this reprocessedspectrum into thermal equilibrium with the backgroundphotons in the radiation bath The timescale δtth(E tinj)on which these processes act on a photon of energy E is

01 1 10 100 1000 10410-7

10-5

0001

0100

10

E [MeV]

K[MeV

-2]

tinj = 104 s

tinj = 106 s

tinj = 108 s

tinj = 1010 s

tinj = 1012 s

FIG 1 The function K(E tinj) plotted as a function ofphoton energy E for several different values of injection timetinj As discussed in the text this function determines theshape of the reprocessed photon spectrum for an instanta-neous injection of photons of energy E at time tinj Thedashed vertical line at E asymp 22 MeV indicates the lowestthreshold energy associated with any photoproduction orphotodisintegration reaction which can significantly alter theabundance of any light nucleus whose primordial abundanceis tightly constrained by observation

roughly

δtth(E tinj) asymp [nB(tinj)σth(E)]minus1

(36)

where nB(tinj) is the number density of baryons at thetime of injection and where σth(E) is the characteristiccross-section for the relevant scattering processes whichinclude Compton scattering and e+eminus pair productionoff nuclei The cross-sections for all relevant individualcontributing processes can be found in Ref [1] Note thatwhile at low energies σth(E) is well approximated by theThomson cross-section this approximation breaks downat higher energies as other processes become relevant

Given these observations the spectrum of the resultingnon-thermal population of photons at time t not onlyrepresents the sum of all contributions from injection atall times tinj lt t but also reflects the subsequent degra-dation of these contributions by the Class-III processeswhich serve to thermalize this population of photons withthe radiation bath This overall non-thermal photonspectrum takes the form

dnγ(E t)

dE=

int t

0

dnγ(E tinj)

dEdtinjG(t tinj)dtinj (37)

where

G(t tinj) equiv eminus(tminustinj)δtth(Etinj)Θ(tminus tinj) (38)

6

is the Greenrsquos function which solves the differentialequation

dG(t tinj)

dt+

G(t tinj)

δtth(E tinj)= δ(tminus tinj) (39)

The population of non-thermal photons described byEq (37) serves as the source for the initial photo-production and photodisintegration reactions ultimatelyresponsible for the modification of light-element abun-dances after BBN We shall therefore henceforth refer tophotons in this population as ldquoprimaryrdquo photons

In what follows we will find it useful to employwhat we shall call the ldquouniform-decay approximationrdquoSpecifically we shall approximate the full exponentialdecay of each dark-sector species χi as if the entirepopulation of such particles throughout the universe wereto decay precisely at the same time τi As we shall seethis will prove critical in allowing us to formulate ourultimate analytic approximations We shall neverthelessfind that the results of our approximations are generallyin excellent agreement with the results of a full numericalanalysis

Within the uniform-decay approximation the contri-bution to dρdt(tinj) from the decay of a single unstableparticle species χ takes the form of a Dirac δ-function

dρ(tinj)

dtinj= ρχ(tinj)εχδ(tinj minus τχ) (310)

where ρχ(tinj) is the energy density of χ at time tinj andwhere εχ is the fraction of the energy density releasedby χ decays which is transferred to photons It thereforefollows that in this approximation the primary-photonspectrum in Eq (37) reduces to

dnγ(E t)

dE= ρχ(τχ)εχK(E τχ)

times eminus(tminusτχ)δtth(Etinj)Θ(tminus τχ) (311)

B Light-Element ProductionDestruction

Generally speaking the overall rate of change of thenumber density na of a nuclear species Na due tothe injection of electromagnetic energy at late times isgoverned by a Boltzmann equation of the form

dnadt

+ 3Hna = C(p)a + C(s)

a (312)

where H is the Hubble parameter and where C(p)a and

C(s)a are the collision terms associated with two different

classes of scattering processes which contribute to thisoverall rate of change We shall describe these individualcollision terms in detail below Since na is affected byHubble expansion it is more convenient to work with thecorresponding comoving number density Ya equiv nanB

where nB denotes the total number density of baryonsThe Boltzmann equation for Ya then takes the form

dYadt

=1

nB

[C(p)a + C(s)

a

] (313)

The collision term C(p)a represents the collective con-

tribution from Class-II processes directly involving thepopulation of primary photons described by Eq (311)

The principal processes which contribute to C(p)a are

photoproduction processes of the form Nb+γ rarr Na+Ncand photodisintegration processes of the form Na + γ rarrNb + Nc where Nb and Nc are other nuclei in thethermal plasma The collision term associated with theseprocesses takes the form

C(p)a =

sumbc

[YbnB

int EC

E(ac)b

dnγ(E t)

dEσ

(ac)b (E)dE

minus YanB

int EC

E(bc)a

dnγ(E t)

dEσ(bc)a (E)dE

] (314)

where the indices b and c run over the nuclei present

in the plasma where σ(ac)b (E) and σ

(bc)a (E) respectively

denote the cross-sections for the corresponding photo-production and photodisintegration processes discussed

above and where E(ac)b and E

(bc)a are the respective

energy thresholds for these processes Expressions forthese cross-sections and values for the correspondingenergy thresholds can be found eg in Ref [3]

By contrast C(s)a represents the collective contribution

from additional secondary processes which involve notthe primary photons themselves but rather a non-thermalpopulation of energetic nuclei produced by interactionsinvolving those primary photons In principle thesesecondary processes include both reactions that producenuclei of species Na and reactions which destroy themIn practice however because the non-thermal populationof any given species Nb generated by processes involvingprimary photons is comparatively small the effect ofsecondary processes on the populations of most nuclearspecies is likewise small As we shall discuss in moredetail in Sect III C the only exception is 6Li whichis not produced in any significant amount during theBBN epoch but which can potentially be produced bysecondary processes initiated by photon injection atsubsequent times Since these processes involve theproduction rather than the destruction of 6Li we focus onthe effect of secondary processes on nuclei which appearin the final state rather than the initial state in whatfollows

The energetic nuclei which participate in secondaryprocesses are the products of the same kinds of reactionswhich lead to the collision term in Eq (314) Thusthe kinetic-energy spectrum dnb(Eb t)dEb of the non-thermal population of a nuclear species Nb produced inthis manner is in large part determined by the energyspectrum of the primary photons In calculating this

7

spectrum one must in principle account for the fact thata photon of energy Eγ can give rise to a range of possibleEb values due to the range of possible scattering anglesbetween the three-momentum vectors of the incomingphoton and the excited nucleus in the center-of-massframe However it can be shown [34] that a reasonable

approximation for the collision term C(s)a for 6Li is

nevertheless obtained by taking Eb to be a one-to-onefunction of Eγ of the form [35]

Eb = E(Eγ) equiv 1

4

[Eγ minus E(bd)

c

] (315)

where E(bd)c is the energy threshold for the primary

process Nc + γ rarr Nb + Nd In this approximationdnb(Eb t)dEb takes the form [3]

dnb(Eb t)

dEb=sumcd

ncb(Eb t)

int EC

Emin(Eb)

[dnγ(Eprimeγ t)

dEprimeγ

times σ(bd)c (Eprimeγ)eminusβb(E

primeγ t)

]dEprimeγ (316)

where b(Eb t) is the energy-loss rate for Nb due toCoulomb scattering with particles in the thermal back-ground plasma The exponential factor βb(E

primeγ t) ac-

counts for the collective effect of additional processeswhich act to reduce the number of nuclei of species NbThe lower limit of integration in Eq (316) is given by

Emin(Eb) = max[Eminus1(Eb) E(bd)c ] where Eminus1(Eb) is the

inverse of the function defined in Eq (315) In otherwords Eminus1(Eb) is the photon energy which correspondsto a kinetic energy Eb for the excited nucleus

In principle the processes which contribute toβa(Eprimeγ t) include both decay processes (in the case inwhich Nb is unstable) and photodisintegration processesof the form Nb+γ rarr Nc+Nd involving a primary photonIn practice the photodisintegration rate due to theseprocesses is much slower that the energy-loss rate due toCoulomb scattering for any species of interest Moreoveras we shall see in Sect III C the only nuclear specieswhose non-thermal population has a significant effect onthe 6Li abundance are tritium (T) and the helium isotope3He Because these two species are mirror nuclei thesecondary processes in which they participate affect the6Li abundance in the same way and have almost identicalcross-sections and energy thresholds Thus in terms oftheir effect on the production of 6Li the populations ofT and 3He may effectively be treated together as if theywere the population of a single nuclear species Althoughtritium is unstable and decays via beta decay to 3Hewith a lifetime of τT asymp 56times 108 s these decays have noimpact on the combined population of T and 3He Wemay therefore safely approximate βb(E

primeγ t) asymp 0 for this

combined population of excited nuclei in what followsThe most relevant processes through which an en-

ergetic nucleus Nb in this non-thermal population canalter the abundance of another nuclear species Nb are

scattering processes of the form Nb + Nf rarr Na + Xin which an energetic nucleus Nb from the non-thermalpopulation generated by primary processes scatters witha background nucleus Nf resulting in the production of anucleus of species Na and some other particle X (whichcould be either an additional nucleus or a photon) Inprinciple processes of the form Nb +Na rarr Nf +X canalso act to reduce the abundance of Na However asdiscussed above this reduction has a negligible impact onYa for any nuclear species Na which already has a sizablecomoving number density at the end of BBN Thuswe focus here on production rather than destructionwhen assessing the impact of secondary processes on theprimordial abundances of light nuclei

With this simplification the collision term associatedwith secondary production processes takes the form

C(s)a =

sumbf

YfnB

int E(EC)

E(aX)bf

[dnb(Eb t)

dEb

times σ(aX)bf (Eb)|v(Eb)|

]dEb (317)

where v(Eb) is the (non-relativistic) relative velocityof nuclei Nb and Nf in the background frame wherednb(Eb t)dEb is the differential energy spectrum of the

non-thermal population of Nb where σ(aX)bf is the cross-

section for the scattering process Nb+Nf rarr Na+X with

corresponding threshold energy E(aX)bf and where E(EC)

is the cutoff in dnb(Eb t)dEb produced by primaryprocesses

C Constraints on Primordial Light-ElementAbundances

The nuclear species whose primordial abundances arethe most tightly constrained by observation mdash and whichare therefore relevant for constraining the late decays ofunstable particles mdash are D 4He 6Li and 7Li The abun-dance of 3He during the present cosmological epoch hasalso been constrained by observation [36 37] Howeveruncertainties in the contribution to this abundance fromstellar sources make it difficult to translate the results ofthese measurements into bounds on the primordial 3Heabundance [38 39] The effect of these uncertainties canbe mitigated in part if we consider the ratio (D+ 3He)Hrather than 3HeH as the former is expected to be largelyunaffected by stellar processing [40ndash42] In this paperwe focus our attention on D 4He 6Li and 7Li as therelationship between the measured abundances of thesenuclei and their corresponding primordial abundances ismore transparent

The observational constraints on the primordial abun-dances of these nuclei can be summarized as followsBounds on the primordial 4He abundance are typicallyphrased in terms of the primordial helium mass fraction

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

5

pair-production off nuclei Note that while the nor-malization of the reprocessed photon spectrum is setby dρ(tinj)dtinj the shape of this spectrum is entirelycontrolled by the temperature Tinj at injection This

temperature behaves like Tinj prop tminus12inj in a radiation-

dominated epoch This implies that as tinj increases thevalue of EC also increases This reflects the fact thatthe thermal bath is colder at later injection times andthus an injected photon must be more energetic in orderfor the pair-production process γ + γBG rarr e+ + eminus tobe effective Numerically the values of EC and EX at agiven injection time tinj are estimated to be [1 3]

EC =m2e

22Tinjasymp(103 MeV

)times(tinj

108 s

)12

EX =m2e

80Tinjasymp(28 MeV

)times(tinj

108 s

)12

(35)

where me is the electron mass and Tinj is the temperatureof the thermal bath at tinj

The reprocessed photon spectrum in Eq (31) is thespectrum which effectively contributes to the photo-production andor photodisintegration of light elementsafter the BBN epoch In order to illustrate how theshape of this spectrum depends on the injection timetinj we plot in Fig 1 the function K(E tinj) whichdetermines the shape of this spectrum as a function ofE for several different values of the injection time tinjSince the ultraviolet cutoff EC in the photon spectrumincreases with tinj injection at later times can initiatephotoproduction and photodisintegration reactions withhigher energy thresholds The dashed vertical line whichwe include for reference represents the lowest thresholdenergy associated with any such reaction which can havea significant impact on the primordial abundance of anylight nucleus which is tightly constrained by observationAs discussed in Sect III B this reaction turns outto be the deuterium-photodisintegration reaction D +γ rarr n + p which has a threshold energy of roughly22 MeV Thus the portion of the photon spectrumwhich lies within the gray shaded region in Fig 1 has noeffect on the abundance of any relevant nucleus Since EClies below this threshold for tinj asymp 104 s electromagneticinjection between the end of BBN and this timescale hasessentially no effect on the abundances of light nuclei

One additional complication that we must take intoaccount in assessing the effect of injection on the abun-dances of light nuclei is that the reprocessed photonspectrum established by Class-I processes immediatelyafter injection at tinj is subsequently degraded by Class-III processes which slowly act to bring this reprocessedspectrum into thermal equilibrium with the backgroundphotons in the radiation bath The timescale δtth(E tinj)on which these processes act on a photon of energy E is

01 1 10 100 1000 10410-7

10-5

0001

0100

10

E [MeV]

K[MeV

-2]

tinj = 104 s

tinj = 106 s

tinj = 108 s

tinj = 1010 s

tinj = 1012 s

FIG 1 The function K(E tinj) plotted as a function ofphoton energy E for several different values of injection timetinj As discussed in the text this function determines theshape of the reprocessed photon spectrum for an instanta-neous injection of photons of energy E at time tinj Thedashed vertical line at E asymp 22 MeV indicates the lowestthreshold energy associated with any photoproduction orphotodisintegration reaction which can significantly alter theabundance of any light nucleus whose primordial abundanceis tightly constrained by observation

roughly

δtth(E tinj) asymp [nB(tinj)σth(E)]minus1

(36)

where nB(tinj) is the number density of baryons at thetime of injection and where σth(E) is the characteristiccross-section for the relevant scattering processes whichinclude Compton scattering and e+eminus pair productionoff nuclei The cross-sections for all relevant individualcontributing processes can be found in Ref [1] Note thatwhile at low energies σth(E) is well approximated by theThomson cross-section this approximation breaks downat higher energies as other processes become relevant

Given these observations the spectrum of the resultingnon-thermal population of photons at time t not onlyrepresents the sum of all contributions from injection atall times tinj lt t but also reflects the subsequent degra-dation of these contributions by the Class-III processeswhich serve to thermalize this population of photons withthe radiation bath This overall non-thermal photonspectrum takes the form

dnγ(E t)

dE=

int t

0

dnγ(E tinj)

dEdtinjG(t tinj)dtinj (37)

where

G(t tinj) equiv eminus(tminustinj)δtth(Etinj)Θ(tminus tinj) (38)

6

is the Greenrsquos function which solves the differentialequation

dG(t tinj)

dt+

G(t tinj)

δtth(E tinj)= δ(tminus tinj) (39)

The population of non-thermal photons described byEq (37) serves as the source for the initial photo-production and photodisintegration reactions ultimatelyresponsible for the modification of light-element abun-dances after BBN We shall therefore henceforth refer tophotons in this population as ldquoprimaryrdquo photons

In what follows we will find it useful to employwhat we shall call the ldquouniform-decay approximationrdquoSpecifically we shall approximate the full exponentialdecay of each dark-sector species χi as if the entirepopulation of such particles throughout the universe wereto decay precisely at the same time τi As we shall seethis will prove critical in allowing us to formulate ourultimate analytic approximations We shall neverthelessfind that the results of our approximations are generallyin excellent agreement with the results of a full numericalanalysis

Within the uniform-decay approximation the contri-bution to dρdt(tinj) from the decay of a single unstableparticle species χ takes the form of a Dirac δ-function

dρ(tinj)

dtinj= ρχ(tinj)εχδ(tinj minus τχ) (310)

where ρχ(tinj) is the energy density of χ at time tinj andwhere εχ is the fraction of the energy density releasedby χ decays which is transferred to photons It thereforefollows that in this approximation the primary-photonspectrum in Eq (37) reduces to

dnγ(E t)

dE= ρχ(τχ)εχK(E τχ)

times eminus(tminusτχ)δtth(Etinj)Θ(tminus τχ) (311)

B Light-Element ProductionDestruction

Generally speaking the overall rate of change of thenumber density na of a nuclear species Na due tothe injection of electromagnetic energy at late times isgoverned by a Boltzmann equation of the form

dnadt

+ 3Hna = C(p)a + C(s)

a (312)

where H is the Hubble parameter and where C(p)a and

C(s)a are the collision terms associated with two different

classes of scattering processes which contribute to thisoverall rate of change We shall describe these individualcollision terms in detail below Since na is affected byHubble expansion it is more convenient to work with thecorresponding comoving number density Ya equiv nanB

where nB denotes the total number density of baryonsThe Boltzmann equation for Ya then takes the form

dYadt

=1

nB

[C(p)a + C(s)

a

] (313)

The collision term C(p)a represents the collective con-

tribution from Class-II processes directly involving thepopulation of primary photons described by Eq (311)

The principal processes which contribute to C(p)a are

photoproduction processes of the form Nb+γ rarr Na+Ncand photodisintegration processes of the form Na + γ rarrNb + Nc where Nb and Nc are other nuclei in thethermal plasma The collision term associated with theseprocesses takes the form

C(p)a =

sumbc

[YbnB

int EC

E(ac)b

dnγ(E t)

dEσ

(ac)b (E)dE

minus YanB

int EC

E(bc)a

dnγ(E t)

dEσ(bc)a (E)dE

] (314)

where the indices b and c run over the nuclei present

in the plasma where σ(ac)b (E) and σ

(bc)a (E) respectively

denote the cross-sections for the corresponding photo-production and photodisintegration processes discussed

above and where E(ac)b and E

(bc)a are the respective

energy thresholds for these processes Expressions forthese cross-sections and values for the correspondingenergy thresholds can be found eg in Ref [3]

By contrast C(s)a represents the collective contribution

from additional secondary processes which involve notthe primary photons themselves but rather a non-thermalpopulation of energetic nuclei produced by interactionsinvolving those primary photons In principle thesesecondary processes include both reactions that producenuclei of species Na and reactions which destroy themIn practice however because the non-thermal populationof any given species Nb generated by processes involvingprimary photons is comparatively small the effect ofsecondary processes on the populations of most nuclearspecies is likewise small As we shall discuss in moredetail in Sect III C the only exception is 6Li whichis not produced in any significant amount during theBBN epoch but which can potentially be produced bysecondary processes initiated by photon injection atsubsequent times Since these processes involve theproduction rather than the destruction of 6Li we focus onthe effect of secondary processes on nuclei which appearin the final state rather than the initial state in whatfollows

The energetic nuclei which participate in secondaryprocesses are the products of the same kinds of reactionswhich lead to the collision term in Eq (314) Thusthe kinetic-energy spectrum dnb(Eb t)dEb of the non-thermal population of a nuclear species Nb produced inthis manner is in large part determined by the energyspectrum of the primary photons In calculating this

7

spectrum one must in principle account for the fact thata photon of energy Eγ can give rise to a range of possibleEb values due to the range of possible scattering anglesbetween the three-momentum vectors of the incomingphoton and the excited nucleus in the center-of-massframe However it can be shown [34] that a reasonable

approximation for the collision term C(s)a for 6Li is

nevertheless obtained by taking Eb to be a one-to-onefunction of Eγ of the form [35]

Eb = E(Eγ) equiv 1

4

[Eγ minus E(bd)

c

] (315)

where E(bd)c is the energy threshold for the primary

process Nc + γ rarr Nb + Nd In this approximationdnb(Eb t)dEb takes the form [3]

dnb(Eb t)

dEb=sumcd

ncb(Eb t)

int EC

Emin(Eb)

[dnγ(Eprimeγ t)

dEprimeγ

times σ(bd)c (Eprimeγ)eminusβb(E

primeγ t)

]dEprimeγ (316)

where b(Eb t) is the energy-loss rate for Nb due toCoulomb scattering with particles in the thermal back-ground plasma The exponential factor βb(E

primeγ t) ac-

counts for the collective effect of additional processeswhich act to reduce the number of nuclei of species NbThe lower limit of integration in Eq (316) is given by

Emin(Eb) = max[Eminus1(Eb) E(bd)c ] where Eminus1(Eb) is the

inverse of the function defined in Eq (315) In otherwords Eminus1(Eb) is the photon energy which correspondsto a kinetic energy Eb for the excited nucleus

