Extragalactic and galactic gamma rays and neutrinos from annihilating dark matter

Rouzbeh Allahverdi,1 Sheldon Campbell,2 and Bhaskar Dutta2

1Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA2Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA

(Received 16 November 2011; published 3 February 2012)

We describe cosmic gamma-ray and neutrino signals of dark matter annihilation, explaining how the

complementarity of these signals provides additional information that, if observable, can enlighten the

particle nature of dark matter. This is discussed in the context of exploiting the separate galactic and

extragalactic components of the signal, using the spherical halo model distribution of dark matter. We

motivate the discussion with supersymmetric extensions of the standard model of particle physics. We

consider the minimal supersymmetric standard model (MSSM) where both neutrinos and gamma-rays are

produced from annihilations. We also consider a gauged B� L, baryon number minus lepton number,

extension of the MSSM, where annihilation can be purely to heavy right-handed neutrinos. We compare

the galactic and extragalactic components of these signals, and conclude that it is not yet clear which may

dominate when looking out of the galactic plane. To answer this question, we must have an understanding

of the contribution of halo substructure to the annihilation signals. We find that different theories with

indistinguishable gamma-ray signals can be distinguished in the neutrino signal. Gamma-ray annihilation

signals are difficult to observe from the galactic center, due to abundant astrophysical sources; but

annihilation neutrinos from there would not be so hidden, if they can be observed over the atmospheric

neutrinos produced by cosmic rays.

DOI: 10.1103/PhysRevD.85.035004 PACS numbers: 98.70.Rz, 12.60.Jv, 95.35.+d, 95.85.Ry

I. INTRODUCTION

Understanding the particle nature of dark matter is amajor problem of modern physics. While indirect gravita-tional evidence of its presence is plentiful, unambiguousidentification of its particle properties is actively sought.Presumably, this population of particles is a relic that wasproduced spontaneously in, and is left over from, the bigbang. It is common for extensions of the standard modelthat contain a dark matter candidate to have a Z2 symmetrythat stabilizes the particle from decay, but does not preventthe particle from self-annihilating. Often in viable theories,the annihilations in the early Universe are important tobring the dark matter’s relic density down to the observeddensity today. Once the rate of expansion of the Universebecomes larger than the dark matter annihilation rate,annihilations become rare, and the dark matter relic issaid to freeze out at this time. However, if dark matterannihilates, then rare annihilations continue to occur today,predominantly in the densest regions of the Universe.Unambiguous identification of cosmic radiation from theseannihilations would not only provide valuable informationabout the particle nature of dark matter, but also informa-tion about the distribution of the matter responsible for thesignal.

The information about dark matter from indirect detec-tion of its cosmic annihilation radiation is complementaryto that gleaned from other current experiments. Indirectdetection experiments constrain the dark matter particlemass, annihilation cross section, and annihilation spec-trum. Meanwhile, direct detection experiments attempt to

observe dark matter-nucleon interactions in a laboratorydetector, and constrain the dark matter mass and itsnucleon-scattering cross section. Particle accelerators tryto detect dark matter production from particle collisions,where the dark matter would be manifested in missingtransverse energy. Again, events of this kind provide in-formation on the dark matter’s particle mass, but also onthe processes that led to the creation of the dark matter.The results from these experiments provide different con-straints for particle physics models, and provide consis-tency checks for one another.Today, indirect detection experiments are looking for

high-energy cosmic rays, gamma rays, and neutrinos pro-duced from dark matter annihilation. The propagation ofproduced cosmic rays within the galaxy is difficult todescribe precisely, complicating the prediction of thesignal’s properties. Also, energy losses do not allow extra-galactic cosmic rays to reach us. However, neutrinos andgamma-rays can be observed from their sources, bothgalactic and extragalactic. This allows their observed sig-nals to probe not only the particle physics that producedthem, but also the distribution of their sources: the matterin our galaxy or extragalactic large-scale structure. Asample of current experiments is the Fermi-LAT [1], cur-rently examining cosmic gamma-rays, and IceCube [2],which is already monitoring high-energy cosmic neutrinos.In their mandates, they both have commitments to analyzetheir data for the presence of dark matter annihilationradiation [3,4]. In this paper, we consider particularmodels from the minimal supersymmetric standardmodel (MSSM) where both neutrinos and gamma-rays

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are produced from annihilations; however, we also discussdifferent well-motivated models where neutrinos may bethe dominant final states. These scenarios would be dis-tinguishable from one another if information were receivedabout dark matter annihilation from observations of bothcosmic gamma-rays and neutrinos.

