Physics Letters B 701 (2011) 530–534

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Physics Letters B

www.elsevier.com/locate/physletb

Constraining decaying dark matter

Ran Huo

Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 May 2011Accepted 19 June 2011Available online 22 June 2011Editor: S. Dodelson

Keywords:Dark matterCosmic microwave backgroundStructure formation

We revisited the decaying dark matter (DDM) model, in which one collisionless particle decays earlyinto two collisionless particles, that are potentially dark matter particles today. The effect of DDM willbe manifested in the cosmic microwave background (CMB) and structure formation. With a systematicmodification of CMB calculation tool camb, we can numerically calculate this effect, and compare itto observations. Further Markov Chain Monte Carlo cosmomc runnings update the constraints in thatmodel: the free streaming length λFS � 0.5 Mpc for nonrelativistic decay, and ( MDDM

keV Y )2 Tdyr � 5 × 10−5

for relativistic decay.© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Decaying dark matter (DDM) in which the decay process hap-pens at an early stage of the universe is natural for many models,and it is sometimes introduced as a way to adjust the DM relicdensity, because after decay the DM relic density will naively belowered by

Ωd =∑

mproduct

moriginalΩDDM. (1)

One example is in [1], in which gravitino, overproduced by reheat-ing after inflation, will decay into the true lightest supersymmetricparticles (LSP) axino as well as an axion.

However, pure gravitational constraints for that decay processare less understood and sometimes even simply ignored. Actuallythe model in [1] with their parameters should be ruled out [2].Here we will present a model independent computation, in whichthe effect is calculated from the first principle, and can be com-pared directly with cosmological observation. Our model have boththe parent particle and the daughter particles interacting veryweakly, so the effect can only be manifested in gravitational effect,such as the cosmic microwave background (CMB) and the largescale structure formation. Our approach is based on a systematicmodification of the CMB codes.

2. Principle

cmbfast [3] and camb [4] are CMB calculation tools whichare based on the photon line of sight integration technique, in-

E-mail address: huor@uchicago.edu.

0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2011.06.053

stead of solving the Boltzmann equation explicitly. They work inthe synchronous gauge, the metric perturbation of which is gaugedcompletely into the spatial 3 × 3 part

gμν = a2(−1

δi j + hij

), (2)

and the graviton hij can be decomposed into Fourier modes hand η

hij(�x) =∫

d3k

(2π)32

ei�k·�x(

kik jh(�k) +(

kik j − 1

3δi j

)6η(�k)

). (3)

Then linearized Einstein equation gives the equations of motionof h and η components [5]

k2η − 1

2

a

ah = 4πGa2δT 0

0 , (4a)

k2η = 4πGa2(ρ + p)θ, (4b)

h + 2a

ah − 2k2η = −8πGa2δT i

i , (4c)

h + 6η + 2a

a(h + 6η) − 2k2η = −24πGa2(ρ + p)σ . (4d)

Here overdot · means derivative to conformal time τ . θ is thepeculiar velocity which is defined by (ρ + p)θ ≡ ikiδT 0

i , and σ

is the shear which is defined by (ρ + p)σ ≡ −(kik j − 13 δi j)Σ

ij =

−(kik j − 13 δi j)(T i

j − 13 δi

j T kk ).

With the h and η metric perturbation, the Boltzmann equationin terms of the fractional perturbation Ψ is [5]

∂Ψ + iqk

(k · n)Ψ + ∂ ln f (q)(η − h + 6η

(k · n)2)

∂τ ε ∂ ln q 2

R. Huo / Physics Letters B 701 (2011) 530–534 531

= 1

f (q)

(∂( f + δ f )

∂τ

)C, (5)

where q = ap is the comoving momentum and is conserved inexpansion if the particle is collisionless, ε = √

q2 + a2m2 is thecomoving energy, and n is the direction of the macroscopic flowof the fluid. f (q) is the unperturbed partition function, and thereal partition function can be defined with fractional perturba-tion f (q, τ , xi,ni) = f (q)(1 + Ψ (q, τ , xi,ni)). Usually f (q) is ther-mal distribution such as Fermi–Dirac distribution or Bose–Einsteindistribution, but in our DDM model for daughter particles it is de-termined by the decay process.

After we plug into the collision term, we will find the formalphoton line-of-sight integration solution to the Boltzmann equa-tion. The anisotropy �T ≡ δT

T∼ 1

4 Ψγ today is given by [3]

�T (�k, n) =τ0∫

0

dτeikμ(τ−τ0)eκ

[(η − αμ2k2)

+ κ

[�T 0 + μve − 1

2P2(μ)(�T 2 + �P 0 + �P 2)

]],

(6)

where the optical depth is κ ≡ − ∫ τ0τ dτ ′κ(τ ′) < 0 and the differ-

ential optical depth is κ = aneσT , which is the common factor forall collision terms. Here ne is the number density of free electronsin coordinate space and σT = 6.65 × 10−25 cm2 is the Thomson

cross section. μ ≡ (k · n), α ≡ h+6η2k2 and ve is the electron velocity.