In principle the processes which contribute toβa(Eprimeγ t) include both decay processes (in the case inwhich Nb is unstable) and photodisintegration processesof the form Nb+γ rarr Nc+Nd involving a primary photonIn practice the photodisintegration rate due to theseprocesses is much slower that the energy-loss rate due toCoulomb scattering for any species of interest Moreoveras we shall see in Sect III C the only nuclear specieswhose non-thermal population has a significant effect onthe 6Li abundance are tritium (T) and the helium isotope3He Because these two species are mirror nuclei thesecondary processes in which they participate affect the6Li abundance in the same way and have almost identicalcross-sections and energy thresholds Thus in terms oftheir effect on the production of 6Li the populations ofT and 3He may effectively be treated together as if theywere the population of a single nuclear species Althoughtritium is unstable and decays via beta decay to 3Hewith a lifetime of τT asymp 56times 108 s these decays have noimpact on the combined population of T and 3He Wemay therefore safely approximate βb(E

primeγ t) asymp 0 for this

combined population of excited nuclei in what followsThe most relevant processes through which an en-

ergetic nucleus Nb in this non-thermal population canalter the abundance of another nuclear species Nb are

scattering processes of the form Nb + Nf rarr Na + Xin which an energetic nucleus Nb from the non-thermalpopulation generated by primary processes scatters witha background nucleus Nf resulting in the production of anucleus of species Na and some other particle X (whichcould be either an additional nucleus or a photon) Inprinciple processes of the form Nb +Na rarr Nf +X canalso act to reduce the abundance of Na However asdiscussed above this reduction has a negligible impact onYa for any nuclear species Na which already has a sizablecomoving number density at the end of BBN Thuswe focus here on production rather than destructionwhen assessing the impact of secondary processes on theprimordial abundances of light nuclei

With this simplification the collision term associatedwith secondary production processes takes the form

C(s)a =

sumbf

YfnB

int E(EC)

E(aX)bf

[dnb(Eb t)

dEb

times σ(aX)bf (Eb)|v(Eb)|

]dEb (317)

where v(Eb) is the (non-relativistic) relative velocityof nuclei Nb and Nf in the background frame wherednb(Eb t)dEb is the differential energy spectrum of the

non-thermal population of Nb where σ(aX)bf is the cross-

section for the scattering process Nb+Nf rarr Na+X with

corresponding threshold energy E(aX)bf and where E(EC)

is the cutoff in dnb(Eb t)dEb produced by primaryprocesses

C Constraints on Primordial Light-ElementAbundances

The nuclear species whose primordial abundances arethe most tightly constrained by observation mdash and whichare therefore relevant for constraining the late decays ofunstable particles mdash are D 4He 6Li and 7Li The abun-dance of 3He during the present cosmological epoch hasalso been constrained by observation [36 37] Howeveruncertainties in the contribution to this abundance fromstellar sources make it difficult to translate the results ofthese measurements into bounds on the primordial 3Heabundance [38 39] The effect of these uncertainties canbe mitigated in part if we consider the ratio (D+ 3He)Hrather than 3HeH as the former is expected to be largelyunaffected by stellar processing [40ndash42] In this paperwe focus our attention on D 4He 6Li and 7Li as therelationship between the measured abundances of thesenuclei and their corresponding primordial abundances ismore transparent

The observational constraints on the primordial abun-dances of these nuclei can be summarized as followsBounds on the primordial 4He abundance are typicallyphrased in terms of the primordial helium mass fraction

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

6

is the Greenrsquos function which solves the differentialequation

dG(t tinj)

dt+

G(t tinj)

δtth(E tinj)= δ(tminus tinj) (39)

The population of non-thermal photons described byEq (37) serves as the source for the initial photo-production and photodisintegration reactions ultimatelyresponsible for the modification of light-element abun-dances after BBN We shall therefore henceforth refer tophotons in this population as ldquoprimaryrdquo photons

In what follows we will find it useful to employwhat we shall call the ldquouniform-decay approximationrdquoSpecifically we shall approximate the full exponentialdecay of each dark-sector species χi as if the entirepopulation of such particles throughout the universe wereto decay precisely at the same time τi As we shall seethis will prove critical in allowing us to formulate ourultimate analytic approximations We shall neverthelessfind that the results of our approximations are generallyin excellent agreement with the results of a full numericalanalysis

Within the uniform-decay approximation the contri-bution to dρdt(tinj) from the decay of a single unstableparticle species χ takes the form of a Dirac δ-function

dρ(tinj)

dtinj= ρχ(tinj)εχδ(tinj minus τχ) (310)

where ρχ(tinj) is the energy density of χ at time tinj andwhere εχ is the fraction of the energy density releasedby χ decays which is transferred to photons It thereforefollows that in this approximation the primary-photonspectrum in Eq (37) reduces to

dnγ(E t)

dE= ρχ(τχ)εχK(E τχ)

times eminus(tminusτχ)δtth(Etinj)Θ(tminus τχ) (311)

B Light-Element ProductionDestruction

Generally speaking the overall rate of change of thenumber density na of a nuclear species Na due tothe injection of electromagnetic energy at late times isgoverned by a Boltzmann equation of the form

dnadt

+ 3Hna = C(p)a + C(s)

a (312)

where H is the Hubble parameter and where C(p)a and

C(s)a are the collision terms associated with two different

classes of scattering processes which contribute to thisoverall rate of change We shall describe these individualcollision terms in detail below Since na is affected byHubble expansion it is more convenient to work with thecorresponding comoving number density Ya equiv nanB

where nB denotes the total number density of baryonsThe Boltzmann equation for Ya then takes the form

dYadt

=1

nB

[C(p)a + C(s)

a

] (313)

The collision term C(p)a represents the collective con-

tribution from Class-II processes directly involving thepopulation of primary photons described by Eq (311)

The principal processes which contribute to C(p)a are

photoproduction processes of the form Nb+γ rarr Na+Ncand photodisintegration processes of the form Na + γ rarrNb + Nc where Nb and Nc are other nuclei in thethermal plasma The collision term associated with theseprocesses takes the form

C(p)a =

sumbc

[YbnB

int EC

E(ac)b

dnγ(E t)

dEσ

(ac)b (E)dE

minus YanB

int EC

E(bc)a

dnγ(E t)

dEσ(bc)a (E)dE

] (314)

where the indices b and c run over the nuclei present

in the plasma where σ(ac)b (E) and σ

(bc)a (E) respectively

denote the cross-sections for the corresponding photo-production and photodisintegration processes discussed

above and where E(ac)b and E

(bc)a are the respective

energy thresholds for these processes Expressions forthese cross-sections and values for the correspondingenergy thresholds can be found eg in Ref [3]

By contrast C(s)a represents the collective contribution

from additional secondary processes which involve notthe primary photons themselves but rather a non-thermalpopulation of energetic nuclei produced by interactionsinvolving those primary photons In principle thesesecondary processes include both reactions that producenuclei of species Na and reactions which destroy themIn practice however because the non-thermal populationof any given species Nb generated by processes involvingprimary photons is comparatively small the effect ofsecondary processes on the populations of most nuclearspecies is likewise small As we shall discuss in moredetail in Sect III C the only exception is 6Li whichis not produced in any significant amount during theBBN epoch but which can potentially be produced bysecondary processes initiated by photon injection atsubsequent times Since these processes involve theproduction rather than the destruction of 6Li we focus onthe effect of secondary processes on nuclei which appearin the final state rather than the initial state in whatfollows

The energetic nuclei which participate in secondaryprocesses are the products of the same kinds of reactionswhich lead to the collision term in Eq (314) Thusthe kinetic-energy spectrum dnb(Eb t)dEb of the non-thermal population of a nuclear species Nb produced inthis manner is in large part determined by the energyspectrum of the primary photons In calculating this

7

spectrum one must in principle account for the fact thata photon of energy Eγ can give rise to a range of possibleEb values due to the range of possible scattering anglesbetween the three-momentum vectors of the incomingphoton and the excited nucleus in the center-of-massframe However it can be shown [34] that a reasonable

approximation for the collision term C(s)a for 6Li is

nevertheless obtained by taking Eb to be a one-to-onefunction of Eγ of the form [35]

Eb = E(Eγ) equiv 1

4

[Eγ minus E(bd)

c

] (315)

where E(bd)c is the energy threshold for the primary

process Nc + γ rarr Nb + Nd In this approximationdnb(Eb t)dEb takes the form [3]

dnb(Eb t)

dEb=sumcd

ncb(Eb t)

int EC

Emin(Eb)

[dnγ(Eprimeγ t)

dEprimeγ

times σ(bd)c (Eprimeγ)eminusβb(E

primeγ t)

]dEprimeγ (316)

where b(Eb t) is the energy-loss rate for Nb due toCoulomb scattering with particles in the thermal back-ground plasma The exponential factor βb(E

primeγ t) ac-

counts for the collective effect of additional processeswhich act to reduce the number of nuclei of species NbThe lower limit of integration in Eq (316) is given by

Emin(Eb) = max[Eminus1(Eb) E(bd)c ] where Eminus1(Eb) is the

inverse of the function defined in Eq (315) In otherwords Eminus1(Eb) is the photon energy which correspondsto a kinetic energy Eb for the excited nucleus

In principle the processes which contribute toβa(Eprimeγ t) include both decay processes (in the case inwhich Nb is unstable) and photodisintegration processesof the form Nb+γ rarr Nc+Nd involving a primary photonIn practice the photodisintegration rate due to theseprocesses is much slower that the energy-loss rate due toCoulomb scattering for any species of interest Moreoveras we shall see in Sect III C the only nuclear specieswhose non-thermal population has a significant effect onthe 6Li abundance are tritium (T) and the helium isotope3He Because these two species are mirror nuclei thesecondary processes in which they participate affect the6Li abundance in the same way and have almost identicalcross-sections and energy thresholds Thus in terms oftheir effect on the production of 6Li the populations ofT and 3He may effectively be treated together as if theywere the population of a single nuclear species Althoughtritium is unstable and decays via beta decay to 3Hewith a lifetime of τT asymp 56times 108 s these decays have noimpact on the combined population of T and 3He Wemay therefore safely approximate βb(E

primeγ t) asymp 0 for this

combined population of excited nuclei in what followsThe most relevant processes through which an en-

ergetic nucleus Nb in this non-thermal population canalter the abundance of another nuclear species Nb are

scattering processes of the form Nb + Nf rarr Na + Xin which an energetic nucleus Nb from the non-thermalpopulation generated by primary processes scatters witha background nucleus Nf resulting in the production of anucleus of species Na and some other particle X (whichcould be either an additional nucleus or a photon) Inprinciple processes of the form Nb +Na rarr Nf +X canalso act to reduce the abundance of Na However asdiscussed above this reduction has a negligible impact onYa for any nuclear species Na which already has a sizablecomoving number density at the end of BBN Thuswe focus here on production rather than destructionwhen assessing the impact of secondary processes on theprimordial abundances of light nuclei

With this simplification the collision term associatedwith secondary production processes takes the form

C(s)a =

sumbf

YfnB

int E(EC)

E(aX)bf

[dnb(Eb t)

dEb

times σ(aX)bf (Eb)|v(Eb)|

]dEb (317)

where v(Eb) is the (non-relativistic) relative velocityof nuclei Nb and Nf in the background frame wherednb(Eb t)dEb is the differential energy spectrum of the

non-thermal population of Nb where σ(aX)bf is the cross-

section for the scattering process Nb+Nf rarr Na+X with

corresponding threshold energy E(aX)bf and where E(EC)

is the cutoff in dnb(Eb t)dEb produced by primaryprocesses

C Constraints on Primordial Light-ElementAbundances

The nuclear species whose primordial abundances arethe most tightly constrained by observation mdash and whichare therefore relevant for constraining the late decays ofunstable particles mdash are D 4He 6Li and 7Li The abun-dance of 3He during the present cosmological epoch hasalso been constrained by observation [36 37] Howeveruncertainties in the contribution to this abundance fromstellar sources make it difficult to translate the results ofthese measurements into bounds on the primordial 3Heabundance [38 39] The effect of these uncertainties canbe mitigated in part if we consider the ratio (D+ 3He)Hrather than 3HeH as the former is expected to be largelyunaffected by stellar processing [40ndash42] In this paperwe focus our attention on D 4He 6Li and 7Li as therelationship between the measured abundances of thesenuclei and their corresponding primordial abundances ismore transparent

The observational constraints on the primordial abun-dances of these nuclei can be summarized as followsBounds on the primordial 4He abundance are typicallyphrased in terms of the primordial helium mass fraction

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

7

spectrum one must in principle account for the fact thata photon of energy Eγ can give rise to a range of possibleEb values due to the range of possible scattering anglesbetween the three-momentum vectors of the incomingphoton and the excited nucleus in the center-of-massframe However it can be shown [34] that a reasonable

approximation for the collision term C(s)a for 6Li is

nevertheless obtained by taking Eb to be a one-to-onefunction of Eγ of the form [35]

Eb = E(Eγ) equiv 1

4

[Eγ minus E(bd)

c

] (315)

where E(bd)c is the energy threshold for the primary

process Nc + γ rarr Nb + Nd In this approximationdnb(Eb t)dEb takes the form [3]

dnb(Eb t)

dEb=sumcd

ncb(Eb t)

int EC

Emin(Eb)

[dnγ(Eprimeγ t)

dEprimeγ

times σ(bd)c (Eprimeγ)eminusβb(E

primeγ t)

]dEprimeγ (316)

where b(Eb t) is the energy-loss rate for Nb due toCoulomb scattering with particles in the thermal back-ground plasma The exponential factor βb(E

primeγ t) ac-

counts for the collective effect of additional processeswhich act to reduce the number of nuclei of species NbThe lower limit of integration in Eq (316) is given by

Emin(Eb) = max[Eminus1(Eb) E(bd)c ] where Eminus1(Eb) is the

inverse of the function defined in Eq (315) In otherwords Eminus1(Eb) is the photon energy which correspondsto a kinetic energy Eb for the excited nucleus

In principle the processes which contribute toβa(Eprimeγ t) include both decay processes (in the case inwhich Nb is unstable) and photodisintegration processesof the form Nb+γ rarr Nc+Nd involving a primary photonIn practice the photodisintegration rate due to theseprocesses is much slower that the energy-loss rate due toCoulomb scattering for any species of interest Moreoveras we shall see in Sect III C the only nuclear specieswhose non-thermal population has a significant effect onthe 6Li abundance are tritium (T) and the helium isotope3He Because these two species are mirror nuclei thesecondary processes in which they participate affect the6Li abundance in the same way and have almost identicalcross-sections and energy thresholds Thus in terms oftheir effect on the production of 6Li the populations ofT and 3He may effectively be treated together as if theywere the population of a single nuclear species Althoughtritium is unstable and decays via beta decay to 3Hewith a lifetime of τT asymp 56times 108 s these decays have noimpact on the combined population of T and 3He Wemay therefore safely approximate βb(E

primeγ t) asymp 0 for this

combined population of excited nuclei in what followsThe most relevant processes through which an en-

ergetic nucleus Nb in this non-thermal population canalter the abundance of another nuclear species Nb are

scattering processes of the form Nb + Nf rarr Na + Xin which an energetic nucleus Nb from the non-thermalpopulation generated by primary processes scatters witha background nucleus Nf resulting in the production of anucleus of species Na and some other particle X (whichcould be either an additional nucleus or a photon) Inprinciple processes of the form Nb +Na rarr Nf +X canalso act to reduce the abundance of Na However asdiscussed above this reduction has a negligible impact onYa for any nuclear species Na which already has a sizablecomoving number density at the end of BBN Thuswe focus here on production rather than destructionwhen assessing the impact of secondary processes on theprimordial abundances of light nuclei

With this simplification the collision term associatedwith secondary production processes takes the form

C(s)a =

sumbf

YfnB

int E(EC)

E(aX)bf

[dnb(Eb t)

dEb

times σ(aX)bf (Eb)|v(Eb)|

]dEb (317)

where v(Eb) is the (non-relativistic) relative velocityof nuclei Nb and Nf in the background frame wherednb(Eb t)dEb is the differential energy spectrum of the

non-thermal population of Nb where σ(aX)bf is the cross-

section for the scattering process Nb+Nf rarr Na+X with

corresponding threshold energy E(aX)bf and where E(EC)

is the cutoff in dnb(Eb t)dEb produced by primaryprocesses

C Constraints on Primordial Light-ElementAbundances

The nuclear species whose primordial abundances arethe most tightly constrained by observation mdash and whichare therefore relevant for constraining the late decays ofunstable particles mdash are D 4He 6Li and 7Li The abun-dance of 3He during the present cosmological epoch hasalso been constrained by observation [36 37] Howeveruncertainties in the contribution to this abundance fromstellar sources make it difficult to translate the results ofthese measurements into bounds on the primordial 3Heabundance [38 39] The effect of these uncertainties canbe mitigated in part if we consider the ratio (D+ 3He)Hrather than 3HeH as the former is expected to be largelyunaffected by stellar processing [40ndash42] In this paperwe focus our attention on D 4He 6Li and 7Li as therelationship between the measured abundances of thesenuclei and their corresponding primordial abundances ismore transparent

The observational constraints on the primordial abun-dances of these nuclei can be summarized as followsBounds on the primordial 4He abundance are typicallyphrased in terms of the primordial helium mass fraction

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

8

Yp equiv (ρ4HeρB)p where ρ4He is the mass density of4He where ρB is the total mass density of baryonicmatter and where the subscript p signifies that it isonly the primordial contribution to ρ4He which is usedin calculating Yp with subsequent modifications to thisquantity due to stellar synthesis etc ignored The 2σlimits on Yp are [43]

02369 lt Yp lt 02529 (318)

The observational 2σ limits on the 7Li abundance are [44]

10times 10minus10 lt

( 7Li

H

)p

lt 22times 10minus10 (319)

where the symbols 7Li and H denote the primordialnumber densities of the corresponding nuclear speciesIn this connection we note that significant tension existsbetween these observational bounds and the predictionsof theoretical calculations of the 7Li abundance whichare roughly a factor of three larger While it is not ouraim in this paper to address this discrepancy electromag-netic injection from the late decays of unstable particlesmay play a role [45ndash47] in reconciling these predictionswith observational data

Constraining the primordial abundance of D is compli-cated by a mild tension which currently exists betweenthe observational results for DH derived from measure-ments of the line spectra of low-metallicity gas clouds [48]and the results obtained from numerical analysis ofthe Boltzmann equations for BBN [49] with input fromPlanck data [50] which predict a slightly lower value forthis ratio We account for these tensions by choosingour central value and lower limit on DH in accord withthe central value and 2σ lower limit from numericalcalculations while at the same time adopting the 2σobservational upper limit as our own upper limit on thisratio Thus we take our bounds on the D abundance tobe

2317times 10minus5 lt

(D

H

)p

lt 2587times 10minus5 (320)

An upper bound on the ratio 6LiH can likewise bederived from observation by combining observationalupper bounds on the more directly constrained quantities6Li7Li and 7LiH By combining the upper boundon 6Li7Li from Ref [51] with the upper bound fromEq (319) we obtain( 6Li

H

)p

lt 2times 10minus11 (321)

While this upper bound is identical to the correspondingconstraint quoted in Ref [3] this is a numerical accidentresulting from a higher estimate of the 6Li7Li ratio (dueto the recent detection of additional 6Li in low-metallicitystars) and a reduction in the upper bound on the 7LiHratio

Having assessed the observational constraints on D4He 6Li and 7Li we now turn to consider the effectthat the late-time injection of electromagnetic radiationhas on the abundance of each of these nuclear speciesrelative to its initial abundance at the conclusion of theBBN epoch

Of all these species 4He is by far the most abundantFor this reason reactions involving 4He nuclei in theinitial state play an outsize role in the production ofother nuclear species Moreover since the abundancesof all other such species in the thermal bath are farsmaller than that of 4He reactions involving these othernuclei in the initial state have a negligible impact on the4He abundance Photodisintegration processes initiateddirectly by primary photons are therefore the only pro-cesses which have an appreciable effect on the primordialabundance of 4He A number of individual such processescontribute to the overall photodisintegration rate of 4Heall of which have threshold energies Ethresh amp 20 MeV

The primordial abundance of 7Li like that of 4Heevolves in response to photon injection primarily as aresult of photodisintegration processes initiated directlyby primary photons At early times when the energyceiling EC in Eq (35) for the spectrum of these photonsis relatively low the process 7Li + γ rarr 4He + T whichhas a threshold energy of only sim 25 MeV dominatesthe photodisintegration rate By contrast at later timesadditional processes with higher threshold energies suchas 7Li+γ rarr 6Li+n and 7Li+γ rarr 4He+2n+p becomerelevant

While the reactions which have a significant impacton the 4He and 7Li abundances all serve to reduce theseabundances the reactions which have an impact on theD abundance include both processes which create deu-terium nuclei and processes which destroy them At earlytimes photodisintegration processes initiated by primaryphotons mdash and in particular the process D + γ rarr n+ pwhich has a threshold energy of only Ethresh asymp 22 MeVmdash dominate and serve to deplete the initial D abundanceAt later times however additional processes with higherenergy thresholds turn on and serve to counteract thisinitial depletion The dominant such process is thephotoproduction process 4He + γ rarr D + n + p whichhas a threshold energy of Ethresh asymp 25 MeV