Constraints on dark matter annihilation exist from ob-servations of gamma-rays from dwarf galaxies and neutri-nos from the Sun. It is apparent from the lack of an obvioussignal from those observations that any existing indirectsignal is faint. Successful observation of this radiation, andproper constraints from the observations, will require pre-cise knowledge of other astrophysical radiation sources,and the development of rigorous predictions of the annihi-lation signal for given realistic models of particle physicsand matter distribution. These predictions will be instru-mental in understanding which kinds of models are and arenot observable, and what constraints will be determined bya given designed experiment. In addition, this research isimportant for the understanding of precisely what infor-mation is available in a given particle model’s signal, ifobserved by a particular experiment. This will not only aidthe analyses of current experiments, but will also informthe design of future experiments.

Implementing the dark matter distribution in terms ofsemi-analytic models allows the basic properties of thedensest regions (the cores of halos) to be well-represented.It is reasonable to understand the predictions produced bythe simplest models first, learning about the physical scalesmost important to the signals. Additional features of thedistribution, as seen in simulations, can be implemented inthe context of these models, and their effects quantified—for example, different models of halo core densities, im-plementation of halo substructure, implementation of dis-tributions of more complex halo shapes, more general halostatistics, etc. It is useful to identify the relevant physicalscales that determine the main properties of the annihila-tion signals. This is an important tool to quantify therobustness of the predictions to uncertainties of ourknowledge of large-scale structures, and tells us aboutthe constraints available to be gained from the experi-ments. This will also guide our understanding of the anni-hilation signals calculated directly from simulation data, asin [5].

We share preliminary results of predictions of the inten-sity spectrum of annihilation gamma rays and neutrinosproduced from within and outside the galaxy for a spheri-cal halo model distribution. In Sec. II, we describe some ofthe particle physics models that motivate our discussion.We explain the calculation of the observed spectrum ofgamma rays from annihilations and show results of thecalculations in Sec. III. Of particular interest is how differ-ent theories can produce the same gamma-ray spectra, eventhough they have different annihilation modes. We showcorresponding results for neutrinos in Sec. IV and discuss

reasons for their consideration in addition to gamma-rays,such as how they break the degeneracy in spectrum fordifferent models. We provide discussion of the results inSec. V.

II. PARTICLE PHYSICS MODELS

In this paper, we discuss particle models that are super-symmetric extensions of the standard model. In thesemodels, R-parity, or some similar parity property, allowsonly an even number of supersymmetric partner particlesto interact on a fundamental interaction vertex. Thisstabilizes the lightest supersymmetric particle, whichbecomes the dark matter candidate.

A. MSSM

In the MSSM, the particle content is restricted to thestandard model, a supersymmetric partner for each stan-dard model degree of freedom, and additional Higgs fields.The supersymmetric charged higgsinos and gauginos mixto produce mass eigenstates called charginos, and theneutral higgsinos and gauginos produce neutralinos. Inaddition to providing a dark matter mass candidate, thismodel also stabilizes radiative corrections to the Higgsmass, and causes unification of the standard model forcesat an energy called the grand unified theory (GUT)scale.The most general allowed parameter space for this

model has more than 100 free parameters. We will restrictour discussion to a small portion of these. In minimalsupergravity (mSUGRA) or the constrained MSSM(CMSSM), supersymmetric masses are unified at theGUT scale with scalar masses having value m0, and m1=2

being the mass of the gauginos [6]. Two Higgs fields eachgain vacuum expectation values, the ratio of which isspecified with the value tan�. We will describe the prop-erties of viable thermally produced dark matter modelswith vanishing soft supersymmetry-breaking trilinear cou-pling parameters A0, and positive mass parameter � thatcouples the two Higgs superfields in the superpotential. Wewill be focusing here on universal masses that are not solarge as to result in a dark matter particle massive enoughto produce significant top quarks from annihilations.In this three-dimensional parameter space of m0, m1=2,

and tan�, the parameter space is typically broken up intofour main regions: the bulk region, the focus point (alsoknown as hyperbolic) region, the co-annihilation region,and the funnel region. In these regions, the dark matterparticle turns out to be the lightest neutralino ~�0

1.