Suffix P means polarization mode and integer suffix labels multi-poles of spherical harmonics.

In DDM model, metric perturbations h and η get significantcontribution from the daughter particles. The perturbation evolu-tion of the daughter particle has the same series with the massiveneutrino

Ψ0 = −qk

εΨ1 + 1

6h

d ln f

d ln q, (7a)

Ψ1 = qk

ε

(Ψ0 − 2

3Ψ2

), (7b)

Ψ2 = qk

ε

(2

5Ψ1 − 3

5Ψ3

)−

(1

15h + 2

5η

)d ln f

d ln q, (7c)

Ψ� = qk

ε

1

2� + 1

(�Ψ�−1 − (� + 1)Ψ�+1

), � � 3, (7d)

Ψ� = qk

εΨ�−1 + � + 1

τΨ�, as truncation. (7e)

The right-hand side source term of Eq. (4) have contributions onlyfrom the first three perturbation modes (δρ, δp ∝ Ψ0, θ ∝ Ψ1 andσ ∝ Ψ2). Because all particle species such as baryon, cold darkmatter (CDM), photon, massless as well as massive neutrino talkto gravity, all their evolution will be modified.

The codes also calculate the transfer functions (TF) as an inter-mediate step in the CMB calculations, which is an indication of thelarge scale structure. The TF is by definition the normalized ratioof perturbation growth factor, from an very early stage to certainlate stage, the normalization is taken with a very large scale whichis out of horizon in the whole evolution

T (k) ≡ δ(k, t f )/δ(k, ti)

δ(k → 0, t f )/δ(k → 0, ti). (8)

3. Modification

We introduce free parameter Ωd which corresponds to thedaughter particles’ energy density today, while still keeping the

nondecay CDM part Ωc . In this way we can treat any combina-tion of decaying and nondecay dark matter. For convenience touse Eq. (1), Ωd is not the whole energy density but only the masscontribution to energy density, namely the kinetic energy of thedaughter particle is not included. Therefore we should have at leastone massive daughter particle for this parameterization. Except foran extreme relativistic decay, the kinetic energy contribution issmall and Ωd represents the energy density very well. We onlyconsider the one-to-two decay process and introduce two mass ra-tios

mp1mo

andmp2mo

, where mp1 and mp2 are separately the masses oftwo product particles and mo is the mass of original particle. Thelast free parameter is the decay lifetime Td . So the complete set ofnew parameters includes Ωd , Td ,

mp1mo

andmp2mo

.Let us go through what will be modified in our DDM model,

compared with the standard �CDM universe. First, the decay pro-cess will affect the expansion of the universe, through changingthe equation of state. Before decay the DDM behaves as the CDMwith a constant equation of state ω = 0, while after decay it doesnot hold and the momentum and energy of daughter particle issubject to redshift, as what happens to massive neutrino. Sincebefore decay the DDM particle can be approximated as being atrest, given the masses of the parent and daughter particles the ini-tial transverse momentum pT is fixed for a two body decay. Thecomoving momentum for each individual daughter particle is de-termined only by the scale factor a∗ at which the decay happens(in this Letter we will always use a ∗ to denote the quantity rightat decay). As the scale factor a(τ ) grows the physical momentump(τ ) decreases, while preserving

q = a(τ )p(τ ) = a∗pT . (9)

With this relation the energy density and pressure can be evalu-ated numerically for product particles in the modification, so is theexpansion process.

The decay is a continuous process for the set of DDM particles,the way for our numerical study is to discretize it into 30 ∼ 60channels, by which we can achieve 0.5% precision for CMB peakheight. Each channel corresponds to daughter particles produced ina small scale factor region a∗ ∼ a∗ +da∗ and has its own Ψ� series.As the universe expands the channels are gradually filled chan-nel by channel in order. The perturbation evolution is described byEq. (7), the only subtlety comes through the factor d ln f

d ln q : the un-perturbed distribution f (q) is no longer the Fermi–Dirac distribu-tion of neutrino, but determined by the decay process. A numberof product particles proportional to dΩ will be redistributed intoq space d3q = 4πq2 dq = 4π p3

T a∗2 da∗ , so up to some factor theunperturbed partition function from decay is

f ≡ dn

d3 p∝ dΩ

d3q= e

− tTd dt

Td

4π p3T a∗2 da∗ = e

− tTd

4π Td p3T a∗a∗ , (10)

and d ln fd ln q can be calculated

d ln f

d ln q= − a∗2

Tda∗ − 3

2+ 3p∗

2ρ∗ . (11)

The last issue for the perturbation evolution differential equa-tion set is the initial condition. The initial values of all perturbationmodes should naturally inherit the values before decay, which forDDM they are the same as CDM and all higher multipoles vanish.So we have

Ψ ∗0 = δCDM, (12)

Ψ ∗� = 0, � � 1. (13)

Including the daughter particles’ contribution to metric pertur-bation finishes our modification.