Unlike 4He 7Li and D the nucleus 6Li is not generatedto any significant degree by BBN However a populationof 6Li nuclei can be generated after BBN as a resultof photon injection at subsequent times The mostrelevant processes are 7Li + γ rarr 6Li + n and thesecondary production processes 4He + 3He rarr 6Li + pand 4He + T rarr 6Li + n where T denotes a tritiumnucleus All of these processes have energy thresholdsEthresh asymp 7 MeV The abundances of 3He and Twhich serve as reactants in these secondary processesare smaller at the end of BBN than the abundance of4He by factors of O(104) and O(106) respectively (forreviews see eg Ref [52]) At the same time thenon-thermal populations of 3He and T generated via the

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

9

Process Associated Reactions Ethresh

D Destruction D + γ rarr n+ p 22 MeV

D Production 4He + γ rarr D + n+ p 261 MeV4He Destruction 4He + γ rarr T + p 198 MeV

4He + γ rarr 3He + n 206 MeV7Li Destruction 7Li + γ rarr 4He + T 25 MeV

7Li + γ rarr 6Li + n 72 MeV7Li + γ rarr 4He + 2n+ p 109 MeV

Primary 6Li Production 7Li + γ rarr 6Li + n 72 MeV

Secondary 6Li Production 4He + 3Herarr 6Li + p 70 MeV4He + Trarr 6Li + n 84 MeV

TABLE I Reactions which alter the abundances of D 4He6Li and 7Li in scenarios involving photon injection at latetimes along with the corresponding energy threshold Ethresh

for each process We note that the excited T and 3Henuclei which participate in the secondary production of 6Liare generated primarily by the processes listed above whichcontribute to the destruction of 4He

photodisintegration of 4He are much larger than the non-thermal population of 4He which is generated via thephotodisintegration of other far less abundant nucleiThus to a very good approximation the reactions whichcontribute to the secondary production of 6Li involve anexcited 3He or T nucleus and a ldquobackgroundrdquo 4He nucleusin thermal equilibrium with the radiation bath

In Table I we provide a list of the relevant reactionswhich can serve to alter the abundances of light nucleias a consequence of photon injection at late times alongwith their corresponding energy thresholds Expressionsfor the cross-sections for these processes are given inRef [3] We note that alternative parametrizations forsome of the relevant cross-section formulae have beenproposed [53] on the basis of recent nuclear experimentalresults though tensions still exist among data fromdifferent sources While there exist additional nuclearprocesses beyond those listed in Table I that in principlecontribute to the collision terms in Eq (313) theseprocesses do not have a significant impact on the Ya ofany relevant nucleus when the injected energy densityis small and can therefore be neglected In should benoted that the population of excited T and 3He nucleiwhich participate in the secondary production of 6Liare generated primarily by the same processes whichcontribute to the destruction of 4He We note thatwe have not included processes which contribute to thedestruction of 6Li The reason is that the collision termsfor these processes are proportional to Y6Li itself andthus only become important in the regime in which therate of electromagnetic injection from unstable-particledecays is large By contrast for reasons that shall bediscussed in greater detail below we focus in what followsprimarily on the regime in which injection is small and6Li-destruction processes are unimportant However wenote that these processes can have an important effecton Y6Li in the opposite regime rendering the bound inEq (321) essentially unconstraining for sufficiently largeinjection rates [3]

Finally we note that the rates and energy thresholdsfor 4He + 3He rarr 6Li + p and 4He + T rarr 6Li + n arevery similar as are the rates and energy thresholds forthe 4He-destruction processes which produce the non-thermal populations of 3He and T [3] In what followswe shall make the simplifying approximation that 3Heand T are ldquointerchangeablerdquo in the sense that we treatthese rates mdash and hence also the non-thermal spectradna(E t)dE of 3He and T mdash as identical Thusalthough T decays via beta decay to 3He on a timescaleτT asymp 108 s we neglect the effect of the decay kinematicson the resulting non-thermal 3He spectrum As we shallsee these simplifying approximations do not significantlyimpact our results

D Towards an Analytic ApproximationLinearization and Decoupling

In order to assess whether a particular injection his-tory is consistent with the constraints discussed in theprevious section we must evaluate the overall changeδYa(t) equiv Ya(t)minusY init

a in the comoving number density Yaof a given nucleus at time t = tnow where Y init

a denotesthe initial value of Ya at the end of BBN In principlethis involves solving a system of coupled differentialequations one for each nuclear species present in thethermal bath each of the form given in Eq (313)

In practice however we can obtain reasonably reliableestimates for the δYa without having to resort to afull numerical analysis This is possible ultimatelybecause observational constraints require |δYa| to bequite small for all relevant nuclei as we saw in Sect III CThe equations governing the evolution of the Ya arecoupled due to feedback effects in which a change inthe comoving number density of one nuclear species Naalters the reaction rates associated with the productionof other nuclear species However if the change in Ya issufficiently small for all relevant species these feedbackeffects can be neglected and the evolution equationseffectively decouple

In order for the evolution equations for a particularnuclear species Na to decouple the linearity criterion|δYb| Yb must be satisfied for any other nuclear speciesNb 6= Na which serves as a source for reactions thatsignificantly affect the abundance of Na at all times afterthe conclusion of the BBN epoch In principle there aretwo ways in which this criterion could be enforced bythe observational constraints and consistency conditionsdiscussed in Sect III C The first is simply that theapplicable bound on each Yb which serves as a sourcefor Na is sufficiently stringent that this bound is alwaysviolated before the linearity criterion |δYb| Yb failsThe second possibility is that while the direct bound onYb may not in and of itself require that |δYb| be smallthe comoving number densities Ya and Yb are neverthelessdirectly related in such a way that the applicable boundon Ya is always violated before the linearity criterion

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

10

|δYb| Yb fails If one of these two conditions is satisfiedfor every species Nb which serves as a source for Na wemay treat the evolution equation for Na as effectivelydecoupled from the equations which govern the evolutionof all other nuclear species We emphasize that Na itselfneed not satisfy the linearity criterion in order for itsevolution equation to decouple in this way

We now turn to examine whether and under whatcircumstances our criterion for the decoupling of theevolution equations is satisfied in practice for all relevantnuclear species In Fig 2 we illustrate the network ofreactions which can have a significant effect on the valuesof Ya for these species The nuclei which appear in theinitial state of one of the primary production processeslisted in Table I are 4He which serves as a source forD and 6Li and 7Li which serves as a source for 6LiWe note that while 3He and T each appear in the initialstate of one of the secondary production processes for 6Liit is the non-thermal population of each nucleus whichplays a significant role in these reactions Since the non-thermal populations of both 3He and T are generatedprimarily as a byproduct of 4He destruction requiringthat the linearity criterion be satisfied for 4He and 7Liis sufficient to ensure that the evolution equations for allrelevant nuclear species decouple

We begin by assessing whether the direct constraintson Y4He and Y7Li themselves are sufficient to enforce thelinearity criterion We take the initial values of thesecomoving number densities at the end of the BBN epochto be those which correspond to the central observationalvalues for Yp and 7LiH quoted in Ref [54] namely

Yp = 02449 and 7LiH = 16times 10minus10 Since neither 4He

nor 7Li is produced at a significant rate by interactionsinvolving other nuclear species the evolution equation inEq (313) for each of these nuclei takes the form

dYbdt

= minus Yb(t)Γb(t) (322)

where the quantity Γb(t) equiv minusC(p)b YbnB ge 0 represents

the rate at which Yb is depleted as a result of photo-disintegration processes This depletion rate varies intime but depends neither on Yb nor on the comovingnumber density of any other nucleus Since Yb decreasesmonotonically in this case it follows that if this comovingquantity lies within the observationally-allowed rangetoday it must also lie within this range at all times sincethe end of the BBN epoch

The bound on Y4He which follows from Eq (318)is sufficiently stringent that |δY4He| Y4He is indeedrequired at all times since the end of BBN for consistencywith observation Thus our linearity criterion is alwayssatisfied for 4He By contrast the bound on Y7Li inEq (319) is far weaker in the sense that |δY7Li| need notnecessarily be small in relation to Y7Li itself Moreoverwhile the contribution to δY6Li from primary production

is indeed directly related to δY7Li we find that theobservational bound on Y6Li is not always violated beforethe linearity criterion |δY7Li| Y7Li fails mdash even ifwe assume Y6Li asymp 0 at the end of BBN The reasonis that not every 7Li nucleus destroyed by primaryphotodisintegration processes produces a 6Li nucleusIndeed 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n+ pcontribute to the depletion of Y7Li as well

Since the energy threshold for 7Li + γ rarr 4He + T islower than the threshold for 7Li + γ rarr 6Li + n therewill be a range of tinj within which injection contributes

to the destruction of 7Li without producing 6Li at allFurthermore even at later injection times tinj amp 49 times105 s when the primary-photon spectrum from injectionincludes photons with energies above the threshold for7Li + γ rarr 6Li + n the 7Li-photodisintegration ratesassociated with this process and the rates associatedwith 7Li + γ rarr 4He + T and 7Li + γ rarr 4He + 2n + pare comparable Consequently only around 30 of7Li nuclei destroyed by primary photodisintegration fortinj amp 49 times 105 s produce a 6Li nucleus in the processThus a large |δY7Li| invariably results in a much smallercontribution to δY6Li Thus 7Li does not satisfy thelinearity criterion when serving as a source for 6Li

That said while our linearity criterion is not trulysatisfied for 7Li we can nevertheless derive meaningfulconstraints on decaying particle ensembles from observa-tional bounds on 6Li by neglecting feedback effects onY7Li in calculating Y6Li Since the Boltzmann equationfor Y7Li takes the form given in Eq (322) Y7Li is alwaysless than or equal to its initial value Y init

7Liat the end

of BBN This in turn implies that the collision term

C(p)6Li

in the Boltzmann equation for 6Li is always lessthan or equal to the value that it would have had ifthe linearity criterion for 7Li had been satisfied Ittherefore follows that the contribution to δY6Li fromprimary production which we would obtain if we were toapproximate Y7Li by Y init

7Liat all times subsequent to the

end of BBN is always an overestimate In this sense thenthe bound on electromagnetic injection which we wouldobtain by invoking this linear approximation for Y7Li

represents a conservative bound Moreover it turns outthat because of the relationship between Y7Li and Y6Lithe bound on decaying ensembles from the destructionof 7Li is always more stringent than the bound fromthe primary production of 6Li Thus adopting the

linear approximation for 7Li in calculating C(p)6Li

does notartificially exclude any region of parameter space forsuch ensembles once the combined constraints from allrelevant nuclear species are taken into account

Motivated by these considerations in what follows weshall therefore adopt the linear approximation in which

Yb asymp Y initb in calculating the collision terms C(p)

a and C(s)a

for any nuclear species Na 6= Nb for which Nb servesas a source As we have seen this approximation isvalid for all species except for 7Li which serves as asource for primary 6Li production Moreover adopting

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

11

4He

D

T

3He

1H

6Li

7Li

com

parat

ively

small

com

para

tivel

y sm

all

comparatively small

FIG 2 Schematic illustrating the network of reactions precipitated by electromagnetic injection after the end of the BBNepoch The nodes in the network represent different nuclear species nodes represented by solid circles correspond to nucleiwhose primordial comoving number densities Ya are reliably constrained by data while nodes represented by dashed circlescorrespond to other nuclear species involved in these reactions An arrow pointing from one node to another indicates thatthe nucleus associated with the node from which the arrow originates serves as a source for the nucleus associated with thenode to which the arrow points A solid arrow indicates that the corresponding reaction can potentially have a non-negligibleimpact on the abundance of the product nucleus while a dashed arrow indicates that the effect of the corresponding reactionis always negligible Note that 1H nuclei mdash ie protons mdash are generated as a byproduct of many of the other reactions shownHowever the impact of these contributions to the 1H abundance is negligible and the corresponding arrows have been omittedfor clarity A checked box superimposed on the arrows emerging from a node indicates that observational constraints enforcethe linearity criterion |δYa| Ya for the corresponding nucleus By contrast an open box indicates that this criterion isnot satisfied The Boltzmann equation for a given nucleus Na effectively decouples in the sense that feedback effects can beneglected in calculating Ya whenever the linearity criterion is satisfied for all Nb 6= Na which serve as a source for Na (thoughnot necessarily for Na itself) Since no other nucleus serves as a source for 4He or 7Li the Boltzmann equations for bothof these species trivially decouple The Boltzmann equation for D also decouples because 4He the one nucleus which servesas a significant source for D is required to satisfy the linearity criterion However the Boltzmann equation for 6Li does notdecouple as one of its source nuclei mdash in particular 7Li mdash does not satisfy the linearity criterion Further details are describedin the text

this approximation for 7Li in calculating C(p)6Li

yields aconservative bound on electromagnetic injection fromdecaying particle ensembles

As discussed above the advantage of working withinthe linear approximation is that the Boltzmann equationsfor all relevant Na effectively decouple and may be solvedindividually in order to yield analytic approximationsfor δYa In the simplest case in which the collision

terms C(p)a and C

(s)a in the Boltzmann equation for Na

include only source terms and not sinks the right side ofEq (313) is independent of Ya itself Thus within thelinear approximation this equation may be integrated

directly yielding

δYa(t) asympint t

t0

dYadtprime

dtprime (323)

where t0 represents the time at the conclusion of theBBN epoch beyond which the initial abundance Ya(t0) =Y inita generated by standard primordial nucleosynthesis

remains essentially fixed in the absence of any subsequent

injection Moreover even in cases in which C(p)a and C

(s)a

include both source and sink terms we may still evaluateδYa in this way provided that observational constraints

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

12

restrict δYa to the region |δYa| Ya and therefore allowus to ignore feedback effects and approximate Ya asymp Y init

a

as a constant on the right side of Eq (313)Since the Boltzmann equation for 6Li contains no non-

negligible sink terms and since observational constraintsrequire that |δYa| Ya for both 4He and D it followsthat δYa is well approximated by Eq (323) for thesespecies Indeed the only relevant nucleus which doesnot satisfy these criteria for direct integration is 7LiNevertheless since 7Li is destroyed by a number ofprimary photodisintegration processes but not producedin any significant amount the Boltzmann equation forthis nucleus takes the particularly simple form specifiedin Eq (322) This first-order differential equation caneasily be solved for Ya yielding an expression for thecomoving number density at any time t ge t0

Ya(t) = Y inita exp

[minusint t

t0

Γa(tprime)dtprime] (324)

When this relation is expressed in terms of δYa ratherthan Ya we find that it may be recast in the morerevealing form

Y inita ln

[1 +

δYa(t)

Y inita

]= minusY init

a

int t

t0

Γa(tprime)dtprime (325)

In situations in which the linearity criterion |δYa(t)| Y inita is satisfied for the nucleus Na itself at all timest ge t0 Taylor expansion of the left side of this equationyields

δYa(t) asymp minusY inita

int t

t0

Γa(tprime)dtprime (326)

which is also the result obtained by direct integrationof the Boltzmann equation for Na in the approximationthat Ya asymp Y init

a Comparing Eqs (325) and (326) we see that if we

were to neglect feedback and take Ya asymp Y inita when

evaluating δYa for a species for which this approximationis not particularly good the naıve result that we wouldobtain for δYa would in fact correspond to the value ofthe quantity Y init

a ln[1+δYa(t)Y inita ] Thus given that a

dictionary exists between the value of δYa obtained fromEq (325) and the value obtained from Eq (326) forsimplicity in what follows we shall derive our analyticapproximation for δYa using Eq (326) and simply notethat the appropriate substitution should be made for thecase of 7Li That said we also find that the constraint onΩχ that we would derive from Eq (326) in single-particleinjection scenarios from the observational bound on Y7Li

differs from the constraint that we would derive fromthe more accurate approximation in Eq (325) by onlyO(10) Thus results obtained by approximating δY7Li

by the expression in Eq (326) are nevertheless fairlyreliable in such scenarios mdash and indeed can be expectedto be reasonably reliable in scenarios involving decayingensembles as well

E Analytic Approximation Contribution fromPrimary Processes

Having discussed how the Boltzmann equations forthe relevant Na effectively decouple in the linear regimewe now proceed to derive a set of approximate analyticexpressions for δYa from these decoupled equations Webegin by considering the contribution to δYa that arisesfrom primary photoproduction or photodisintegrationprocesses The contribution from secondary processeswhich is relevant only for 6Li will be discussed inSect III F

Our ultimate goal is to derive an approximate analyticexpression for the total contribution to δYa due theinjection of photons from an entire ensemble of decayingstates However our first step in this derivation shallbe to consider the simpler case in which the injection isdue to the decay of a single unstable particle species χwith a lifetime τχ We shall work within the uniform-decay approximation in which the non-thermal photonspectrum takes the particularly simple form in Eq (311)In this approximation the lower limit of integration inEq (323) may be replaced by τχ while the upper limitcan be taken to be any time well after photons at energies

above the thresholds E(ac)b and E

(bc)a for all relevant

photoproduction and photodisintegration processes havethermalized Thus we may approximate the change inthe comoving number density of each relevant nuclearspecies as

δYa asympint infinτχ

dYadt

dt (327)

In evaluating each δYa we may also take advantageof the fact that the rates for the relevant reactionsdiscussed in Sect III C turn out to be such that the first(source) and second (sink) terms in Eq (314) are neversimultaneously large for any relevant nuclear speciesIndeed the closest thing to an exception occurs during avery small time interval within which the source and sinkterms for D are both non-negligible Thus dependingon the value of τχ and its relationship to the timescalesassociated with these reactions we may to a very goodapproximation treat the effect of injection from a singledecaying particle as either producing or destroying Na

With these approximations the integral over t inEq (327) may be evaluated in closed form In particularwhen the source term in Eq (314) dominates we findthat δYa is given by

δYa asympsumbc

Ybεχρχ(τχ)

nB(τχ)

timesint EC

E(ac)b

dEK(E τχ)S(ac)b (E) (328)

where we have defined

S(ac)b (E) equiv

σ(ac)b (E)

σth(E) (329)

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

13

By contrast when the sink term dominates we find thatδYa is given by

δYa asymp minussumbc

Yaεχρχ(τχ)

nB(τχ)

timesint EC

E(bc)a

dEK(E τχ)S(bc)a (E) (330)

While the expressions for δYa in Eqs (330) and (328)pertain to the case of a single unstable particle withinthe uniform-decay approximation it is straightforwardto generalize these results to more complicated scenariosIndeed within the linear approximation the total changeδYa in Ya which results from multiple instantaneousinjections over an extended time interval is well ap-proximated by the sum of the individual contributionsfrom these injections In the limit in which this set ofdiscrete injections becomes a continuous spectrum thissum becomes an integral over the injection time tinjThus in the continuum limit δYa is well approximatedby

δYa asympint infin

0

dYadtinj

dtinj (331)

where dYadtinj is the differential change in Ya due to aninfinitesimal injection of energy in the form of photonsat time tinj

The approximation in Eq (331) allows us to accountfor the full exponential time-dependence of the electro-magnetic injection due to particle decay in calculatingδYa for any given nucleus By extension Eq (331)gives us the ability to compare the results for δYa ob-tained both with and without invoking the uniform-decayapproximation thereby providing us insight into howreliably δYa can be computed with this approximation

As an example let us consider the effect of a singledecaying particle χ with lifetime τχ on the comovingnumber density of 4He Within the uniform-decayapproximation δY4He is given by Eq (330) because thesink term in Eq (314) dominates By contrast when thefull exponential nature of χ decay is taken into accountthe corresponding result is

δYa asymp minussumbc

int infin0

dtinj

[Yaεχ

nB(tinj)

dρχ(tinj)

dtinj

timesint EC

E(bc)a

dEK(E tinj)S(bd)c (E)

] (332)

where dρχ(tinj)dtinj denotes the rate of change in theenergy density of χ per unit time tinj Prior to tMREthe energy density of an unstable particle with anextrapolated abundance Ωχ may be written

ρχ(t) = ρcrit(tnow)

(tnow

tMRE

)2(tMRE

t

)32

Ωχeminustτχ

(333)

where ρcrit(tnow) is the critical density of the universeat present time and where tMRE is the time of matter-radiation equality The corresponding rate of change inthe energy density properly evaluated in the comovingframe and then transformed to the physical frame is

dρχ(tinj)

dtinj= minusρcrit(tnow)

(tnow

tMRE

)2(tMRE

tinj

)32

times Ωχτχeminustinjτχ (334)

The corresponding expressions for continuum injectionin cases in which the source term in Eq (314) domi-nates are completely analogous and can be derived in astraightforward way