In the bulk region, bothm0 andm1=2 are relatively small.

The neutralino is nearly pure bino (the gauginowhich is thesupersymmetric partner of the weak hypercharge gaugeboson), and annihilates predominantly to bottom anti-bottom quark pairs b �b, secondarily to tau anti-tau leptonpairs �þ�� (more so at larger tan�). These processes in the

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bulk region give the correct annihilation cross sectionto account for the relic density, if it were thermallyproduced.

Generically, larger values of m0 and m1=2 result in

theories with larger mass dark matter that have smallerannihilation cross sections, and therefore would result inmore thermally produced dark matter in the Universe thanis observed today. However, when considered carefully, wesee other parameter space does result in the correct relicdensity, due to different mechanisms [7], according to theparameter space of interest.

The focus point region [8] has a branch where m1=2

remains small and m0 is allowed to increase. As m0 doesso, the lightest neutralino gains a larger Higgsino compo-nent, which opens up additional annihilation channels.Here, annihilation dominantly produces WþW� bosons,with a small branching fraction also producing ZZ bosonpairs for small to moderate tan�. For large tan�, theHiggsino component of the lightest neutralino is againsmall in this region, but Bino annihilation is enhanced byan increased coupling to the pseudoscalar Higgs A andannihilation is again dominated by b �b and �þ��. Someparts of this parameter space have restrictions from theresults of the CMS [9] and ATLAS [10] experiments at theLarge Hadron Collider.

There is a threshold where m0 becomes too small andone of the supersymmetric partners of the tau (stau ~�)becomes the lightest supersymmetric particle, which iselectrically charged and therefore cosmologically disal-lowed. This threshold increases with m1=2. Near this

boundary, the ~� mass is only slightly larger than the ~�01

mass, enhancing the co-annihilation interaction cross sec-tion between these particles. The ~�’s present in the earlyUniverse co-annihilate with the ~�0

1’s, and reduces the

neutralino density to the correct value. This parameterspace is the stau-neutralino co-annihilation region [11].When A0 > 0, there is parameter space at low m1=2 where

a supersymmetric partner of the top quark (stop ~t) becomeslighter than ~�0

1. The stop-neutralino co-annihilation region

[12] is near this boundary. In these parameter spaces, ~�01 is

again nearly pure bino and mostly b �b and some �þ�� areproduced from annihilations. Because there are no ~� or ~tparticles present today, they no longer contribute to anni-hilations and the effective annihilation cross section ofthe neutralinos is reduced from its value at freezeout.Additionally, at low tan�, annihilation is dominated byt-channel sfermion exchange, which is helicity-suppressed[13]. The presence of a strong p-wave annihilation com-ponent brings the annihilation cross section up to itsneeded value at freezeout, but slow relative motions ofthe particles today do not allow the p-wave to contribute.In these cases, the annihilation cross sections are quitesmall, which make the rates of annihilations low and theintensity of annihilation radiation much more difficult todetect. The situation becomes better at large tan� where

annihilation via A is a stronger component, lifting much ofthe helicity suppression.The final parameter space, the heavy Higgs or A anni-

hilation funnel regions, occurs where the mass of one of theHiggs bosons is near half the ~�0

1 mass, resulting in a Breit-

Wigner resonance enhancement of the annihilations atfreezeout interaction energies [14]. Since the resonancedoes not enhance the cross section today, the annihilationcross sections are again lower in the present epoch fordark matter models of this parameter space.

B. Gauged Uð1ÞB�L Model

Another paradigm we wish to discuss, which is interest-ing in the context of neutrino radiation production, is theUð1ÞB�L extension of the MSSM [15]. Here, baryonnumber B minus lepton number L is a gauged chargewith associated gauge boson Z0 that couples to baryonsand leptons, according to their B� L charges with gaugecoupling g0. This extension requires the presence of right-handed neutrinos Nc for anomaly cancellation, providing anatural framework to explain neutrino masses and oscilla-tions. In order for this new internal symmetry to be sponta-neously broken, we must introduce two new Higgssuperfields H0

1 and H02, standard model neutral and oppo-

sitely charged under B� L for anomaly cancellation. Theyare coupled by a new mass parameter �0 in a new termadded to the MSSM superpotential. The physical neutrinos� are light, but Nc heavy, by the type I see-saw mechanism[16]. This requires a Majorana mass for the Nc, which doesnot obey the B� L symmetry; however, the Nc can have aYukawa coupling to another a Higgs field with leptonnumber �2, which we identify with H0