532 R. Huo / Physics Letters B 701 (2011) 530–534

Fig. 1. CMB anisotropy C T T� , C E E

� , C T E� modes and z = 0 total matter TF for various DDM model. The “�CDM” model (black) has the best fit cosmological parameters which

apply implicitly to the following models. The “Fiducial DDM” model (red) has Ωd = 0.226 completely replacing CDM, two mass ratiosmp1mo

= 0.1 andmp2mo

= 0, and decay

lifetime Td = 109 s. Then we separately vary Ωd to be half of the fiducial value while the other half is still the CDM (yellow), mass ratiomp1mo

= 0.01 (brown) and decay

lifetime Td = 1010 s (pink). We also show an example of very “Relativistic DDM” model (blue), which has Ωd = 10−5 and CDM component the same as the �CDM model, twomass ratios

mp1mo

= 10−5 (so that ΩDDM = 1) andmp2mo

= 0, and decay lifetime Td = 109 s. We then vary Ωd to be 10−4 (cyan) andmp1mo

to be 10−6 (green). (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this Letter.)

4. Sample calculation

Generally speaking, because DDM induces no modification ofthe photon–electron–baryon plasma, the CMB anisotropy spectrumis only directly affected by the Sachs–Wolfe (SW) effect throughmetric perturbation. Relativistic particles contribute more thanmassive particles to the peculiar velocity and the shear, becausethey are not suppressed by v

c or v2

c2 factors. So the SW effect willbe enhanced dominantly by the relativistic decay products, and theacoustic peaks will raise and move to higher multipoles �, as whathappens to a common SW effect.

As for the matter power spectrum, structure formation requiresDM particles to condensate into clumps, in order to amplify thedensity perturbation. Therefore the DM particles must be mov-ing slowly to be gravitationally captured, just like a slow incidentcomet will be captured by the sun to form an elliptic orbit but notthe fast moving one with a hyperbolic orbit. If DDM is dominantand nondecay CDM doesn’t exist or is negligible, after decay onsmall scale the free streaming effect will prevent structures fromgrowing, and the TF will be much lower than the nondecay case. Ifthere is still sizable CDM, the small scale power is not completelyerased, but only get smaller.

R. Huo / Physics Letters B 701 (2011) 530–534 533

Fig. 2. Decay lifetime and mass ratio regions for nonrelativistic decay scenario, where the DM is pure DDM and one product is massless (left) or two products haveidentical masses (right). The black gridded regions which contain 95% MCMC points (red dots) correspond to λFS � 0.49 Mpc (left) or λFS � 0.50 Mpc (right), and the greengridded regions which contain 99.7% points correspond to λFS � 0.72 Mpc (left) or λFS � 0.73 Mpc (right), comparing to the yellow gridded regions of previous constraintλFS � 1 Mpc [8]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

Fig. 3. Decay lifetime and ΩDDM region for relativistic decay scenario. Here we take h2 = 0.5. The black line which contains 95% MCMC points (red dots) is the constraint(

MDDMkeV Y )2 Td

yr � 5.3 × 10−5, and the green line which contains 99.7% points is (MDDM

keV Y )2 Tdyr � 1.5 × 10−4, comparing to the yellow line of previous constraint (

MDDMkeV Y )2 Td

yr �7.5 × 10−2 [9]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

For certain decay lifetime between nucleosynthesis and recom-bination, there are two different approaches which may make theDDM model work. One is that the decay may happens to the dom-inant part of the dark matter, but the daughter particle has massclose to the parent particle and is very nonrelativistic, at least afterredshift at the late recombination and structure formation epoch,so that the decay effect is minimized. The other is that the daugh-

ter particle is light and can be all the way relativistic, but only atiny part of the dark matter today comes from it, so that its effectis constrained. We will separately call them nonrelativistic scenarioand relativistic scenario. A combination of the two works for sure,but their individual effects are primarily interesting.

We present our sample calculations in Fig. 1. As expected, aswe go to more DDM component, larger mass hierarchy and longer

534 R. Huo / Physics Letters B 701 (2011) 530–534

decay lifetime, we see greater SW effect and greater free stream-ing effect. The TF is also calculated in [6]. In models with no CDMwe see very suppressed TF on small scales, and the oscillationstructure of which is given by baryon acoustic oscillation. The twovariants of “Relativistic DDM” model coincide nearly exactly, whichimplies in that limit only ΩDDM matters.