In Fig 3 we compare the results obtained for δY4He

within the uniform-decay approximation to the resultsobtained with the full exponential time-dependence ofχ decay taken into account In particular within the(Ωχ τχ) plane we display contours of the corresponding4He mass fraction Yp obtained within the uniform-decay approximation (upper panel) and through the fullexponential calculation (lower panel) For concretenessin calculating these contours we have assumed an initialvalue Yp = 02449 for the 4He mass fraction at the endof BBN following Ref [54]

Comparing the two panels of Fig 3 we see thatthe results for Yp obtained within the uniform-decayapproximation are indeed very similar throughout mostof the parameter space shown to the results obtainedthrough the full calculation Nevertheless we observethat discrepancies do arise For example we notethat the constraints obtained within the uniform-decayapproximation are slightly stronger for particles withlifetimes within the range 107 s τχ 108 s and slightlyweaker for τχ above this range than the constraints ob-tained through the full calculation These discrepanciesare ultimately due to the fact that the reprocessed photonspectrum in Eq (31) depends on the temperature ofthe radiation bath Thus there is a timescale tσmax forwhich the reaction rate for a particular photoproductionor photodisintegration process is maximized for fixeddρ(tinj)dtinj Since all of the energy density ρχ initiallyassociated with the decaying particle is injected preciselyat τχ within the uniform-decay approximation thisapproximation yields slightly more stringent constraintsthan those obtained through the full calculation whenτχ sim tσmax By the same token the constraints obtainedwithin the uniform-decay approximation are slightly lessstringent than those obtained through the full calculationwhen τχ differs significantly from tσmax

We also observe that for lifetimes τχ 107 s theuniform-decay approximation likewise yields constraintsthat are weaker than those obtained through the fullexponential calculation The reason for this is thatthe upper energy cutoff EC in the reprocessed photon

spectrum is proportional to t12inj For sufficiently early

injection times EC lies below the threshold energy E(bc)a

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

14

021

022

023

024

021

022

023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

021

022023

024

021

022023

024

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

FIG 3 Contours of the primordial 4He mass fractionYp obtained in the presence of a single decaying particlespecies χ with lifetime τχ and extrapolated abundance Ωχplotted within the (Ωχ τχ) plane Note that Ωχ representsthe abundance that χ would have had at present time ifit had not decayed The upper panel shows the resultsobtained within the uniform-decay approximation while thelower panel shows the corresponding results obtained withthe full exponential time-dependence of χ decay taken intoaccount Comparing the results shown in the two panels wesee that the uniform-decay approximation indeed provides areasonable approximation for Yp

for the photodisintegration reactions which contributeto δY4He Injection at such early times therefore has

essentially no impact on Yp This implies that within theuniform-decay approximation a particle with a lifetimein this regime likewise has no effect on Yp By contrastwithin the full exponential calculation injection occurs ata significant rate well after τχ leading to a non-negligiblechange in Yp even when the lifetime of the decayingparticle is short

In summary the results shown in Fig 3 indicatethat other than in the regime where τχ is sufficiently

short that EC lies below the threshold energy for 4Hedestruction the result for δY4He obtained within theuniform-decay approximation is very similar to the resultobtained through the full exponential calculation for thesame τχ and Ωχ We find this to be the case for theother relevant nuclei as well Thus having shown thatthe results for δYa obtained within the uniform-decayapproximation accord well with those obtained throughthe full exponential calculation at least for sufficientlylong τχ we shall adopt this approximation in deriving ourconstraints on ensembles of electromagnetically-decayingparticles As we shall see the advantage of workingwithin the uniform-decay approximation is that withinthis approximation it is possible to write down a simpleanalytic expression for δYa However as we shall discussin more detail below we shall adopt an alternativestrategy for approximating δYa within the regime inwhich τχ is short and the results obtained within theuniform-decay approximation do not agree with thoseobtained through the full exponential calculation

In order to write down our analytic expressions forδYa we shall make one additional approximation we

shall treat the ratio of cross-sections S(bd)c (E) as not

varying significantly as a function of energy between the

threshold energy E(bc)a and EC When this approxima-

tion holds we may treat this ratio as a constant andpull it outside the integral over photon energies In orderto justify this approximation we begin by noting that

the cross-sections σ(bc)a (E) for primary processes typically

peak at a value slightly above E(bc)a but fall precipitously

with E beyond that point (see eg Ref [3]) In the

vicinity of the peak however the variation of σ(bc)a (E)

is relatively gentle Since the thermalization cross-section σth(E) for primary photons varies much lessrapidly across the relevant region of E the functional

dependence of S(bc)a (E) on E is principally determined

by the behavior of σ(bc)a (E) Moreover because K(E τχ)

falls rapidly with E the dominant contribution to theenergy integral in Eq (330) arises from photons justabove threshold even within the approximation that

S(bc)a (E) is constant Thus to a good approximation

we may replace S(bd)c (E) by a constant on the order of

its peak value and take this quantity outside the energyintegral

Within this approximation it is now possible toanalytically evaluate the integral in Eq (328) The formof the result depends on the relationship between the

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

15

threshold energy E(bc)a for the scattering process and the

energy scales EC(tinj) and EX(tinj) which determine theshape of the photon spectrum at time tinj = τχ There

are three cases of interest the case in which E(bc)a gt

EC(τχ) the case in which EC(τχ) gt E(bc)a gt EX(τχ)

and the case in which EX(τχ) gt E(bc)a Moreover the

relations in Eq (35) imply that each of these casescorresponds to a specific range of τχ In particular the

respective lifetime regimes are τχ lt t(bc)Ca t

(bc)Ca lt τχ lt

t(bc)Xa and t

(bc)Xa lt τχ where we have defined

t(bc)Ca asymp (108 s)times

[E

(bc)a

103 MeV

]2

t(bc)Xa asymp (108 s)times

[E

(bc)a

28 MeV

]2

(335)

Within the τχ lt t(bc)Ca regime none of the photons

produced by χ decay exceed the threshold for the photo-disintegration process The contribution to δYa withinthe uniform-decay approximation is therefore formallyzero As discussed above this is the regime in whichthe uniform-decay approximation fails to reproduce theresults obtained through the full exponential calcula-tion Thus in order to derive a meaningful boundon decaying particles with lifetimes in this regime weinstead model the injection of photons using the fullcontinuum expression in Eq (332) with dρ(tinj)dtinj

given by Eq (334) However in order to arrive at asimple analytic expression for δYa we include only the

contribution from injection times in the range t(bc)Ca lt

tinj lt t(bc)Ca + τχ and drop terms in τχt

(bc)Ca beyond

leading order in the resulting expression With theseapproximations in this regime we find

δYa =sumbc

A(bc)a Ωχ

(τχ

t(bc)Ca

)eminust

(bc)Ca τχ (336)

where the proportionality constant A(bc)a for each con-

tributing reaction is independent of the properties of thedecaying particle This treatment ensures that we obtaina more reliable estimate for the contribution to δYa fromparticles with τχ in this regime

Within the remaining two lifetime regimes the contri-bution to δYa within the uniform-decay approximationis non-vanishing Thus within these regimes we obtainour approximation for δYa by integrating Eq (328) as

discussed above For t(bc)Ca lt τχ lt t

(bc)Xa we find

δYa =sumbc

B(bc)a Ωχ

1minus β

(τχ

t(bc)Xa

)minus12 (337)

where B(bc)a is a proportionality constant and where β equiv

EXEC asymp 027 is the τχ-independent ratio of the energyscales in Eq (35) Likewise for t

(bc)Xa lt τχ we find

δYa =sumbc

B(bc)a Ωχ

(τχ

t(bc)Xa

)minus14

times

2minus (1 + β)

(τχ

t(bc)Xa

)minus14 (338)

We emphasize that the proportionality constant B(bc)a

for a given process in Eq (338) is the same as thecorresponding proportionality constant in Eq (337)

However in general B(bc)a differs from the corresponding

proportionality constant A(bc)a appearing in Eq (336)

Strictly speaking Eq (338) does not hold for arbi-trarily large τχ since photons produced by extremely latedecays are not efficiently reprocessed by Class-I processesinto the spectrum in Eq (31) In order to accountfor this in what follows we shall consider each termin the sum in Eq (338) to be valid only for injection

times tinj lt t(bc)fa where t

(bc)fa is a characteristic cutoff

timescale associated with the reaction Photons injectedafter this cutoff timescale are assumed to have no effecton Ya For most reactions it is appropriate to take

t(bc)fa asymp 1012 s as this is the timescale beyond which

certain crucial Class-I processes effectively begin to shutoff and the reprocessed photon spectrum is no longerreliably described by Eq (31)

F Analytic Approximation Contribution fromSecondary Processes

The approximate analytic expressions for δYa whichwe have derived in Sect III E are applicable to all ofthe primary photoproduction or photodisintegration pro-cesses relevant for constraining electromagnetic injectionfrom unstable-particle decays after BBN However sincesecondary production can contribute non-negligibly tothe production of 6Li we must derive analogous expres-sions for δYa in the case of secondary production as wellMoreover since secondary production is fundamentallydifferent from primary production in terms of particlekinematics there is no reason to expect that theseexpressions should have the same functional dependenceon τχ as that exhibited by the expressions in Eqs (336)ndash(338) Indeed as we shall see they do not

We begin this undertaking by observing that withinthe uniform-decay approximation the contribution toδYa from secondary production is given by

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

16

δYa asympsumbcdf

int infinτχ

dt

Yf εχρχ(τχ)

int E(EC)

E(aX)bf

dEb

[nc(t)σ

(aX)bf (E)|v(Eb)|b(Eb t)

int EC

Emin(Eb)

dEγK(Eγ τχ)σ(bd)c (Eγ)e

minus (tminusτχ)

δtth(Eγτχ)

]

(339)

We may simplify this expression by noting that the time-dependence of the energy-loss rate b(Eb t) of excitednuclei due to Coulomb scattering is primarily due tothe dilution of the number density of electrons ne(t)Since ne(t) scales with t in the same manner as nc(t)the quantity

Υ(aX)bcf (Eb) equiv

nc(t)σ(aX)bf (E)|v(Eb)|b(Eb t)

(340)

is to a good approximation a comoving quantity and

hence independent of t Thus Υ(aX)bcf (E) may be pulled

outside the time integral in Eq (339) Within thisapproximation the contribution to δYa reduces to

δYa asympsumbcdf

Yf εχρχ(τχ)

nB(τχ)

int E(EC)

E(aX)bc

dEb

[Υ

(aX)bcf (Eb)

timesint EC

Emin(Eb)

dEγK(Eγ τχ)S(bd)c (Eγ)

] (341)

In order to proceed further we must first assess the

dependence of the quantities S(ab)c (Eγ) and Υ

(aX)bcf (Eb) on

the respective energy scales Eγ and Eb However we find

that Υ(aX)bcf (Eb) varies reasonably slowly over the relevant

range of Eb [3 35] Thus to a good approximation thisquantity may also be pulled outside the integral over Eb

By contrast we find that S(ab)c (Eγ) cannot reliably

be approximated as a constant over the range of Eγrelevant for secondary production This deserves furthercomment mdash especially because we have approximated

S(ab)c (Eγ) as a constant in deriving the expressions in

Eqs (336)ndash(338) for primary production As we shallnow make clear there are important differences betweenthe kinematics of primary and secondary production

which enable us to approximate S(ab)c (Eγ) as independent

of Eγ in the former case but not in the latterFor primary production as discussed in Sect III E

the rapid decrease of K(Eγ τχ) with Eγ suppresses the

contribution to δYa from photons with Eγ E(ac)b

The dominant contribution to δYa therefore comes froma narrow region of the spectrum just above threshold

within which S(ac)b (Eγ) varies reasonably slowly while

photons with energies well above E(ac)b have little col-

lective impact on δYa Thus in approximating theoverall contribution to δYa from primary production it

is reasonable to treat S(ac)b (Eγ) as a constant

By contrast for secondary production mdash or at leastfor the secondary production of 6Li the one nuclear

species in our analysis for which secondary productioncan have a significant impact on δYa mdash photons with Eγ

well above the threshold energy E(bd)c for any relevant

primary process play a more important role One reason

for this is that the energy threshold E(aX)bc for each

secondary processes which contributes meaningfully to6Li production corresponds to a primary-photon energy

Eminus1[E(aX)bf ] well above the associated primary-process

threshold E(bd)c Indeed the kinetic-energy thresholds

for 4He + 3Herarr 6Li + p and 4He + Trarr 6Li +n given inTable I are quite similar and both correspond to photon

energies of roughly Eminus1[E(aX)bf ] sim O(50 MeV) mdash energies

well above the peak in S(bd)c (Eγ) Thus it is really the

energy threshold for the secondary process which setsthe minimum value of Eγ relevant for the secondary

production of 6Li Since S(bd)c (Eγ) varies more rapidly

with Eγ at these energies than it does around its peakvalue it follows that this variation cannot be neglectedin determining the overall dependence of δYa on τχ inthis case

There is however another reason why the variation

of S(bd)c (Eγ) with Eγ cannot be neglected in the case

of secondary production mdash a deeper reason which isrooted more in fundamental differences between primaryand secondary production than in the values of theparticular energy thresholds associated with processespertaining to 6Li The overall contribution to δYa fromprimary production in Eq (328) involves a single integralover Eγ Thus for primary production the fall-off inK(Eγ τχ) itself with Eγ is sufficient to suppress thepartial contribution to δYa from photons with Eγ E

(bd)c By contrast the overall contribution to δYa from

secondary production involves integration not only overEγ but also over Eb In this case the fall-off in K(Eγ τχ)with Eγ is not sufficient to suppress the partial con-tribution to δYa from photons with energies well abovethreshold Thus for any secondary production processan accurate estimate for δYa can only be obtained when

the variation of S(bd)c (Eγ) with Eγ mdash a variation which

can be quite significant at such energies mdash is taken intoaccount

We must therefore explicitly incorporate the functional

dependence of S(bd)c (Eγ) on Eγ into our calculation of

δYa We recall that S(bd)c (Eγ) as we have defined it

in Eq (329) represents the ratio of the cross-section

σ(bd)c (Eγ) for the primary process Nc + γ rarr Nb + Nd

which produces the population of excited nuclei to thecross-section σth(Eγ) for the Class-III processes which

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

17

serve to thermalize the primary-photon spectrum Thecross-sections for the two primary processes 4He + γ rarr3He + n and 4He + γ rarr T + p relevant for secondary6Li production expressed as a functions of the photonenergy Eγ both take the form [3]

σ(bd)c (Eγ) asymp σ

(bd)c0

[Eγ

E(bd)c

minus 1

] [Eγ

E(bd)c

]minus92

(342)

where σ(bd)c0 asymp 206 mb and σ

(bd)c0 asymp 198 mb for these two

processes respectively By contrast σth(Eγ) includestwo individual contributions The first contribution isdue to e+eminus pair-production off nuclei via Bethe-Heitlerprocesses of the form γ + Na rarr X + e+ + eminus whereNa denotes a background nucleus and X denotes somehadronic final state The cross-section for this process is

σBH(Eγ) =3α

8πσT

[28

9ln

(2Eγme

)minus 218

27

] (343)

where α is the fine-structure constant where me is theelectron mass and where σT asymp 661 mb is the Thomsoncross-section The second contribution to σth(Eγ) is dueto Compton scattering for which the cross-section is

σCS(Eγ) =3me

8EγσT

[1minus 2me

Eγminus 4m2

e

E2γ

ln

(1 +

2Eγme

)]

+4me

Eγ+

1

2

[1minus

(1 +

2Eγme

)minus2]

(344)

Given the dependence of these cross-sections on Eγ

we find that over the photon-energy range E(bd)c lt Eγ

1 GeV the ratio S(bd)c (Eγ) for each relevant process is

well approximated by a simple function of the form

S(bd)c (Eγ) =

σ(bd)c (Eγ)

σBH(Eγ) + σCS(Eγ)

asymp S(bd)c0

(Eγ

E(bd)c

minus 1

)(Eγ

E(bd)c

)minus4

(345)

where S(bd)c0 is a constant In Fig 4 we show a comparison

between the exact value of S(bd)c (Eγ) for the individual

process 4He + γ rarr T + p and our approximation inEq (345) over this same range of Eγ We see from thisfigure that our approximation indeed provides a good fit

to S(bd)c (Eγ) over most of this range Moreover while

the discrepancy between these two functions becomesmore pronounced as Eγ sim O(1 GeV) we emphasize that

S(bd)c (Eγ) is comparatively negligible at these energies

Thus although the primary-photon spectrum can includephotons with energies O(1 GeV) mdash indeed for τχ asymp1012 s the timescale beyond which Eq (33) ceases toprovide a reliable description of this spectrum we findthat EC sim O(10 GeV) mdash these photons have littleimpact on δYa Thus for the purposes of approximating

50 100 500 100010-6

10-5

10-4

0001

0010

0100

1

Eγ [MeV]

Sc(bd) (Eγ)

Approximation

Full Function

ℰ-1[Ebf(aX)]

FIG 4 The exact value of the function S(bd)c (Eγ) for the

process 4He + γ rarr T + p (red curve) and our approximationto this function in Eq (345) (black dashed curve) plotted

over the range E(bd)c lt Eγ ltsim 1 GeV The gray vertical line

corresponds to the minimum photon energy Eminus1[E(aX)bf ] for

the production of an excited T nucleus above the kinetic-energy threshold for the secondary process 4He+Trarr 6Li+nThus photons in the gray shaded region play no role in thesecondary production of 6Li We note that the corresponding

S(bd)c (Eγ) curves and thresholds for 4He + γ rarr 3He + n (the

other primary process relevant for secondary 6Li production)are nearly identical to those shown here

δYa any discrepancy between Eq (345) and the exact

expression for S(bd)c at such energies may safely be

ignored

Armed with our approximation for S(bd)c (Eγ) in

Eq (345) it is now straightforward to evaluate theintegral in Eq (341) and obtain an approximate analyticexpression for δYa Just as it does for primary photopro-duction and photodisintegration the functional depen-dence of δYa on τχ for secondary production depends onthe relationship between τχ and a pair of characteristictimescales determined by the energy thresholds for therelevant processes As we have discussed above it is

Eminus1[E(aX)bf ] rather than E

(bd)c which sets the minimum

Eγ required of a photon in order for it to contribute to

the secondary production of 6Li It therefore follows thatthe characteristic timescales for the secondary production

of this nucleus are determined by Eminus1[E(aX)bf ] rather than

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

18

E(bd)c Thus for secondary production we define

t(bcf)Ca asymp (108 s)times

(Eminus1[E

(aX)bf ]

103 MeV

)2

t(bcf)Xa asymp (108 s)times

(Eminus1[E

(aX)bf ]

28 MeV

)2

(346)

For τχ lt t(bcf)Ca within the uniform-decay approxima-

tion δYa is formally zero as it is in the correspondinglifetime regime for primary production Thus in order toderive a meaningful constraint on unstable particles withlifetimes within this regime we follow the same procedureas we employed in order to calculate the contributionto δYa from primary production for a particle with

τχ lt t(bc)Ca We model the injection of photons using the

expression for secondary production appropriate for con-tinuum injection with dρ(tinj)dtinj given by Eq (334)In analogy to our treatment of the primary process in thisregime we include only the contribution from injection

times in the range t(bcf)Ca lt tinj lt t

(bcf)Ca + τχ Dropping

terms beyond leading order in the dimensionless variable

wCχ equiv τχt(bcf)Ca we find

δYa =sumbcf

A(bcf)a ΩχwCχe

minuswminus1Cχ (347)

where the proportionality constant A(bcf)a for each combi-

nation of primary and secondary processes is independentof τχ We note that the dependence of δYa on τχ inthis expression is exactly the same as in Eq (336) By

contrast for t(bcf)Ca lt τχ lt t

(bcf)Xa we find

δYa =sumbcf

B(bcf)a Ωχ

[(5minus 3θ)minus 20β3w

minus32Xχ

+15β4(1 + θ)wminus2Xχminus 12β5θw

minus52Xχ

] (348)

where we have defined wXχ equiv τχt(bcf)Xa in analogy with

wCχ and where

θ equiv E(bd)c

Eminus1[E

(aX)bf

] (349)

represents the ratio of the photon-energy threshold forprimary production to the minimum photon energyneeded to produce an excited nucleus of species Nbwith a kinetic energy above the energy threshold for thesecondary process Numerically we find θ asymp 046 for4He + γ rarr T + p followed by 4He + T rarr 6Li + nwhile θ asymp 052 for 4He + γ rarr 3He + n followed by

4He + 3Herarr 6Li + p Finally for t(bcf)Xa lt τχ we find

δYa =sumbcf

B(bcf)a Ωχ

[16

21(9minus 5θ)w

minus14Xχ

minus4(1 + 5β3)wminus32Xχ +

15

7(1 + θ)(1 + 7β4)wminus2

Xχ

minus4

3θ(1 + 9β5)w

minus52Xχ

] (350)