2. This Higgs willgain a vacuum expectation value around 1 TeV, producingthe Nc Majorana mass and generating the appropriateneutrino spectrum. Thus, by defining supersymmetricpartners for each of the introduced new fields and puttingthem in chiral supermultiplets, the minimal Uð1ÞB�L

extension to the MSSM has superpotential [17]

W ¼ WMSSM þ yDNcHuLþ fH0

2NcNc þ�0H0

1H02; (1)

where Hu is the Higgs superfield of the MSSM that givesmass to the up-type quarks, and L is the superfield con-taining the left-handed leptons. Note that flavor and theweak isospin SUð2ÞL indices have been suppressed.There exists parameter space in this framework where

the LSP is a supersymmetric partner of Nc, the right-handed sneutrino ~N. If the Nc mass is less than the ~Nmass, then annihilations could produce a large number ofNc, which would then decay according to the particularmodel considered. In any case, one would expect manydirect neutrinos to be produced, while photons would onlybe produced secondarily.The particular model we will consider will be a parame-

ter space where the ~N has a mass of 150 GeV. In thiscase, the dominant annihilation channels are the s-wave

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processes ~N ~N ! NcNc and ~N� ~N� ! Nc�Nc� viat-channel exchange of B� L neutralinos through its cou-pling with the gaugino ~Z0. The Nc, taken to have mass135 GeV, then decay exclusively to � and standard modelHiggs h, which we took to have mass mh ¼ 120 GeV. Atthis mass, the Higgs decays mostly toWW� bosons, and tob �b quarks, each of which produce secondary photons andneutrinos. We will also discuss cases where ~N is heavier.

III. GAMMA-RAY RESULTS

The production of gamma-rays due to extragalactic darkmatter annihilation is estimated using the halo modeldistribution of dark matter used in [18], where each halois specified by its mass M and observed redshift z. TheSheth-Tormen halo mass function dn

dM [19] is a good ap-

proximation of the halo distributions seen in simulations.We take typical halos to be spherical, with a Navarro-Frenk-White (NFW) density profile [20]

�hðrjM; zÞ ¼ �sðM; zÞ½r=rsðM; zÞ�½1þ r=rsðM; zÞ�2

extending out to virial radius

RvirðM; zÞ ¼�

3M

4��virh�iðzÞ�1=3

with �vir ¼ 180 and h�iðzÞ being the background matterdensity at redshift z. The distribution of scale radii rs is

described by the distribution of halo concentrations c �Rvir

rs, which is approximately described by the physically

motivated model of [21] over the mass scales probed bysimulations. We set the minimum halo mass scale at10�6M�. Cosmological parameters used were from theWMAP7 data [22], and we use the linear power spectrumof [23] to describe large-scale fluctuations in the matterdistribution. These inputs describe the simulations wellenough to give us a good approximation of the annihilationsignal. However, as we will establish, the signal (producedfrom dark matter distributed according to our halo model)is sensitive to low-mass halo properties and halo coreproperties, which are beyond the reach of current simula-tions, and the knowledge of which will require a goodunderstanding of interactions with baryonic matter. Weconsider all photons emitted since the epoch of re-ionization, which we estimate to have occurred at redshiftzmax ¼ 10.

Given this description of the densest regions of large-scale structure, the mean extragalactic intensity of gamma-rays from the annihilation of dark matter particles, each ofmass m, at constant s-wave relative-velocity-weighted an-nihilation cross section �v with annihilation spectrumdN

dEis

I;EGðEÞ ¼ �vZ dz

HðzÞWðð1þ zÞE; zÞh�2iðzÞ: (2)

The intensity window function is

WðE; zÞ ¼ 1

8�m2

1

ð1þ zÞ3dN

dE

ðEÞe��ðE;zÞ; (3)

where �ðE; zÞ is the cosmic opacity to gamma-rays [24],

and the mean square matter density is determined from

h�2iðzÞ ¼Z

dMdn

dMðM; zÞ

Zd3r�2

hðrjM; zÞ: (4)