5. Parameter constraint

We have further run our modified camb as calculation tool forthe Markov Chain Monte Carlo (MCMC) program cosmomc [7] forcosmological parameter evaluation. We are equipped with all cur-rent public data except for the SDSS DR8: 4 CMB data sets whichare WMAP 7 years, ACBAR, CBI and BOOMERANG [10]; 3 matterpower spectrum data sets which are SDSS DR7 LRG, SDSS DR4and 2dFGRS [11], as well as supernova data. We are constrainedto minimal �CDM plus DDM model, which is a flat universe ofΩk = 0, no hot dark matter Ων = 0, standard cosmological con-stant w = −1, no tensor mode r = 0 and no spectral index runningnrun = 0. The channels we used are CMB, HST, mpk, BBN, AgeTophat Prior and SN, which are consistent with that model andindependent from each other. In addition to the 4 decay param-eters there are 7 other parameters subject to Monte Carlo: Ωbh2,Ωch2, θ , zrei, ns , log A, A S Z .

Although a running for more general parameter space is pos-sible, limited by our computing facility we are still working inthe nonrelativistic scenario and the relativistic scenario, withoutexploring the intermediate region. First we consider the nonrela-tivistic decay scenario. We further constrain that all dark matterundergoes decay, which trade Ωd into Ωc . Then we focus on somecertain relation between two daughter particles’ mass for furthersimplification. Corresponding to [1] we do the case that one prod-uct is massless, the other one which we focused on is that the twoproducts have the same mass. The results are shown in Fig. 2.

We find the allowed region can be described concisely by thefree streaming length constraint λFS � 0.5 Mpc at 95% confidencelevel. Here the free streaming length is defined as the length mea-sured today, where a DDM particle decays exactly at its expecta-tion lifetime Td , and the daughter particle travels before matter-radiation equality. Normalized to a0 = 1, we have

λFS ≡τeq∫

τd

v(τ )dτ =aeq∫

ad

q√m2a2 + q2

da√8πG

3 (ρm0a + ρr0)

. (14)

In the identical products’ masses case the free streaming length isthe same for the two particles; and in the one massless productparticle case it is defined for the massive particle, since it con-tributes to the structure formation. In fact we also checked theintermediate case such as one daughter particle is three timesheavier than the other, in which the free streaming length cannotbe defined without ambiguity, and the numerical allowed region isa reasonable average of the two particles’ regions corresponding tothe same free streaming length at the same confidence level.

Then we consider the relativistic decay scenario. Here we useEq. (1) to trade Ωd into ΩDDM of parent particle if it doesn’t de-cay, by doing so we are able to explore a large hierarchy regionof relativistic mass ratio, the exact value of which has little effectas shown in the previous sample calculations. Another way for ex-pression is ΩDDMh2 = 274.2 MDDM

keV Y , where Y ≡ nDDMs and nDDM and

s are separately the DDM number and entropy densities.

The allowed region can be analytically expressed as ( MDDMkeV Y )2 ×

Tdyr � 5 × 10−5 at 95% confidence level, which is shown in Fig. 3.Thanks to the high precision WMAP data, it improves the previousone by 3 orders of magnitude.

6. Discussion

We see no supporting evidence to introduce the DDM, minimal�CDM is still among the best fit. We are not using the small scalesuch as the halo structure data, so we do not face the small scalestructure problem of [8]. Here the scale of structure discrepancy isfurther constrained to be below 0.5 Mpc.

From high energy physics model building perspective, our re-sults put interesting constraint on the gauge mediated supersym-metry breaking (GMSB) scenario in general, in which gravitino isusually a very light LSP. Generically in that kind of model thegravitino mass is in hierarchy with all the other LSP candidate,and combined with large reheating temperature after inflation itis overproduced, probably by many orders. Our results covers allthe gravitino related decay process which is suppressed by thePlanck scale, where gravitino is either the parent particle or oneof the daughter particle. It is very hard to introduce a decay pro-cess which can solve the so-called “gravitino problem” and doesnot contradicts with observation, and the constraint is done in themost inevitable way, that the process can even escape electromag-netic and hadronic constraints.

Going beyond the gravitino model in GMSB, this constraint canstill judge other exotic models which satisfy collisionless daugh-ter particles condition. Moreover, we would argue that this puregravitational bound should apply to any DDM model even with in-teractions other than gravitation, since gravitational effect shouldalways be the weakest.

Acknowledgements

RH wish to thank Carlos E.M. Wagner, Wayne Hu, Tower Wang,Jonathan Feng, Haibo Yu for useful discussion and information. RHis supported partially by Robert R. McCormick Fellowship of theUniversity of Chicago.

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