Thus to summarize we have derived a set of sim-ple analytic approximations for the change δYa in theabundance of a given nuclear species Na due to the lateinjection of photons by a decaying particle χ For pri-mary photoproduction or photodisintegration processeswe find that δYa is given by Eq (336) Eq (337) orEq (338) depending on the lifetime τχ of the particleLikewise for secondary production we find that δYa isgiven by Eq (347) Eq (348) or Eq (350) within thecorresponding lifetime regimes

G The Fruits of Linearization Light-ElementConstraints on Ensembles of Unstable Particles

We now turn to the task of extending these resultsto the case of an ensemble of decaying particles χi withlifetimes τi and extrapolated abundances Ωi Indeed wehave seen that if the linearity criterion is satisfied bothfor Na itself and for all of its source nuclei Nb 6= Naall feedback effects on Ya can be neglected Thus inthis regime the overall change δYa in the abundance ofa light nucleus is well approximated by the sum of theindividual contributions associated with the individual χifrom each pertinent process Indeed it is only because wehave entered a linear regime that such a direct sum is nowappropriate These then are the fruits of linearization

While certain nuclei in our analysis mdash namely 6Liand 7Li mdash do not have the property that the linear-ity criterion is always satisfied for both the nucleusitself and its source nuclei we emphasize that we arenevertheless able to derive meaningful bounds on thecomoving number densities of these nuclei As noted inSect III D artificially adopting the linearity criterion forY7Li in the Boltzmann equation for 6Li always yields aconservative bound on δY6Li regardless of the injectionhistory For 7Li the issue is that feedback effects on thephotodisintegration rate due to changes in the comovingnumber density of 7Li itself are not necessarily smallThus strictly speaking δY7Li is not well approximatedby a direct sum of the contributions from the individualχi However since this direct sum is simply an approxi-mation of the integral in Eq (325) it is equivalent to thequantity Y init

7Liln[1+δY7Li(t)Y

init7Li

] Thus for all relevantNa we find that an ensemble of decaying particles makesan overall contribution to δYa mdash or in the case of7Li to the quantity Y init

a ln[1 + δYa(t)Y inita ] mdash which

can be approximated as a direct sum of the individual

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

19

contributions from the individual ensemble constituentsIndeed this yields either a reliable estimate for the truevalue of δYa or a reliably conservative bound on δYa

In principle each of these individual contributions toa given δYa can involve a large number of reactions withdifferent energy thresholds and scattering kinematics

each with its own distinct fit parameters A(bc)a B

(bc)a

t(bc)Xa t

(bc)Ca etc In practice however the number of

reactions which contribute significantly to δYa for anyone of the four relevant nuclear species is quite small ascan be seen from Table I Moreover it is often the casethat many if not all of the reactions which have a non-negligible impact on a given δYa have very similar energythresholds and scattering kinematics When this is thecase the contribution to δYa from these processes canto a very good approximation collectively be modeledusing a single set of fit parameters For example theonly reactions which have a significant impact on δY4He

are the primary photodisintegration processes 4He+γ rarrT + p and 4He + γ rarr 3He + n which have very similarenergy thresholds and scattering kinematics Thus afit involving a single set of parameters yields an accurateapproximation for δY4He The reactions which contributeto δY7Li differ more significantly in terms of their energythresholds and scattering kinematics Nevertheless as weshall see the collective effect of these processes is also wellmodeled by a single set of fit parameters By contrastfor D there are two processes with qualitatively differentenergy thresholds and scattering kinematics which mustbe modeled using separate sets of fit parameters primaryphotodisintegration via D + γ rarr n + p and primaryphotoproduction via 4He + γ rarr D + n + p Likewisefor 6Li two sets of fit parameters are required one forprimary photoproduction via 7Li + γ rarr 6Li + n and onefor secondary production via both 4He + 3He rarr 6Li + pand 4He + Trarr 6Li + n

In general then we require at most two distinct setsof parameters in order to model the overall contributionto δYa for each relevant nucleus due to electromagneticinjection from an ensemble of decaying particles Thusin general we may express this overall contribution asthe sum of two terms

δYa = δY (1)a + δY (2)

a (351)

For each relevant nuclear species one of these termsis associated with a primary process primary photo-production in the case of 6Li and primary photodisin-

tegration in the case of 4He 7Li and D Thus δY(1)a

takes a universal form for all relevant nuclear species Inparticular Eqs (347)ndash(350) suggest that we may modelδY

(1)a with a function of the form

δY (1)a =

sumtAaltτilttBa

AaΩi

(τitCa

)eminustCaτi

+sum

tBaltτilttXa

BaΩi

[1minus β

(tXaτi

)12]

+sum

tXaltτilttfa

BaΩi

(tXaτi

)14[

2minus (1 + β)

(tXaτi

)14]

(352)

where Aa Ba tCa tXa etc are model parameters whoseassignments we shall discuss below By contrast theform of the second term in Eq (351) differs depending onthe nucleus in question For 4He and 7Li we simply have

δY(2)a = 0 For D this term is associated with primary

production and thus takes exactly the same functional

form as δY(1)a In other words we have

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[1minus β

(tlowastXaτi

)12]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

(tlowastXaτi

)14[

2minus (1 + β)

(tlowastXaτi

)14]

(353)

where Alowasta Blowasta tlowastCa tlowastXa etc represent an additional setof model parameters distinct from the parameters Aa

Ba tCa tXa etc in Eq (352) For 6Li the term δY(2)a is

associated with secondary production and therefore takesthe form

δY (2)a =

sumtlowastAaltτiltt

lowastBa

AlowastaΩi

(τitlowastCa

)eminust

lowastCaτi

+sum

tlowastBaltτilttlowastXa

BlowastaΩi

[(5minus 3θ)minus 20β3

(tlowastXaτi

)32

+ 15β4(1 + θ)

(tlowastXaτi

)2

minus 12β5θ

(tlowastXaτi

)52]

+sum

tlowastXaltτilttlowastfa

BlowastaΩi

[16

21(9minus 5θ)

(tlowastXaτi

)14

minus 4(1 + 5β3)

(tlowastXaτi

)32

+15

7(1 + θ)(1 + 7β4)

(tlowastXaτi

)2

minus 4

3θ(1 + 9β5)

(tlowastXaτi

)52]

(354)

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

20

Nucleus Process tBa(s) tCa(s) tXa(s) Aa Ba δY mina δY max

a

4He Destruction 89times 106 32times 106 42times 107 minus33times 10minus1 minus15times 100 minus30times 10minus3 mdash7Li Destruction 45times 105 57times 104 16times 106 minus27times 10minus10 minus38times 10minus9 minus55times 10minus11 mdash

DDestruction 22times 105 46times 104 11times 106 minus12times 10minus4 minus12times 10minus3

minus89times 10minus7 16times 10minus6

Production 19times 107 80times 106 11times 108 18times 10minus2 82times 10minus2

6LiPrimary Production 18times 106 60times 105 10times 107 16times 10minus10 98times 10minus10

mdash 18times 10minus11

Secondary Production 89times 107 19times 107 46times 108 80times 10minus8 18times 10minus7

TABLE II Values for the fit parameters appearing in Eqs (352)ndash(354) which characterize the contribution to the change δYain the comoving number density of nucleus Na due to the electromagnetic decays of unstable particles after BBN An asteriskappearing after the name of the process indicates that the values listed in the table correspond to the parameters Alowasta Blowasta tlowastCa

tlowastXa etc associated with δY(2)a for the nucleus in question rather than the parameters Aa Ba tCa tXa etc associated with

δY(1)a The observational upper and lower limits δY min

a and δY maxa on the value of δYa are also shown

We note that for simplicity and compactness of notationwe have implicitly taken εi = 1 for each χi in formulatingthe expressions appearing in Eqs (352)ndash(354) How-ever it is straightforward to generalize these results tothe case in which εi differs from unity for one or more ofthe χi In particular the corresponding expressions for

δY(1)a and δY

(2)a in this case may be obtained by replacing

Ωi for each species with the product Ωiεi Moreoverwe remind the reader that for the case of 7Li a moreaccurate estimate of δYa can be obtained by replacing

δY(1)a with Y init

a ln[1 + δY(1)a (t)Y init

a ] in Eq (352)

We now determine the values of the parameters inEq (352) We begin by noting that the parameter tAafor each relevant nucleus Na may be assigned essentiallyany value between the end of the BBN epoch and thetimescale at which the first reaction which contributes toδYa becomes efficient Thus for simplicity we choose auniversal value tAa = 104 s for all Na which correspondsto the timescale at which the first reaction in Table I mdashnamely the photodisintegration process D + γ rarr n + pmdash effectively turns on For tfa we likewise choose auniversal value tfa = 1012 s for all Na As discussedin Sect VI the reason for this choice is that manyof the Class-I processes which serve to establish thereprocessed photon spectrum in Eq (31) start to becomeinefficient around this timescale As a result the photonspectrum that results from electromagnetic injection attimes tinj amp 1012 s differs from that in Eq (31) and

the analytic approximation for δY(1)a in Eq (352) is no

longer reliable The values of tlowastAa and tlowastfa appearing in

Eqs (353) and (354) are determined in the same way

Given these results we now determine the valuesof the remaining parameters in Eqs (352)ndash(354) bydemanding that the constraint contour we obtain fromeach of these equations be consistent with the contourobtained from a more complete numerical analysis ofthe Boltzmann system in the limiting case in whichour decaying ldquoensemblerdquo comprises only a single particlespecies χ In this single-particle case our analyticexpression for δYa becomes a function of only twovariables the abundance Ωχ and lifetime τχ of χ Wedetermine the constraint contours corresponding to the

observational limits quoted in Sect III C by surveying(Ωχτχ) space and numerically solving the full coupledsystem of Boltzmann equations for δYa at each pointThe undetermined parameters in Eqs (352)ndash(354) arethen chosen such that our analytic expression provides agood fit to the corresponding constraint contour for eachnucleus We list the values for all parameters obtainedin this manner in Table II

In deriving these constraint contours it is necessaryto translate the observational constraints in Eqs (318)ndash(321) on the relevant Ya at the end of the BBN epochinto bounds of the form

δY mina lt δYa lt δY max

a (355)

on the corresponding subsequent change δYa in each YaIn so doing we must specify a set of initial values ofYa at the end of BBN For Y4He and Y7Li we takeYp = 02449 and 7LiH = 16 times 10minus10 as discussed inSect III D For YD we take the value which correspondsto the central theoretical prediction DH = 2413times 10minus5

for the primordial DH ratio derived in Ref [48] Finallysince 6Li is not produced in any significant amount duringBBN we take Y6Li = 0 The values for the observationalupper and lower limits δY min

a and δY maxa on δYa which

correspond to these choices for the initial Ya are listed inTable II alongside the values for our fit parameters

In Fig 5 we display contours showing the results of ournumerical calculation of δYa in (Ωχ τχ) space for eachrelevant nucleus More specifically we display contoursof Yp for 4He whereas we display contours of log10 Ya for

D 7Li and 6Li Moreover for purposes of illustrationwe display separate contours for D production anddestruction mdash ie contours evaluated considering either

production or destruction processes in C(p)D only with the

cross-sections for the opposite set of processes artificiallyset to zero Likewise we also display separate contoursfor primary and secondary 6Li production We empha-size however that in assessing our overall constraints ondecaying ensembles we consider these processes togetherallowing for the possibility of a cancellation between

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

21

-6

-5

-472

-467

-6

-5

-472

-467

4 5 6 7 8 9

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

D Destruction

-35

-41

-45

-462

-35

-41

-45

-462

6 7 8 9 10 11

-50

-40

-30

-20

log10(τχs)

log 10(Ω

χ)

D Production

02

022 023

0237

02

022 023

0237

6 7 8 9 10 11

-30

-25

-20

-15

-10

-05

log10(τχs)

log 10(Ω

χ)

4He Destruction

-13-11

-103

-1003

-13-11

-103

-1003

4 5 6 7 8 9 10 11

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

7Li Destruction

-1045

-1073

-116

-145

-1045

-1073

-116

-145

4 5 6 7 8 9

-20

-15

-10

-05

00

05

log10(τχs)

log 10(Ω

χ)

Primary 6LiProduction

-95

-99

-103

-1073

-95

-99

-103

-1073

6 7 8 9 10 11

-50

-45

-40

-35

-30

-25

log10(τχs)

log 10(Ω

χ)

Secondary 6LiProduction

FIG 5 Numerical results for δYa for each relevant nucleus in the presence of a single electromagnetically-decaying particle χwith extrapolated abundance Ωχ and lifetime τχ In all panels except that pertaining to 4He destruction we plot contours oflog10 Ya while for 4He destruction we plot contours of Yp Note that for later reference we have plotted separate contours forD production and destruction likewise we show separate contours for primary and secondary 6Li production Regions of the(τχΩχ) plane which are found by numerical analysis to be excluded are those above andor to the right of the gray dashedline in each panel while the solid colored curve in each panel demarcates the analogous region found to be excluded by fittingthe free parameters in our analytic approximations for δYa in Eqs (352)ndash(354) to our numerical results We see that the fullnumerical results and our analytic approximations agree very well in all cases except that of primary 6Li production whereour analytic approximation diverges from the numerical result at large Ωχ and thus provides an even more conservative boundon the allowed parameter space

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

22

the two individual contributions to δYa (in the case ofD) or a reinforcement (in the case of 6Li) The graydashed line in each panel of Fig 5 indicates the upperbound on Ωχ obtained through a numerical calculationof δYa which follows from the observational limits inEqs (318)ndash(321) By contrast the solid curve in eachpanel represents the corresponding constraint contourobtained through our analytic approximation for δYa inEq (351)

We see from Fig 5 that in all cases except for thatof primary 6Li production the constraint contours weobtain using the linear and uniform-decay approxima-tions are quite similar to those obtained in the morerealistic case in which we account for the fact that χhas a constant decay rate and in which we account

for feedback effects in the collision terms C(p)a and C(s)

a

for each relevant nucleus Moreover as discussed inSect III D we see that for primary 6Li production theconstraint contour obtained using these approximationsindeed represents a conservative bound on Ωχ Wealso note that with this sole exception the results ofour analytic approximations are in good agreement withthe corresponding results in Ref [3] within the regimein which Ωχ is small and the linear approximationholds Once again we note that the effects of additionalprocesses which we do not incorporate into this analysiscan have important effects on the Ya in the regime inwhich Ωχ mdash and therefore the overall magnitude ofelectromagnetic injection mdash is large These include forexample photodisintegration processes which contributeto the depletion of a previously generated population of6Li as well as additional processes which contribute tothe production or destruction of D [3] However sincethe regions of (Ωχ τχ) space in which these effects on

D and 6Li are relevant are excluded by constraints onthe primordial abundances of other nuclei our neglectingthese processes will have essentially no impact on theoverall constraints we derive on ensembles of decayingparticles

In Fig 6 we plot our analytic approximations to theconstraint contours obtained for each relevant nucleustogether in (Ωχ τχ) space Note that the 6Li contourincludes both primary and secondary contributions toδY6Li In determining the D contour we have includedcontributions from both photoproduction and photodis-integration processes the interference between whichis responsible for the ldquofunnelrdquo region at τχ asymp 1065 swithin which the bound on Ωχ is considerably weakenedHowever since the upper and lower bounds on δYD derivefrom different considerations different colors are used todistinguish the part of the contour which correspondsto the upper bound (dark blue) from the part whichcorresponds to the lower bound (light blue) We notethat in all cases these contours are in good agreementwith the corresponding results in Ref [3] within the

4 6 8 10 12-6

-5

-4

-3

-2

-1

0

log10(τχs)

log 10(Ω

χ)

FIG 6 Compilation of constraints on electromagnetic in-jection from bounds on the abundances of light nuclei Theindividual contours shown indicate the constraints stemmingfrom considerations of D production (dark blue) and destruc-tion (light blue) 7Li destruction (green) 4He destruction(magenta) and 6Li production (red) the latter including bothprimary and secondary contributions In addition we alsoplot constraints stemming from limits on potential distortionsof the CMB-photon spectrum both micro-type (orange) and y-type (black) as will be discussed in Sect IV In each case theregions of parameter space above each contour are excluded

regime in which Ωχ is small and the linear approximationholds

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

23

IV SPECTRAL DISTORTIONS IN THE CMB

In addition to altering the abundances of light nucleielectromagnetic injection from the decays of unstableparticles in the early universe can also have observableeffects on the spectrum of CMB photons distortingthat spectrum away from a pure blackbody profileObservational limits on these distortions therefore alsoconstrain such decays Once again while the constraintson a single decaying particle species are well known thecorresponding constraints on an ensemble of decayingparticles are less well established Our aim in this sectionis to derive approximate analytic expressions for theCMB distortions which can arise due to entire ensemblesof decaying particles mdash expressions analogous to those wederived in Sect III for the change δYa in the comovingnumber density in each relevant nucleus after BBN

One of the subtleties which arises in constrainingdistortions in the CMB-photon spectrum from ensemblesof decaying particles with a broad range of lifetimes isthat both the manner in which that spectrum is distortedand the magnitude of that distortion depend on thetimescale over which the injection takes place At earlytimes energetic photons produced by these decays arerapidly brought into both kinetic and thermal equilib-rium by the Class-III processes discussed in Sect IIAs a result the spectrum of photons retains a black-body shape However at later times photon-number-changing processes such as double-Compton scatteringand bremsstrahlung ldquofreeze outrdquo in the sense that therates for these processes drop below the expansion rate Hof the universe Once this occurs injected photons are nolonger able to attain thermal equilibrium with photons inthe radiation bath Nevertheless they are able to attainkinetic equilibrium with these photons through elasticCompton scattering As a result the photon spectrumno longer retains its original blackbody shape rather it isdistorted into a Bose-Einstein distribution characterizedby a pseudo-degeneracy parameter micro

Eventually at an even later time tEC asymp 9times109 s elasticCompton scattering also freezes out Thus photonsinjected at times t amp tEC attain neither kinetic northermal equilibrium with photons in the radiation bathNevertheless the injected photons continue to interactwith electrons in the radiation bath at an appreciablerate until the time of last scattering tLS asymp 119times 1013 sDuring this window photons in the radiation bath areup-scattered by electrons which acquire significant energyfrom these interactions The ultimate result is a photonspectrum which is suppressed at low frequencies andenhanced at high frequencies This spectral distortionis characterized by a non-zero Compton y parameteryC Finally after last scattering any additional photonsinjected from particle decay simply contribute to thediffuse photon background

In order to derive analytic expressions for the con-straints on the parameters micro and yC in the presenceof a decaying ensemble we shall once again make use

of the uniform-decay approximation as well as a linearapproximation similar to the approximation we invokedin calculating the δYa in Sect III The validity of suchan approximation follows from the fact that both micro andyC are tightly constrained by observation The moststringent limits on these parameters which are currentlythose from COBEFIRAS data [55] are

|micro| lt 90times 10minus5

yC lt 12times 10minus5 (41)

Moreover proposed broad-spectrum experiments suchas PIXIE [56] could potentially increase the sensitivityto which these distortions can be probed by more thanan order of magnitude These stringent constraintson the shape of the CMB-photon spectrum imply thatthe contribution δργ to the photon energy density frominjection during the relevant epoch must be quite smallin the sense that δργ ργ Thus to a very goodapproximation we may ignore feedback effects on theoverall photon number density nγ and energy density ργwhen analyzing distortions in that spectrum This allowsus to replace these quantities by their equilibrium values

This linear approximation significantly simplifies theanalysis of the resulting CMB spectrum [6 7 57 58]For example in this approximation we may study theevolution of the CMB-photon spectrum using the methodof Greenrsquos functions [25 58 59] In this approachwe may express a generic spectral distortion whichconstitutes a shift ∆Iν(tnow) in the present-day intensityof photons with frequency ν relative to the expectedintensity in the standard cosmology in the form

∆Iν(tnow) =

intG(ν t tnow)

d

dt

(Q

ργ

)dt (42)

where Q is the energy density injected at time t andwhere G(ν t tprime) is a Greenrsquos function which relatesthe intensity of photons with frequency ν injected attime t to the photon intensity at that same frequencyat a later time tprime gt t In general G(ν t tnow) iswell approximated by a sum of terms representing anoverall temperature shift a micro-type distortion and y-typedistortion [25] This Greenrsquos-function formalism providesa tool for numerically estimating the overall distortion tothe CMB-photon spectrum from any arbitrary injectionhistory provided that the overall injected energy is smallWe shall draw on this formalism extensively in derivingour analytic approximations for δmicro and δyC

A The micro Parameter

The time-evolution of the pseudo-degeneracy parame-ter micro is governed by an equation of the form [6 7]

dmicro

dt=

dmicroinj

dtminus micro(ΓDC + ΓBR) (43)