Note that it is possible that �v depends on the relativevelocity of the annihilating particles. The most prominentexample of this, which appears in particle models, is thepresence of p-wave annihilation [25]. However, this anni-hilation component must be very large to affect the ob-served annihilation signal, and thermal freeze-out modelsfor which this is true have very small cross sections and aredifficult to see [18]. Since, in this paper, we are focusing onmodels where dark matter is thermally produced, it isreasonable to restrict our initial investigations to pures-wave annihilation. However, if a special case scenario,such as resonant annihilation [26,27] were present, then thecalculation needs to be modified, as described in [18].If we take our own Milky Way Galaxy dark matter halo

to be a typical halo of our large-scale structure model atmass MG ¼ 2� 1012M�, then it has scale radius rs;G ¼38:0 kpc, virial radius Rvir;G ¼ 412 kpc, and concentrationcG ¼ 10:8. The important parameter here for our calcula-tion is the scale radius, since the contribution to the anni-hilation signal due to dark matter outside this radius is verysmall; therefore, the virial radius definition (and hence thevalue of concentration) does not significantly affect theprediction of the galactic annihilation signal. We estimatethe solar system’s position in the halo as being R� ¼8:0 kpc from the galactic center.With this description, the intensity of gamma-rays, due

to dark matter annihilation in the galactic halo in thedirection of angle c from the galactic center (assumedcoincident with the halo center), is typically written as

I;GðE; c Þ ¼ �v

8�m2

dN

dE

ðEÞJðc Þ; (5)

where the J-factor is the line of sight integration of thesquare dark matter density from the solar system outthe halo [28]

Jðc Þ�Z rmaxðc Þ

0dr

��h

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2�2rR�cosc þR2�

qjMG;0

��2

(6)

with

rmaxðc Þ ¼ R� cosc þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2vir;G � R2�sin2c

q: (7)

In Fig. 1, we see the contributions of the galactic andextragalactic components of annihilation to the gamma-rayintensity for different lines of sight in the halo. The particle

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physics model used in this example is the focus pointregion in mSUGRA parameter space where the dark matterparticle is the lightest neutralino having mass m ¼150 GeV at tan� ¼ 10, A0 ¼ 0, and sign�> 0. Here,annihilation is predominantly to WþW� boson pairs. Theparticle annihilation cross section and annihilation spec-trum were calculated using the computer programDarkSUSY 5.0.5 [29]. The cross section for this model is�v ¼ 1:9� 10�26 cm3=s.

In our dark matter density distribution models, the ga-lactic component is dominant at the peak of the signalwhen looking toward the galactic center, but the contribu-tions of the components are comparable when looking outof the galactic plane or away from the galactic center. It isconceivable that with slightly different choices of distribu-tion parameters, the relative importance of each may bealtered considerably. The relative strength of the galactic toextragalactic intensity at a given photon energy is

I;EGðEÞI;GðE; c Þ ¼

Zdz

� h�2iðzÞHðzÞð1þ zÞ3Jðc Þ

�

��dN

dEðð1þ zÞEÞdN

dEðEÞ

�e��ðð1þzÞE;zÞ:

The important parameters then appear in the first factor ofthe integrand. Figure 2 plots the extragalactic and galactic

contributions to this factor in units of �2c=H0, where �c is

the cosmological critical density to collapse and H0 is theHubble constant. The extragalactic part is relatively flat inscale, with an area under the curve of around48 000�2

c=H0. The convolution with the annihilation spec-trum and opacity could modify the importance of thisfactor, depending on the details of those functions. Onemay wonder what mass scale of halos most contributes tothe mean square density h�2i. In Fig. 3, we can see that themass integrand goes very nearly likeM�1 all the way to themaximum mass scale, suggesting that all mass scalescontribute nearly equally to the intensity. If the massdependence of the Sheth-Tormen mass function correctlydescribes the halo distribution down to low scales, and

0 2 4 6 8 10z

4400

4600

4800

5000

5200

2 z

H z 1 z 3

c2

H0

0 30 60 90 120 150 180103

104

105

106

J c2

H0

FIG. 2. Left: The magnitude of extragalactic intensity is approximately proportional to the area under this curve, around48 000�2

c=H0. Right: The corresponding contribution to the galactic intensity.

10 610 3 1 103 106 109 1012 1015M M10 12

10 910 610 3

1103106109

nM

3r h2 c

2

M

FIG. 3. The mass integrand of the mean square density atredshift z ¼ 0, 0.5, and 1 from bottom to top, respectively.