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

24

The first term in Eq (43) is a source term which arisesdue to the injection of energetic photons Within thelinear approximation this source term takes the genericform

dmicroinj

dtasymp 1

2143

[3

ργ

dργdtminus 4

nγ

dnγdt

] (44)

where nγ and ργ respectively denote the overall numberdensity and energy density of photons We emphasizethat this expression is completely general and applieseven in absence of any source of photon injection inwhich case the two terms in Eq (44) cancel

The second term in Eq (43) is a sink term associatedwith processes which serve to bring the population ofphotons in the radiation bath into thermal equilibriumand thereby erase micro As discussed above the domi-nant such processes are double-Compton scattering andbremsstrahlung the respective thermalization rates forwhich are well approximated by power-law expressionsof the form [6]

ΓDC = ΓMREDC

(t

tMRE

)minus94

ΓBR = ΓMREBR

(t

tMRE

)minus138

(45)

where tMRE once again denotes the time of matter-radiation equality The coefficients in this expression aregiven by

ΓMREDC asymp

(573times 10minus39 GeV

) (ΩBh

2)

times(

1minus Yp2

)(Tnow

27 K

)32

ΓMREBR asymp

(157times 10minus36 GeV

) (ΩBh

2)32

times(

1minus Yp2

)(Tnow

27 K

)minus54

(46)

where h is the Hubble constant ΩB is the present-dayabundance of baryons and Tnow asymp 27 K is the present-day temperature of the CMB For completeness we notethat the expression for ΓBR in Eq (45) represents anoverestimate of the true bremsstrahlung rate In particu-lar a full frequency-dependent treatment of the evolutionof the photon spectrum reveals that bremsstrahlung isless efficient at early times [13] As a result the truebremsstrahlung rate is actually smaller than the resultin Eq (45) would suggest However we shall find belowthat the effect of bremsstrahlung on micro is subleading incomparison to the effect of double-Compton scatteringeven with our overestimate for ΓBR Given this we shallchoose to ignore the effect of bremsstrahlung entirelyat a later stage of our analysis Thus our use of theexpression in Eq (45) for the bremsstrahlung rate willultimately have no effect on our overall results

Up to this point our expressions governing the evo-lution of micro are completely general and may be ap-plied to any injection history provided that the total

energy density injected is sufficiently small that thelinear approximation is valid In order to derive anapproximate analytic expression for the overall change δmicroin micro due to an ensemble of unstable particles we proceedin essentially the same manner as we did in derivingexpressions for the changes δYa in the comoving numberdensities of light nuclei in Sect III We begin by derivingan expression for δmicro in the presence of a single decayingparticle species χ with mass mχ and lifetime τχ Whenwe take into account the full exponential nature of thedecay we find

dργdt

= minus4Hργ + εχΓχρχ

dnγdt

= minus3Hnγ +N

(γ)χ Γχρχmχ

(47)

where N(γ)χ is the average number of photons produced

per decay and where εχ is the fraction of the energyreleased by χ decays which is transferred to photonsBy contrast within the uniform-decay approximation inwhich dρχdt is given by Eq (310) we have

dργdt

= minus4Hργ + εχρχ(τχ)δ(τχ minus t)

dnγdt

= minus3Hnγ +N

(γ)χ ρχ(τχ)

mχδ(τχ minus t) (48)

In this approximation a simple analytic expression forδmicro may be obtained by directly integrating Eq (43) fromthe time te asymp 169 times 103 s at which electron-positronannihilation freezes out to the time tEC at which elasticCompton scattering freezes out In particular we findthat a particle species with a lifetime τχ anywhere withinthis range gives rise to a micro-type distortion of size

δmicro asymp η

2143

[90ζ(3)εχ

π4

(τχs

)12

minus 4N (γ)χ

( mχ

MeV

)minus1]

times( mp

MeV

) ΩχΩB

exp

[8

5ΓMRE

BR t138MRE

(tminus58EC minus τminus58

χ

)+

4

5ΓMRE

DC t94MRE

(tminus54EC minus τminus54

χ

)] (49)

where mp is the proton mass where ΩB is the totalpresent-day abundance of baryons where η equiv nBnγ asymp62times10minus10 is the baryon-to-photon ratio of the universeand where Ωχ once again refers to the extrapolatedabundance of χ For τχ outside this range δmicro asymp 0

Since the bremsstrahlung term in the exponential inEq (49) is generically negligible in comparison with thedouble-Compton-scattering term we may safely neglectit in what follows Moreover for mχ and τχ within ourregimes of interest the second term in the prefactor inEq (49) is subleading and can likewise be neglectedThus we find that δmicro is well approximated by

δmicro asymp(48times 10minus7

) ΩχεχΩB

(τχs

)12

eminus(tmicroτχ)54 (410)

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

25

where we have defined

tmicro equiv(

4

5ΓMRE

DC

)45

t95MRE asymp 66times 108 s (411)

This latter quantity represents the timescale at whichthe damping due to double-Compton scattering becomesinefficient and contributions to δmicro become effectivelyunsuppressed

The dependence of δmicro on τχ has a straightforwardphysical interpretation The exponential suppressionfactor reflects the washout of micro by double-Comptonscattering at early times τχ tmicro The additionalpolynomial factor reflects the fact that the equilibriumphoton density grows more rapidly with time thandoes the matter density at earlier times This impliesthat particles decaying at an earlier time yield smallerperturbations to the photon spectrum

B The Compton y Parameter

As discussed above elastic Compton scattering servesto maintain kinetic equilibrium among photons in theradiation bath after double-Compton scattering andbremsstrahlung freeze out However elastic Comptonscattering eventually freezes out as well at time t simtEC An injection of photons which occurs after tEC butbefore the time tLS of last scattering contributes to thedevelopment of a non-vanishing Compton y parameteryC Within the linear approximation a reliable estimatefor the rate of change of yC may be obtained from therelation [60]

dyCdtinj

asymp 1

4ργ(tinj)

dρ(tinj)

dtinj (412)

where dρ(tinj)dtinj denotes the rate at which energydensity is injected into the photon bath We note that incontrast with the corresponding evolution equation for microin Eq (43) this equation contains no damping term

In deriving our analytic approximation for the changeδyC from an ensemble of unstable particle species weonce again begin by considering the case of a single suchspecies χ When we take into account the full exponentialnature of the decay we obtain

dyCdtinj

asymp 1

4ργ(tinj)εχΓχρχ(tinj) (413)

The corresponding result within the uniform-decay ap-proximation is

dyCdtinj

asymp 1

4ργ(tinj)εχρχ(τχ)δ(τχ minus tinj) (414)

In integrating this latter expression over t in order toobtain our approximate expression for δyC we mustaccount for the fact that the epoch during which in-jection contributes to yC straddles the time tMRE of

matter-radiation equality Using the appropriate time-temperature relations

T asymp TMRE times

(tMREt)

12 t tMRE

(tMREt)23 t amp tMRE

(415)

before and after the transition to matter-dominationwhere TMRE is the temperature at matter-radiationequality we find

δyC asymp15ζ(3)ηmp

2π4TMRE

ΩχεχΩB

times

(

τχtMRE

)12

τχ lt tMRE(τχtMRE

)23

τχ gt tMRE

(416)where η once again denotes the baryon-to-photon ratioof the universe

C The Fruits of LinearizationCMB-Distortion Constraints on Ensembles of

Unstable Particles

Having derived analytic approximations for the spec-tral distortions δmicro and δyC due to a single decayingparticle species within the uniform-decay approximationwe now proceed to generalize these results to the case ofan arbitrary injection history

Within the linear approximation the overall contri-bution to δmicro from a decaying ensemble is simply thesum of the individual contributions from the constituentparticles χi with lifetimes in the range te τi tECLikewise the overall contribution to yC is a sum ofindividual contributions from χi with tEC τi tLSHowever injection from constituents with lifetimes τi simtEC gives rise to intermediate-type distortions whichare distinct from the purely micro-type or purely y-typedistortions discussed above While these intermediate-type distortions are not merely a superposition of micro-typeand y-type distortions [58] isolating and constrainingsuch intermediate-type distortions would require a moreprecise measurement of the CMB-photon spectrum thanCOBEFIRAS data currently provide We thereforefind that mdash at least for the time being mdash the neteffect of injection at times t sim tEC may indeed bemodeled through such a superposition with δmicro and δyCgiven by modified versions of Eqs (410) and (416)The appropriate modifications of these expressions canbe inferred from the approximate forms of the Greenrsquosfunctions in Ref [25] In particular we model the overallcontributions to δmicro from the decaying ensemble by an

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

26

Distortion t0a(s) tBa(s) t1a(s) αa Aa Ba

micro-type 33times 106 24times 109 12times 1010 10times 100 54times 10minus3 31times 10minus1

yc-type 33times 106 68times 107 54times 109 12times 100 37times 10minus5 27times 10minus1

TABLE III Values for the parameters appearing in Eqs (417) and (418) which characterize the spectral distortions δmicro andδyC due to the electromagnetic decays of unstable particles decaying after electron-positron annihilation with a = micro y

expression of the form

δmicro =sum

teltτilttBmicro

AmicroΩi

(τit0micro

)12

eminus(t0microτi)54

+sum

tBmicroltτilttMRE

BmicroΩi

(τit1micro

)12 [1minus eminus(t1microτi)

αmicro]

+sum

tMREltτilttLS

BmicroΩi

(τit2micro

)23 [1minus eminus(t2microτi)

4αmicro3]

(417)

Likewise for δyC we have

δyC =sum

teltτilttBy

AyΩi

(τit0y

)12

eminus(t0yτi)54

+sum

tByltτilttMRE

ByΩi(τit1y)12

1 + (t1yτi)αy

+sum

tMREltτilttLS

ByΩi(τit2y)23

1 + (t2yτi)4αy3

(418)

In these equations we have introduced two sets of pa-rameters Amicro Bmicro αmicro t0micro etc and Ay By αy t1y etcin order to characterize δmicro and δyC respectively We

note however that the timescales t2micro equiv t14MREt

341micro and

t2y equiv t14MREt

341y are not independent model parameters

but merely shorthand notation for quantities which arecompletely determined by t1micro and t1y respectively Wealso note that this parametrization accounts not only formicro-type distortions at times t tEC and y-type distortionsat times t amp tEC but also for subdominant contributionsto δyC prior to tEC and to δmicro after tEC Finally we notethat in formulating Eqs (417) and (418) we have onceagain implicitly taken εi = 1 for all χi just as we didin Eqs (352)ndash(354) The corresponding expressions forδmicro and δyC may nevertheless be obtained for the case inwhich εi differs from unity for one or more of the χi byonce again replacing Ωi for each species with the productΩiεi

We now determine the values of the parameters inEqs (417) and (418) in a manner similar to thatin which we determined the values of the parametersin Eqs (352)ndash(354) Specifically we demand thatour analytic approximations yield constraint contours

consistent with those obtained from numerical analysisin the case in which the ensemble comprises a singledecaying particle with lifetime τχ and abundance Ωχ Inparticular we survey (Ωχ τχ) space and compute thegeneric spectral distortions numerically at each pointusing the Greenrsquos-function formalism of Refs [25 59]The generic spectra are then fit by a superposition ofa micro-type and a y-type distortion in order to determinea corresponding pair of values for δmicro and δyC Theundetermined parameters in Eqs (417) and (418) arethen chosen such that our analytic expressions for δmicroand δyC provide good fits to the corresponding constraintcontours Our best-fit values for all of these parametersare quoted in Table III

In Fig 7 we display contours showing the results ofour numerical calculations for δmicro (left panel) and δyC(right panel) within the (Ωχ τχ) plane The gray dashedline in each panel demarcates the upper boundary ofthe region within the (Ωχ τχ) plane which is consistentwith the limits on these distortions quoted in Eq (41)By contrast the solid curve in each panel representsthe corresponding constraint contour obtained throughour analytic approximation in Eqs (417) and (418)As we see the results obtained through our analyticapproximation represent a good fit to our numericalresults for both δmicro and δyC

The constraint contours obtained from our analytic ap-proximations for both micro- and y-type distortions are alsoincluded in Fig 6 in order to facilitate comparison withthe constraint contours associated with the primordialabundances of light nuclei We note that the constrainton δmicro is more stringent than the constraint on yC forτχ tEC as expected while the opposite is true forτχ amp tEC We also see that both of these constraintsare subdominant in comparison with the constraints onthe primordial abundances of light nuclei except at latetimes τχ amp 1010 s

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

27

-3

-405-5

-6

-3

-405-5

-6

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

μ

-3

-492

-7

-9

-3

-492

-7

-9

6 7 8 9 10 11 12

-5

-4

-3

-2

log10(τχs)

log 10(Ω

χ)

yC

FIG 7 Contours of log10(δmicro) (left panel) and log10(δyC) (right panel) obtained from the analytic approximations in Eqs (417)and (418) in the presence of a single electromagnetically-decaying particle with extrapolated abundance Ωχ and lifetime τχplotted within the (Ωχ τχ) plane The gray dashed line in each panel demarcates the upper boundary of the region within the(Ωχ τχ) plane which is consistent with the observed limits quoted in Eq (41) while the solid curve represents the constraintcontour obtained by fitting the corresponding analytic approximation in Eqs (417) and (418) to these numerical results

V MODIFICATIONS TO THE IONIZATIONHISTORY OF THE UNIVERSE

As we have seen in Sect IV electromagnetic injectionfrom late-decaying particles can give rise to certaindistortions in the CMB-photon spectrum However suchinjections can affect the CMB in other ways as well Inparticular when such injection occurs during recombi-nation the resulting photons and electrons contributeto the heating and ionization of neutral hydrogen andhelium as they cool thereby expanding the surface oflast scattering This in turn leads to alterations inthe pattern of CMB anisotropies including a dampingof correlations between temperature fluctuations as wellas an enhancement of correlations between polariza-tion fluctuations at low multipole moments [9] Theseconsiderations therefore place additional constraints ondecaying particle ensembles

In a nutshell these effects arise because the ratesfor many of the Class-I processes discussed in Sect IIare still sizable at times t sim tLS These processestherefore continue to rapidly redistribute the energies ofphotons injected during the recombination epoch to lowerenergies thereby allowing for efficient photoionizationof neutral hydrogen and helium at a rate that exceedsthe rate of cosmic expansion However there is only asomewhat narrow time window around tLS during whichan injection of photons can have a significant impacton the surface of last scattering At times t tLSany additional ionization of particles in the plasma iseffectively washed out By contrast at times t tLS

the relevant Class-I processes effectively shut off andphotoionization is suppressed

To a good approximation then we may regard anymodification of the ionization history of the universe asbeing due to injection at t sim tLS Thus the crucialquantity which is constrained is the overall injectionrate of energy density in the form of photons or otherelectromagnetically-interacting particles at tLS Sincethe universe is already matter-dominated by the re-combination epoch the injection rate associated with asingle decaying particle species χ with lifetime τχ andextrapolated abundance Ωχ at tLS is given by

dργdt

∣∣∣∣t=tLS

= ρcrit(tnow)

(tnow

tLS

)2

εχΩχτχeminustLSτχ

(51)where εχ once again denotes the fraction of the energydensity released by χ decays which is transferred tophotons Thus we model the constraint on such adecaying particle from its effects on the ionization historyof the universe by an inequality of the form

εχΩχτχeminustIHτχ ΓIH (52)

where tIH and ΓIH are taken to be free parametersWe next determine the values of ΓIH and tIH by fitting

these parameters to the constraint contours obtained inRefs [12 61] from a numerical analysis of the injec-tion and deposition dynamics The constraint contoursobtained from subsequent numerical studies by other

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

28

authors [62] are quite similar In particular we find

ΓIH asymp 62times 10minus26 sminus1

tIH asymp 14times 1013 s (53)

Note that recent results from the final Planck datarelease [63] will affect the numerical values for ΓIH andtIH slightly but not dramatically

In Fig 8 we compare the results of our analytic ap-proximation with the numerical results of Refs [12 61]We observe that our approximation for the constraintcontour is in good agreement with these numerical resultsfor τχ amp 1015 s and deviates significantly from thiscontour only when τχ sim tLS However as discussed inRef [12] the results to which we perform our fit yieldan artificially conservative bound on Ωχ for lifetimesτχ sim tLS as the production of additional Lyman-αphotons due to excitations from the ionizing particles wasintentionally neglected Other analyses of the ionizationhistory [64] which include this effect yield constraintcontours which more closely resemble that obtained fromour analytic approximation

The constraint in Eq (52) may be generalized toan ensemble of decaying particles in a straightforwardmanner Indeed since this constraint is ultimately abound on the overall injection rate at t sim tLS thecorresponding constraint on an ensemble of decayingparticles χi takes the formsum

i

εiΩiτieminustIHτi ΓIH (54)

with values of ΓIH and tIH once again given in Eq (53)

We emphasize that the constraint in Eq (54) stemssolely from the effect of electromagnetic injection fromthe decays of the ensemble constituents on the surfaceof last scattering Since this effect arises primarily frominjection at t sim tLS the constraints are most relevantfor ensembles in which the constituents have lifetimesroughly around this timescale However electromagneticinjection can affect the ionization history of the universein other ways as well For example energy deposited inthe intergalactic medium as a result of particle decays attimescales well after tLS serves both to elevate the gastemperature and to increase the ionization fraction ofhydrogen These effects can shift the onset of reionizationto earlier times [61 65] and impact the 21-cm absorp-tionemission signal arising from hyperfine transitionsin neutral hydrogen [66ndash68] These considerations arerelevant primarily for particles with lifetimes τi tnow

mdash particles which would in principle contribute to thepresent-day dark-matter abundance Since our primaryfocus in this paper is on ensembles whose constituentshave lifetimes τi tnow we do not investigate theseadditional considerations here However we note thatthey can play an important role in constraining ensemblesof particles with longer lifetimes such as those whicharise within the context of the DDM framework

12 14 16 18 20 22 24-14

-12

-10

-8

-6

-4

-2

0

log10(τχs)

log 10(Ω

χ)

tLS tnow

FIG 8 Constraints on a decaying particle with lifetimeτχ and extrapolated abundance Ωχ from modifications ofthe ionization history of the universe These constraintscorrespond to the case in which εχ = 1 and the entiretyof the energy density released by χ decays is transferredto photons The gray dashed contour represents the boundobtained in Refs [12 61] from numerical analysis The brownshaded region above the contour is excluded The solidcurve represents the constraint contour obtained by fittingthe corresponding analytic approximation in Eq (52) to theseresults Dashed vertical lines indicating the time tLS of lastscattering and the present age of the universe tnow have alsobeen included for reference We see that our approximationfor the constraint contour is in good agreement with thesenumerical results for τχ amp 1015 s and deviates significantlyfrom this contour only when τχ sim tLS

VI CONTRIBUTIONS TO THE DIFFUSEPHOTON BACKGROUND

At times t tLS the universe becomes transparent tophotons with a broad range of energies O(keV) Eγ O(TeV) [9] Photons within this energy range whichare injected on timescales t tLS are not reprocessedby Class-I processes and do not interact with CMBphotons at an appreciable rate Thus as discussed inSect IV such photons therefore simply free-stream untilthe present epoch and contribute to the diffuse photonbackground The total diffuse background of photonsobserved from the location of Earth includes both agalactic contribution from processes occurring within theMilky-Way halo (at redshifts z asymp 0) and an extra-galacticcontribution from processes which occurred within themore distant past The decays of unstable particles withlifetimes τi tnow contribute essentially exclusively tothe extra-galactic component of this total backgroundBy contrast particles with lifetimes τi amp tnow mdash

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

29

particles which would contribute non-negligibly to thepresent-day abundance of dark matter mdash in principlecontribute to both components Since our primary focusin this paper is on particles with lifetimes in the regimeτi tnow we focus on the extra-galactic componentof the diffuse photon background However we alsonote that since the local dark-matter density within theMilky-Way halo is several orders of magnitude largerthan the corresponding cosmological density the galacticcomponent of the diffuse photon background is typicallythe more important of the two for constraining particleswith lifetimes τi amp tnow

The overall diffuse extra-galactic contribution dΦdEγto the differential flux of photons per unit energy ob-served by a detector on Earth from the decays of anensemble of unstable constituent particles χi is simplythe direct sum of the individual contributions dΦidEγfrom these constituents In order to evaluate theseindividual contributions we begin by noting that thedecays of a given species χi averaged over all possibledecay channels produce a characteristic injection spec-trum of photons dNidEγ The spectrum of photonsarriving at present time at a detector on Earth is theintegrated total contribution from decays occurring atdifferent redshifts A photon injected at redshift z withenergy Eprimeγ is redshifted to energy Eγ = (1+z)minus1Eprimeγ whenit arrives at the location of Earth Thus we find that theobserved flux of photons observed at a detector on Earthfrom the decay of an individual decaying species χi maybe written in the form

dΦidEγ

=1

4π

intdΩ

int infin0

dz

ρi(z)

(1 + z)3

Γimi

d`

dzeminusκ(z)

timesintdEprimeγδ

[Eprimeγ minus (1 + z)Eγ

] dNidEprimeγ

(61)

where dΩ is the differential solid angle of the detectorwhere ρi(z) is the energy density of χi at redshift zwhere κ(z) is the optical depth of the universe to photonsemitted at redshift z and where `(z) is the line-of-sightdistance away from Earth expressed as a function of z

We focus here primarily on photons with energies inthe range O(keV) Eγ O(TeV) for which theuniverse is effectively transparent at all times t amp tLS

and opaque at times t tLS Moreover for simplicitywe shall approximate the optical depth for such photonswith a function of the form

κ(z) asymp

0 z lt zLS

infin z gt zLS (62)

where zLS equiv z(tLS) asymp 1100 is the redshift at the time oflast scattering In other words we shall assume that theuniverse is completely transparent to all photons withinthis energy range after last scattering and completelyopaque beforehand Moreover since tLS tMREwe approximate the line-of-sight distance `(z) in a flat

universe dominated by matter and dark energy as

`(z) =c

Hnow

int z

0

dz

F12(z) (63)

where c is the speed of light where Hnow is the present-day value of the Hubble parameter and where we havedefined

F(z) equiv Ωm(1 + z)3 + ΩΛ (64)

where Ωm and ΩΛ respectively denote the present-dayabundances of massive matter and dark energy the for-mer including contributions from both dark and baryonicmatter The energy density ρi(z) of a decaying particlecan be expressed in terms of its extrapolated abundanceΩi as

ρi(z) = Ωiρcrit(tnow)(1 + z)3eminusΓit(z) (65)

where the time t(z) which corresponds to redshift z is

t(z) =1

Hnow

int z

0

dz

(1 + z)F12(z) (66)

With these substitutions Eq (61) becomes

dΦidEγ

=cρcrit(tnow)ΩiΓi

4πHnowmi

intdΩ

int zLS

0

dz

[1

F12(z)

times eminusΓit(z)dNidEprimeγ

∣∣∣∣Eprimeγ=(1+z)Eγ

] (67)

Finally in assessing our prediction for dΦdEγ fora decaying particle ensemble we must account for thefact that at a realistic detector the photon energiesrecorded by the instrument differ from the actual energiesof the incoming photons due to the non-zero energyresolution of the detector We account for this effectby convolving the spectrum of incoming photons witha smearing function Rε(Eγ minus Eprimeγ) which represents theprobability for the detector to register a photon energyEγ given an actual incoming photon energy Eprimeγ Forconcreteness we consider a Gaussian smearing functionof the form

Rε(Eγ minus Eprimeγ) =1radic

2πεEprimeγexp

[minus

(Eγ minus Eprimeγ)2

2ε2Eprime2γ

] (68)

where ε is a dimensionless parameter which sets theoverall energy-dependent standard deviation σ(Eprimeγ) =εEprimeγ of the distribution Thus we have our final resultfor the extra-galactic photon flux

dΦ

dEγ=sumi

intdEprimeγ

dΦidEprimeγ

Rε(Eγ minus Eprimeγ) (69)

with the individual fluxes dΦidEprimeγ given by Eq (67)

and with Rε(Eγ minus Eprimeγ) given by Eq (68)

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

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[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

30

Limits on additional contributions to the extra-galacticphoton flux at energies Eγ amp 800 keV can be derivedfrom measurements of the total gamma-ray flux byinstruments such as COMPTEL EGRET and Fermi-LAT For photon energies in the range 800 keV Eγ 30 MeV COMPTEL data on the diffuse extra-galacticbackground spectrum [69] are well modeled by a power-law expression of the form

dΦ

dEγdΩasymp (499times 10minus3)

(Eγ

MeV

)minus240

MeVminus1cmminus1sminus1srminus1 (610)

Likewise for energies in the range 30 MeV Eγ 141 GeV EGRET data [70] are well modeled by thepower-law expression

dΦ

dEγdΩasymp (735times 10minus3)

(Eγ

MeV

)minus235

MeVminus1cmminus1sminus1srminus1 (611)

At even higher energies the spectrum of the extra-galactic diffuse photon background can be inferred fromFermi-LAT data Within the energy range 100 MeV Eγ 820 GeV these data are well described by a power-law expression with an exponential suppression at highenergies [71]

dΦ

dEγdΩasymp (415times 10minus3)

(Eγ

MeV

)minus232

exp

(minusEγ

279 GeV

)MeVminus1cmminus1sminus1srminus1 (612)

Finally we note that while we focus here primarily onphotons with energies above 800 keV estimates of theextra-galactic photon background at lower energies havebeen computed from BAT [72] and INTEGRAL [73] dataand from observations of Type-I supernovae [74]

The expression in Eq (69) is completely general andapplicable to any ensemble of particles which decays tophotons with energies within the transparency windowO(keV) Eγ O(TeV) To illustrate how thisexpression can be evaluated in practice we consider aconcrete example involving a specific set of injectionspectra dNidEγ for the ensemble constituents In par-ticular we consider a scenario in which each constituentχi decays with a branching fraction of effectively unity toa pair of photons via the process χi rarr γ + γ Under theassumption that the χi are ldquocoldrdquo (ie non-relativistic)by the time tLS each of these photons has energy Eγ asympmi2 in the comoving frame at the moment of injectionWe thus have

dNidEγ

asymp 2δ(Eγ minus

mi

2

) (613)

While higher-order processes will generically also giverise to a continuum spectrum which peaks around Eγ simO(10minus3mi) we focus here on the contribution from the

photons with Eγ asymp mi2 since line-like features arisingfrom monochromatic photon emission are far easier todetect than continuum features Thus for the injectionspectrum in Eq (613) the total contribution to thedifferential extra-galactic diffuse photon flux from thedecays of the constituents within this ensemble is givenby

dΦidEγ

asymp cρcrit(tnow)

2πHnow

intdΩ

intdEprimeγ

Rε(Eγ minus Eprimeγ)

Eprimeγ

timessumi

ΩiΓimiFminus12

(mi

2Eprimeγminus 1

)

times exp

[minusΓit

(mi

2Eprimeγminus 1

)] (614)

VII APPLICATIONS TWO EXAMPLES

In Sects IIIndashVI we examined the observational limitson electromagnetic injection from particle decays afterthe BBN epoch and formulated a set of constraintswhich can be broadly applied to any generic ensembleof unstable particles In this section we examine theimplications of these constraints by applying them to apair of toy ensembles which exhibit a variety of scalingbehaviors for the constituent masses mi extrapolatedabundances Ωi and lifetimes τi For simplicity weshall focus primarily on the ldquoearly-universerdquo regime inwhich 1 s τi tLS for all of the χi Within thisregime the leading constraints on injection are thosefrom modifications of the abundances of light nuclei andfrom distortions in the CMB

We consider two different classes of mass spectra forour decaying ensemble constituents The first consists ofspectra in which the masses of these decaying particlesare evenly distributed on a linear scale according to thescaling relation

mk = m0 + k∆m (71)

where k = 0 1 N minus 1 and where m0 and ∆m aretaken to be free parameters which describe the massof the lightest ensemble constituent and the subsequentmass splittings respectively Mass spectra of this ap-proximate linear form are realized for example forthe KK excitations of a particle propagating in a flatextra spacetime dimension whose total length is smallcompared with 1m0 The second class of mass spectrawe consider are spectra in which the particle masses areevenly distributed on a logarithmic scale according to thescaling relation

mk = m0 ξk (72)

where m0 once again denotes the mass of the lightestensemble constituent and where ξ is a dimensionless freeparameter A mass spectrum of this sort is expected

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

31

for example for axion-like particles in axiverse construc-tions [19]

A variety of scaling relations for the decay widths Γiof the χi across the ensemble can likewise be realized indifferent scenarios For simplicity in this paper we focuson the case in which each χi decays with a branchingfraction of effectively unity to a pair of photons via theprocess χi rarr γ + γ Moreover we shall assume that thescaling relation for Γi takes the form

Γi = Cim3i

Λ2 (73)

where Λ is a parameter with dimensions of mass andwhere Ci is a dimensionless coefficient mdash a coefficientwhich in principle can also have a non-trivial dependenceon mi Such a scaling relation arises naturally forexample in situations in which each of the χi decays viaa non-renormalizable Lagrangian operator of dimensiond = 5 in an effective theory for which Λ is the cutoffscale For purposes of illustration we focus on the casein which the coefficient Ci = Cχ is identical for all Ci Inthis case the Γi are specified by a single free parameterthe ratio Mlowast equiv Λ

radicCχ We emphasize that it is possible

mdash and indeed in most situations even expected mdash thatCχ 1 Thus unlike Λ itself the parameter Mlowast mayexceed the reduced Planck mass MP without necessarilyrendering the theory inconsistent However we remarkthat for values of Mlowast in this regime the Weak GravityConjecture [75] imposes restrictions on the manner inwhich the decay operator in Eq (73) can arise

Finally we consider a range of different possible scalingrelations for the extrapolated abundances Ωi across theensemble In particular we consider a family of scalingrelations for the Ωi of the form

Ωi = Ω0

(mi

m0

)γ (74)

in which the abundance Ω0 of the lightest ensembleconstituent and the scaling exponent γ are both takento be free parameters In principle γ can be eitherpositive or negative However since τi prop mminus3

i andsince the leading constraints on injection at timescalestinj tLS are typically less stringent for earlier tinj themost interesting regime turns out to be that in whichγ ge 0 and the initial energy density associated with eachχi is either uniform across the ensemble or decreases withincreasing τi

In summary then each of our toy ensem-bles is characterized by a set of six parametersNm0∆mMlowastΩ0 γ for the mass spectrum inEq (71) and Nm0 ξMlowastΩ0 γ for the mass spectrumin Eq (72) Note that for any particular choice ofthese parameters the lifetimes of the individual ensembleconstituents span a range τNminus1 le τi le τ0

A Results for Uniform Mass Splittings

We begin by examining the results for a toy ensemblewith a mass spectrum given by Eq (71) For such amass spectrum the density nm of states per unit mass isuniform across the ensemble however it is also usefulto consider the density nτ of states per unit lifetimewhich is more directly related to the injection history Inthe continuum limit in which the difference between thelifetimes of sequential states χi and χi+1 in the ensembleis small and the lifetime τ may be approximated as acontinuous variable we find that

nτ =M

23lowast

3 ∆mτ43 (75)

The density of states per unit lifetime is therefore greatestwithin the most unstable regions of the ensemble More-over we also note that the density of states per unit ln(τ)also decreases with ln(τ) These considerations turn outto have important implications for the bounds on suchensembles as shall become apparent below

In Fig 9 we show the constraints on an ensemblecomprising N = 3 unstable particles in which the lightestensemble constituent has a mass m0 = 10 GeV Weconsider this ensemble to be ldquolow-densityrdquo in that it has adensity of states per unit τ which is small throughout therange of parameters shown (This will later be contrastedwith analogous results in Fig 10 for a ldquohigh-densityrdquoensemble) The contours shown in the left and centerpanels of Fig 9 represent upper bounds on the totalextrapolated abundance Ωtot as functions of Mlowast for fixed∆m The contours shown in the left panel correspond tothe choice of ∆m = 15 GeV those in the center panelcorrespond to the choice of ∆m = 95 GeV In the rightpanel we show the corresponding contours as functionsof ∆m for fixed Mlowast = 10195 GeV The blue greenand red curves shown in each panel represent the boundson Ωtot from constraints on the primordial abundancesof D 7Li and 6Li respectively We use a uniformcolor for the D contour mdash a contour which reflects thesum of contributions to δYD from both production anddestruction processes The dotted dashed and solidcurves of each color correspond respectively to the choicesγ = 0 1 2 for the scaling exponent in Eq (74) Thecorresponding black curves shown in each panel representthe bounds arising from limits on distortions of the CMB-photon spectrum obtained by taking whichever bound(either that from δmicro or that from δyC) is more stringentat every point

In interpreting the results shown in Fig 9 we note thateach choice of parameters specifies a particular range oflifetimes τNminus1 le τi le τ0 for the ensemble constituentsThis range of lifetimes varies across the range of Mlowastshown in the left panel from 103 times 104 s le τi le658 times 105 s for Mlowast = 10165 GeV to 103 times 1010 s leτi le 658 times 1011 s for Mlowast = 10195 GeV Across thisentire range of Mlowast values the range of τi values for theensemble is reasonably narrow spanning only roughly a

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

32

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 15 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 95 GeV

15 20 25 30

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 9 Compilation of constraints on electromagnetic injection for ldquolow-densityrdquo ensembles with mass spectra of the formgiven in Eq (71) in which the mass splitting ∆m between successive χi is constant The contours shown correspond to thechoice of N = 3 with m0 = 10 GeV The curves shown in each panel indicate the upper bound on the total abundance Ωtot ofthe ensemble from constraints on the primordial abundances of D (blue curve) 7Li (green curve) and 6Li (red curve) and fromlimits on distortions of the CMB-photon spectrum (black curve) The dotted dashed and solid curves correspond respectivelyto the choices γ = 0 1 2 for the scaling exponent in Eq (74) In the left and center panels we plot this upper bound asa function of the coupling-suppression scale Mlowast with fixed ∆m = 15 GeV and with fixed ∆m = 95 GeV respectively In theright panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV Comparing the results shown in the centerpanel with those of Fig 6 we see that the bounds on a decaying ensemble differ not only quantitatively but even qualitativelyfrom the bounds on its individual constituents

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 158 GeV

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

Δm = 10 GeV

05 10 15 20

-6

-5

-4

-3

-2

log10(ΔmGeV)

log 10(Ω

tot)

M = 10195 GeV

FIG 10 Same as in Fig 9 but for ldquohigh-densityrdquo ensembles with N = 20 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ∆m = 158 and with fixed ∆m = 10 GeVrespectively In the right panel we plot the bound as a function of ∆m with fixed Mlowast = 10195 GeV

single order of magnitude As a result the bounds onΩtot for a decaying ensemble shown in the left panel ofFig 9 as functions of Mlowast closely resemble the boundson Ωχ for a single decaying particle species shown inFig 6 as a function of τχ Moreover since the τi forall particles in the ensemble are roughly comparablethe results are not particularly sensitive to γ whichdetermines how Ωtot is distributed across the ensembleThe most stringent constraint on Ωtot is the boundassociated with the primordial D abundance over almostthe entire range of Mlowast shown The only exception occursin the regime where Mlowast is large and the lifetimes of all of

the ensemble constituents are extremely long in whichcase the constraint associated with CMB distortionssupersedes the D constraint We note that for valuesof Ωtot which lie above the D contour D may be eitheroverabundant or underabundant We also note that aldquofunnelrdquo similar to that appearing in Fig 6 is visiblein the D constraint contours for all values of γ shownin the figure and that this funnel occurs where Mlowast issuch that the lifetimes of the ensemble constituents areapproximately τi asymp 1065 s

By contrast the results shown in the center panel ofFig 9 correspond to the regime in which the range of life-

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

33

times spanned by the ensemble is quite broad extendingover several orders of magnitude in τi In particular therange of lifetimes varies across the range of Mlowast shownin this panel from 103 times 104 s le τi le 822 times 107 s forMlowast = 10175 GeV to 822 times 107 s le τi le 658 times 1011 sfor Mlowast = 10195 GeV We note that the range of Mlowastshown in this panel has been truncated relative to theleft panel in order to ensure that the characteristic decaytimescales for all ensemble constituents occur well afterthe BBN epoch For an ensemble with such a broad rangeof lifetimes the effect of distributing Ωtot over multipleparticle species is readily apparent Indeed the shapes ofthe constraint contours in the center panel of Fig 9 bearlittle resemblance to the shapes of the contours in Fig 6Thus comparing the results shown in the center panel ofFig 9 with those in Fig 6 we see that the bounds ona decaying ensemble indeed differ not only quantitativelybut even qualitatively from the bounds on its individualconstituents

For example two separate funnels arise in the Dconstraint contour for γ = 2 Each of these funnelsrepresents not merely a cancellation between the produc-tion and destruction contributions to YD from a singleparticle but rather a cancellation among the individualcontributions from all constituent particle species inthe ensemble The funnel on the left arises due to acancellation between the net negative contributions toYD from χ1 and χ2 and the net positive contributionfrom χ0 By contrast the funnel on the right arises dueto a cancellation between a net negative contribution toYD from χ2 and positive contributions from χ0 and χ1with the latter contribution suppressed because τ1 asymp1065 s The positions of these funnels then reflectthe collective nature of the ensemble mdash and they haveimportant consequences Indeed within certain regionsof parameter space within which the D constraints areweakened by cancellations among δYD contributions fromdifferent ensemble constituents the constraint associatedwith the 6Li abundance actually represents the leadingbound on Ωtot for the ensemble

Finally in the right panel of Fig 9 we show howthe bounds on Ωtot vary as a function of ∆m for fixedMlowast = 10195 GeV We choose this benchmark for Mlowastbecause the CMB-distortion constraints and the leadingconstraints from light nuclei are comparable for thischoice of Mlowast In interpreting the results displayed inthis panel it is important to note that τ0 is independentof ∆m By contrast the τi for all other χi decrease withincreasing ∆m Since broadly speaking the impact thata decaying particle has on both CMB distortions and onthe primordial abundances of light nuclei tends to de-crease as τi decreases the bounds on decaying ensemblesgenerally grow weaker as ∆m increases This weakeningof the constraints generally grows more pronounced asγ increases and as a larger fraction of the abundanceis carried by the heavier ensemble constituents Forγ = 0 on the other hand the fraction of Ωtot carriedby the lightest ensemble constituent χ0 is independent of

∆m As a result many of the constraint contours shownin the right panel of Fig 9 become essentially flat forsufficiently large ∆m as the lifetimes of the remainingχi become so short that the impact of their decays onthe corresponding cosmological observables is negligiblein comparison with those of χ0

Thus far we have focused on the regime in whichthe spacing between the lifetimes τi of the ensembleconstituents is large mdash in other words the regime inwhich the density of states per unit τ is small across theensemble However it is also interesting to consider theopposite regime in which the density of states per unit τis sufficiently large over the entire range τNminus1 le τ le τ0that the ensemble effectively acts as a continuous sourceof electromagnetic injection over this interval This rangeof lifetimes is completely determined in our toy model byMlowast m0 and the quantity mNminus1 = (N minus 1)∆m + m0Thus we may make a direct comparison between thebounds on such ldquohigh-density ensemblesrdquo and the boundson ldquolow-densityrdquo ensembles with the same τi range byvarying N and ∆m oppositely while holding Mlowast m0and mNminus1 fixed Towards this end in Fig 10 weshow the upper bounds on Ωtot for an ensemble with ahigher density of states per unit lifetime The constraintcontours in the left and center panels of the figurerepresent the same choices of m0 and mNminus1 as in thecorresponding panels of Fig 9 but for N = 20 ratherthan N = 3 For these parameter choices the ensembleis effectively in the ldquohigh-densityrdquo regime over the entirerange of Mlowast shown in each panel In the right panelof Fig 10 we once again show constraint contours asfunctions of ∆m for fixed Mlowast

As one might expect the primary effect of distributingthe abundance of the ensemble more evenly across thesame range of lifetimes is that features in the constraintcontours associated with particular χi are in large partsmoothed out While the D funnels in the left panel arestill present this reflects the fact that the range of τifor the ensemble is sufficiently narrow that distributingthe abundance more democratically across this range haslittle impact on the constraint contours A single Dfunnel also appears in the constraint contour for γ = 2 inthe center panel of Fig 10 The presence of this featurereflects the fact that for γ = 2 the vast majority of Ωtot

is carried by the most massive mdash and therefore mostunstable mdash ensemble constituents Moreover the densityof states per unit lifetime is also much higher for theseshorter-lived states than it is for the rest of the ensembleThe funnel can be understood as corresponding to thevalue of Mlowast for which the lifetimes of these states whichare smaller than but roughly similar to τNminus1 satisfyτi sim τNminus1 asymp 1065 s Once again however we note thatthe location of the funnel is shifted slightly away fromthis value of Mlowast due to the collective contribution to YDfrom the longer-lived constituents in the ensemble Asimilar feature is also apparent in the constraint contourfor γ = 2 in the right panel of Fig 10

We can gain further insight into the results displayed

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

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[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

34

in Fig 10 by examining the continuum limit in which∆m rarr 0 In this limit the sums appearing inEqs (352)ndash(354) and in Eqs (417) and (418) can berecast as integrals over the continuous lifetime variable τmdash integrals which may be evaluated in a straightforwardmanner yielding analytic expressions for δYa δmicro andδyC For example in the case in which the τi span theentire range from tAa to tfa for each relevant nucleus we

find that for any γ ge 0 the term δY(1)a in Eq (351) takes

the form

δY (1)a = AaΞ(tCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

5+2γ6

5 + 2γminus 3xminus

1+γ3

1 + γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

5+2γ6

5 + 2γminus 24xminus

7+4γ12

7 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(76)

where Γ(n z) is the incomplete gamma function where

we have defined

Ξ(t) equiv (γ + 1)ΩtotM2(1+γ)3lowast

3(mγ+1Nminus1 minusm

γ+10

)t(1+γ)3

(77)

where Aa and Ba are defined in Table II and where thelimits of integration are

xminAa = tAatCa xmax

Aa = tBatCaxminBa = tBatXa xmax

Ba = 1xminCa = 1 xmax

Ca = tfatXa (78)

In cases in which τi does not span the entire range fromtAa to tfa the values of the quantities in Eq (78) shouldbe replaced by values which restrict the overall range oflifetimes to max[τNminus1 tAa] le τi le min[τ0 tfa]

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and must be evaluated separately For D the

term δY(2)a in Eq (351) has exactly the same form as

Eq (76) but with Alowasta in place of Aa Blowasta in place of Ba

etc For 6Li we find for all γ ge 0 that the δY(2)a term

takes the form

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ

3minus 2

3 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

17+2γ6

17 + 2γminus 45β4(1 + θ)xminus

7+γ3

7 + γ+

120β3xminus11+2γ

6

11 + 2γminus 3(5minus 3θ)xminus

1+γ3

1 + γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

17+2γ6

17 + 2γminus 45(1 + θ)(1 + 7β4)xminus

7+γ3

7(7 + γ)+

24(1 + 5β3)xminus11+2γ

6

11 + 2γminus 64(9minus 5θ)xminus

7+4γ12

7(7 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(79)

where the limits of integration are given by expressions analogous to those appearing in Eq (78)Expressions for the CMB-distortion parameters δmicro and δyC may be obtained in a similar fashion In particular for

the case in which the τi span the entire range from te to tLS we find that δmicro takes the form

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 2

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

1minus 2γx

1minus2γ6 minus 1

αmicroΓ

(2γ minus 1

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

1minus γx

1minusγ3 minus 3

4αmicroΓ

(γ minus 1

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(710)

for all γ ge 0 except for the special case in which γ = 1 For this special case the replacement

minus 3

1minus γx

1minusγ3 minusrarr lnx (711)

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

35

should be made in the fourth line of Eq (710) Likewise we find that δyC takes the form

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 2

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 1minus2γ

6

6αmicro minus 2γ + 12F1

(1 1 +

1minus 2γ

6αy 2 +

1minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

1minusγ3

1minus γ 2F1

(1γ minus 1

4αy 1 +

γ minus 1

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(712)

where 2F1(a b c z) denotes the ordinary hypergeometricfunction where Aa Ba and αa are defined in Table IIIand where the limits of integration are

xmin0a = tet0a xmax

0a = tBat0axmin

1a = tBat1a xmax1a = tMREt1a

xmin2a = tMREt2a xmax

2a = tLSt2a (713)

Once again in cases in which τi does not span the entirerange from te to tLS the values of the quantities inEq (713) should be replaced by values which restrictthe overall range of lifetimes to max[τNminus1 te] le τi lemin[τ0 tLS]

B Results for Exponentially Rising Mass Splittings

We now consider the results for a toy ensemble with amass spectrum given by Eq (72) The density of statesper unit mass is inversely proportional to m in this caseand the corresponding density of states per unit lifetimeis

nτ =1

3 ln(ξ)τ (714)

which likewise decreases with τ However we note thatthe density of states per unit ln(τ) in this case remainsconstant across the ensemble

In Fig 11 we show the bounds on Ωtot for a ldquolow-densityrdquo ensemble with a mass spectrum given byEq (72) and the same benchmark parameter choicesN = 3 and m0 = 10 GeV as in Fig 9 In the left andcenter panels the value of ξ has been chosen such thatthe range of τi for the ensemble constituents is the sameas in Fig 9 at both ends of the range of Mlowast shownFor the mass spectrum considered here this range of τidepends on m0 on Mlowast and on the quantity ξNminus1 In theright panel of Fig 11 we show how the bounds on Ωtot

vary as a function of ξ for fixed Mlowast = 10195 GeV Thecontours shown in Fig 11 exhibit the same overall scalingbehaviors as those in Fig 9 However there are salientdifferences which reflect the fact that the density of statesper unit ln(τ) is uniform in this case across the ensemble

In Fig 12 we show the corresponding results for anensemble with m0 = 10 GeV and a mass spectrum

again given by Eq (72) but with a sufficiently high nτthroughout the range of Mlowast or ξ shown in each panelthat the ensemble is effectively always within the ldquohigh-densityrdquo regime We find that taking N = 10 is sufficientto achieve this The left and center panels of Fig 12correspond to the same ranges of Mlowast and the same valueof ξNminus1 as the left and center panels of Fig 11 The rightpanel once again shows how the bounds on Ωtot vary asa function of ξ for fixed Mlowast = 10195 GeV Once againwe see that the principal consequence of increasing nτis that constraint contours become increasingly smoothand featureless

Once again as we did for the case of an ensemble with auniform mass splitting we may derive the correspondingexpressions for δYa δmicro and δyC in the case of anexponentially rising mass splitting in the continuum limitby direct integration of the expressions in Eqs (352)ndash(354) and in Eqs (417) and (418) For the massspectrum in Eq (72) this limit corresponds to takingξ rarr 1 For the case in which the τi span the entire rangefrom tAa to tfa for each relevant nucleus we find that for

γ gt 0 the term δY(1)a in Eq (351) takes the form

δY (1)a = AaΞ(tCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xmaxAa

xminAa

+ BaΞ(tXa)

[6βxminus

3+2γ6

3 + 2γminus 3xminus

γ3

γ

]∣∣∣∣∣xmaxBa

xminBa

+ BaΞ(tXa)

[6(1 + β)xminus

3+2γ6

3 + 2γminus 24xminus

3+4γ12

3 + 4γ

]∣∣∣∣∣xmaxCa

xminCa

(715)

where the limits of integration are defined as in Eq (78)and where we have defined

Ξ(t) equiv γΩtotM2γ3lowast

3(mγNminus1 minusm

γ0

)tγ3

(716)

By contrast for the special case in which γ = 0 thereplacement

minus 3xminusγ3

γminusrarr lnx (717)

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

36

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 2

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 447

04 06 08 10 12

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 11 Same as in Fig 9 but for ldquolow-densityrdquo ensembles with mass spectra of the form given in Eq (72) in which themasses of the χi are evenly distributed on a logarithmic scale In the left and center panels we plot the upper bound on Ωtot

as a function of the coupling-suppression scale Mlowast with fixed ξ = 2 and with fixed ξ = 447 respectively In the right panelwe plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

165 170 175 180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 12

180 185 190 195

-6

-5

-4

-3

-2

log10(MGeV)

log 10(Ω

tot)

ξ = 139

010 015 020 025

-6

-5

-4

-3

-2

log10(ξ)

log 10(Ω

tot)

M = 10195 GeV

FIG 12 Same as in Fig 11 but for ldquohigh-densityrdquo ensembles with N = 10 In the left and center panels we plot the upperbound on Ωtot as a function of the coupling-suppression scale Mlowast with fixed ξ = 12 and with fixed ξ = 139 respectively Inthe right panel we plot the bound as a function of ξ with fixed Mlowast = 10195 GeV

should be made in the second term in square brackets

appearing in the second line of Eq (715) while Ξ(t) isgiven not by Eq (716) but rather by

Ξ(t) equiv Ωtot

3 ln(mNminus1m0) (718)

For D and 6Li the term δY(2)a in Eq (351) is non-

vanishing and once again must be evaluated separately

For D the term δY(2)a in Eq (351) has exactly that

same form as Eq (715) but with Alowasta in place of Aa Blowastain place of Ba etc For 6Li we find that the δY

(2)a term

takes the form

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

37

δY (2)a = AlowastaΞ(tlowastCa)Γ

(γ3minus 1 xminus1

) ∣∣∣∣∣xlowastmaxAa

xlowastminAa

+ BlowastaΞ(tlowastXa)

[72β5θxminus

15+2γ6

15 + 2γminus 45β4(1 + θ)xminus

6+γ3

6 + γ+

120β3xminus9+2γ

6

9 + 2γminus 3(5minus 3θ)xminus

γ3

γ

]∣∣∣∣∣xlowastmaxBa

xlowastminBa

+ BlowastaΞ(tlowastXa)

[8θ(1 + 9β5)xminus

15+2γ6

15 + 2γminus 45(1 + θ)(1 + 7β4)xminus

6+γ3

7(6 + γ)+

24(1 + 5β3)xminus9+2γ

6

9 + 2γminus 64(9minus 5θ)xminus

3+4γ12

7(3 + 4γ)

]∣∣∣∣∣xlowastmaxCa

xlowastminCa

(719)

where the limits of integration are given by expressions analogous to those appearing in Eq (78) Once again as was

the case with δY(1)a the expression for δY

(2)a in Eq (719) requires modification for the special case in which γ = 0

In particular the replacement specified in Eq (717) should likewise be made in the last term in square brackets on

the second line of Eq (719) Moreover for γ = 0 the quantity Ξ(t) is given by Eq (718) rather than by Eq (716)The corresponding expressions for δmicro and δyC for the case in which the τi span the entire range from te to tLS can

likewise be computed by direct integration In particular we find that δmicro is given by

δmicro =4Amicro

5Ξ(t0micro)Γ

(4γ minus 6

15 xminus54

) ∣∣∣∣∣xmax0micro

xmin0micro

+BmicroΞ(t1micro)

[6

3minus 2γx

3minus2γ6 minus 1

αmicroΓ

(2γ minus 3

6αmicro xminusαmicro

)]∣∣∣∣∣xmax1micro

xmin1micro

+BmicroΞ(t2micro)

[3

2minus γx

2minusγ3 minus 3

4αmicroΓ

(γ minus 2

4αmicro xminus

4αmicro3

)]∣∣∣∣∣xmax2micro

xmin2micro

(720)

for all γ ge 0 except for the special case in which γ = 2 For this special case the replacement

minus 3

2minus γx

2minusγ3 minusrarr lnx (721)

should be made in the fourth line of Eq (720) Likewise we find that δyC is given by

δyC =4Ay

5Ξ(t0y) Γ

(4γ minus 6

15 xminus54

)∣∣∣∣xmax0y

xmin0y

+ByΞ(t1y)6xα+ 3minus2γ

6

6αmicro minus 2γ + 32F1

(1 1 +

3minus 2γ

6αy 2 +

3minus 2γ

6αyminusxαy

)∣∣∣∣xmax1y

xmin1y

+ByΞ(t2y)3x

2minusγ3

2minus γ 2F1

(1γ minus 2

4αy 1 +

γ minus 2

4αyminusxminus

4αy3

)∣∣∣∣xmax2y

xmin2y

(722)

The limits of integration in Eqs (720) and (722) are

defined as in Eq (78) and Ξ(t) is defined as in Eq (716)for γ gt 0 and as in Eq (718) for γ = 0 Once again wenote that in cases in which the τi do not span the relevantrange of timescales during which particle decays canaffect one of these cosmological observables the limitsof integration in Eqs (715)ndash(722) should be replacedby the values which restrict the overall range of lifetimes

appropriately

In summary the results shown in Figs 9ndash12 illus-trate some of the ways in which the constraints oninjection from unstable-particle decays can be modifiedin scenarios in which the injected energy density isdistributed across an ensemble of particles with a rangeof lifetimes These results also illustrate that a non-trivial interplay between the contributions from different

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

38

decaying particle species within the ensemble can haveunexpected and potentially dramatic effects on the upperbound on Ωtot mdash as when cancellations among positiveand negative contributions to δYD from a broad rangeof particles within the ensemble result in a significantweakening of this upper bound Thus we see thatthe bounds on a decaying ensemble can exhibit collectiveproperties and behaviors that transcend those associatedwith the decays of its individual constituents

VIII CONCLUSIONS DISCUSSION ANDSUMMARY OF RESULTS

In this paper we have considered ensembles of unstableparticle species and investigated the cosmological con-straints which may be placed on such ensembles due tolimits on electromagnetic injection since the conclusionof the BBN epoch Indeed as we have discussed suchinjection has the potential to modify the primordialabundances of light nuclei established during BBN Suchinjection can also give rise to spectral distortions inthe CMB alter the ionization history of the universeand leave observable imprints in the diffuse photonbackground For each of these individual considerationswe have presented an approximate analytic formulationfor the corresponding constraint which may be appliedto generic ensembles of particles with lifetimes spanninga broad range of timescales 102 s τi tnow For easeof reference these analytic approximations along withthe corresponding equation numbers are compiled in Ta-ble IV In deriving these results we have taken advantageof certain linear and uniform-decay approximations Wehave also demonstrated how these results can be appliedwithin the context of toy scenarios in which the massspectrum for the decaying ensemble takes one of twocharacteristic forms realized in certain commonly-studiedextensions of the SM

Several comments are in order First it is worth notingthat while the values of the parameters in Table II werederived assuming a particular set of initial comovingnumber densities Ya for the relevant nuclei at the endof the BBN epoch the corresponding parameter valuesfor different sets of initial comoving number densities Y primeamay be obtained in a straightforward manner from thevalues in Table II Provided that Y primea and Ya are notsignificantly different the characteristic timescales tBatCa and tXa for each process are to a good approximationunchanged and a shift in the initial comoving numberdensities can be compensated by an appropriate rescalingof the relevant normalization parameters Aa and Ba Forexample a shift in the initial helium mass fraction at theend of BBN can be compensated by a rescaling of thefit parameters AlowastD and BlowastD associated with D productionand of the parameters A4He and B4He associated with4He destruction by Y primepYp as well as a rescaling of the

parameters Alowast6Liand Blowast6Li

associated with secondary 6Li

production by (Y primepYp)2 The quadratic dependence on

Y primepYp exhibited by the last of these rescaling factorsreflects the fact that the reactions through which thenon-thermal populations of 3He and T nuclei are initiallyproduced and the reactions through which these nuclei inturn contribute to 6Li production both involve 4He in theinitial state Similarly a shift in the initial 7Li abundancecan be compensated by a rescaling of the fit parametersA7Li and B7Li associated with 7Li destruction and ofthe parameters A6Li and B6Li associated with primary6Li production by Y prime7Li

Y7Li while a shift in the initialD abundance can be compensated by a rescaling of theparameters AD and BD associated with D destruction byY primeDYD

Second we remark that in this paper we have fo-cused primarily on constraints related to electromagneticinjection In situations in which a significant fractionof the energy liberated during unstable particle decaysis released in the form of hadrons the limits obtainedfrom bounds on the primordial abundances of light nucleimay differ significantly from those we have derived hereDetailed analyses of the limits on hadronic injection inthe case of a single decaying particle species have beenperformed by a number of authors [4 76ndash78] It wouldbe interesting to generalize these results to the case of adecaying ensemble as well

Third as we have seen there are two fundamentallydifferent timescale regimes in which the correspondingphysics is subject to very different leading boundsan ldquoearlyrdquo regime 102 s ltsim tinj

ltsim 1012 s and a ldquolaterdquo

regime tinjgtsim 1012 s This distinction is particularly

important when considering the implications of ourresults within the context of the Dynamical Dark Matterframework [23 24] Within this framework the morestable particle species within the ensemble provide thedominant contribution to the dark-matter abundance atpresent time while the species with shorter lifetimeshave largely decayed away However the masses decaywidths abundances etc of the constituents of a realisticDDM ensemble are governed by the same underlying setof scaling relations Thus in principle one might hope toestablish bounds on the ensemble as a whole within theDDM framework by constraining the properties of thoseensemble constituents which decay on timescales τ tLS

mdash ie ldquoearlyrdquo timescales on which the stringent limitson CMB distortions and the primordial abundances oflight nuclei can be brought to bear

In practice however it turns out that these consid-erations have little power to constrain most realisticDDM scenarios On the one hand both the lifetimesτi and extrapolated abundances Ωi of the individualensemble constituents typically decrease monotonicallywith i in such scenarios mdash either exponentially [79]or as a power law [24 80 81] On the other handas indicated in Fig 8 the constraint associated withthe broadening of the surface of last scattering dueto ionization from particle decays becomes increasinglystringent as the lifetime of the particle decreases downto around τi sim tLS This implies that for an ensemble

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

Phys Rev D 80 043526 (2009) [arXiv09061197 [astro-phCO]]

[11] D P Finkbeiner S Galli T Lin and T R SlatyerPhys Rev D 85 043522 (2012) [arXiv11096322 [astro-phCO]]

[12] T R Slatyer Phys Rev D 87 123513 (2013)[arXiv12110283 [astro-phCO]]

[13] R Khatri and R A Sunyaev JCAP 1206 038 (2012)[arXiv12032601 [astro-phCO]]

[14] T Banks D B Kaplan and A E Nelson Phys Rev D49 779 (1994) [hep-ph9308292]

[15] B de Carlos J A Casas F Quevedo and E RouletPhys Lett B 318 447 (1993) [hep-ph9308325]

[16] T Banks M Berkooz and P J Steinhardt Phys RevD 52 705 (1995) [hep-th9501053]

[17] E Witten Phys Lett 149B 351 (1984)[18] P Svrcek and E Witten JHEP 0606 051 (2006) [hep-

th0605206][19] A Arvanitaki S Dimopoulos S Dubovsky N Kaloper

and J March-Russell Phys Rev D 81 123530 (2010)[arXiv09054720 [hep-th]]

[20] G D Coughlan W Fischler E W Kolb S Raby andG G Ross Phys Lett 131B 59 (1983)

[21] N Arkani-Hamed T Cohen R T DrsquoAgnolo A HookH D Kim and D Pinner Phys Rev Lett 117 251801(2016) [arXiv160706821 [hep-ph]]

[22] T Cohen R T DrsquoAgnolo and M Low arXiv180802031[hep-ph]

39

Constraint Analytic formulation

Primordial abundances of light nucleiAnalytic approximations for δYa given inEq (351) with the corresponding limits andparameter values given in Table II

Spectral distortions in the CMBAnalytic approximation for δmicro and δyC given inEqs (417) and (418) with the parameter valuesgiven in Table III and limits given in Eq (41)

Ionization history of the universeConstraint specified by Eq (54) with the valuesof ΓIH and tIH given in Eq (53)

Diffuse extra-galactic photon backgroundDifferential signal flux given by Eq (69)with differential background fluxes given inEqs (610)ndash(612)

TABLE IV Summarycompilation of our main results

of unstable particles in which the collective abundance ofthe cosmologically stable constituents is around ΩDM asymp026 the Ωi for those constituents with lifetimes τi tLS are constrained to be extremely small Thus theconstraints associated with CMB distortions and withthe primordial abundances of light nuclei in the ldquoearlyrdquoregime are not particularly relevant for constraining theproperties of DDM ensembles in which the abundancesscale with lifetimes in this way Nevertheless we notethat in ensembles in which τi and Ωi do not increasemonotonically with i (such as those in Ref [81]) theconstraints associated with these considerations couldindeed be relevant This is currently under study [82]

Acknowledgments

The research activities of KRD are supported in partby the Department of Energy under Grant DE-FG02-

13ER41976 (de-sc0009913) and by the National ScienceFoundation through its employee IRD program The re-search activities of JK are supported in part by NationalScience Foundation CAREER Grant PHY-1250573 Theresearch activities of PS are supported in part by theDepartment of Energy under Grant de-sc0010504 inpart by the Vetenskapsradet (Swedish Research Council)through contract No 638-2013-8993 and the Oskar KleinCentre for Cosmoparticle Physics and in part by theDepartment of Energy under Grant DE-SC007859 andthe LCTP at the University of Michigan The researchactivities of BT are supported in part by National ScienceFoundation Grant PHY-1720430 The opinions andconclusions expressed herein are those of the authorsand do not represent any funding agencies

[1] M Kawasaki and T Moroi Astrophys J 452 506(1995) [astro-ph9412055]

[2] S Sarkar Rept Prog Phys 59 1493 (1996) [hep-ph9602260]

[3] R H Cyburt J R Ellis B D Fields and K A OlivePhys Rev D 67 103521 (2003) [astro-ph0211258]

[4] R H Cyburt J Ellis B D Fields F LuoK A Olive and V C Spanos JCAP 0910 021 (2009)[arXiv09075003 [astro-phCO]]

[5] M Kawasaki K Kohri T Moroi and Y Takaesu PhysRev D 97 023502 (2018) [arXiv170901211 [hep-ph]]

[6] W Hu and J Silk Phys Rev D 48 485 (1993)[7] W Hu and J Silk Phys Rev Lett 70 2661 (1993)[8] J A Adams S Sarkar and D W Sciama Mon Not

Roy Astron Soc 301 210 (1998) [astro-ph9805108][9] X L Chen and M Kamionkowski Phys Rev D 70

043502 (2004) [astro-ph0310473][10] T R Slatyer N Padmanabhan and D P Finkbeiner

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