10 2 10 1 100 101 102E GeV10 14

10 1310 1210 1110 1010 9

E2I GeV cm2 s sr

10 2 10 1 100 101 102E GeV10 14

10 1310 1210 1110 1010 9

E2I GeV cm2 s sr

10 2 10 1 100 101 102E GeV10 14

10 1310 1210 1110 1010 9

E2I GeV cm2 s sr

FIG. 1 (color online). The gamma-ray signal from annihilating dark matter in the directions c ¼ 30�, 90�, and 150� from thegalactic center, respectively. The dark matter shown here is a 150 GeV neutralino in the focus point region of mSUGRA withtan� ¼ 10. The dotted (blue online) line is the extragalactic component, the dot-dashed (red online) line is the galactic component.The solid line is the net signal.

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those low-mass halos have density profiles well-describedby NFW, with concentrations described by the modelspecified, then all mass scales are important contributorsto annihilations.

However, let us consider for the moment the effect of theneglected substructure. At Milky Way size halos, it isexpected that substructure will increase the annihilationrate by a factor on the order of 100, depending on theminimum halo mass scale [30]. By definition, the smallesthalos will not have any subhalos, and larger halos will havemore and more substructure. Thus, one would expect thelargest halos to contribute the most to intensity purely onthe basis of their substructure.

For the galactic contribution, if our galactic halo is well-described by an NFW (or similar) profile, then the value ofscale radius rs has a significant effect on how concentratedthe dark matter is to the galactic core. Based on observa-tions of stellar velocities, it is generally estimated that ourgalactic halo has a somewhat smaller scale radius than thetypical radius we used [31]. This would result in an in-crease in the predicted galactic intensity.

The scaling of the density at the core is also important.On the right-hand plot of Fig. 2, we see how the intensityformally diverges as the line of sight approaches the galac-tic center for the NFW profile. Observing a signal fromtoward the galactic center would help to better understandhow the density is distributed there in our halo, and wouldallow us to test various ideas about the effects that thecentral black hole and baryonic cooling have on the profile.

It is expected that the substructure observed in thesimulations would increase the galactic signal by a factorof a few, not as significantly as the extragalactic intensity[30]. Therefore, it is not unreasonable to suppose that theextragalactic annihilation could dominate over the galacticsignal for most lines-of-sight that are not too close to thegalactic center.

In summary, our estimation of the most crucial elementsin these calculations, which have the greatest effects on theresult, is:

(i) the halo scale radius, the galactic value of which hasan important effect on the galactic signal component,

and the halo distribution of which affects the extra-galactic signal; and

(ii) the inclusion of subhalos, not yet taken into account,will also increase the predicted signal, and willdepend on the scale of minimum halo mass.

Thus, one can conclude from this discussion that thegalactic and extragalactic annihilation signals in Fig. 1are of comparable intensity, due to our value of rs;G, andthe lack of substructure effects.Keeping in mind these uncertainties, it is still interesting

to compare these calculations to the experimental mea-surements. The extragalactic signal for this model peaks atE2I � 10�9 GeV=cm2=s=sr while the extracted extraga-

lactic gamma-rays reported by the Fermi Gamma-raySpace Telescope is E2

I � 5� 10�7 GeV=cm2=s=sr at

that energy [32]. At higher dark matter particle masses mwith the same annihilation operators, the gamma-ray peakenergy increases proportional to m, but the intensity Idecreases like m�3 (two factors from the number density,one from the annihilation spectrum). However, the extra-galactic background intensity is measured to drop moreslowly, consistent with a power law scaling E�2:41

.

Unless the annihilation at the galactic core is very bright,it will be difficult to observe those dark matter annihilationgamma-rays originating from there because there are somany other bright sources of astrophysical gamma-rays inthat region that have theoretical uncertainties associatedwith them. A less contaminated signal, for example, wouldbe the consideration of the mean annihilation signal awayfrom the core. The galactic and extragalactic componentsfor this are shown in Fig. 4 for the same focus point model.For comparison, we also show total signals for dark matter,of the same mass, that annihilates toWþW�, b �b, or �þ��,at the same annihilation cross section as the focus pointmodel. The annihilation spectra for these models weresimulated with the event generator Pythia 6.135 [33]. Thesources of photons in these models are from decayingpions or radiating charged fermions. The W and b spectraare more dispersed to lower energies because they aremore likely to decay to hadronic showers where each

10 2 10 1 100 101 102E GeV10 14

10 1310 1210 1110 1010 9

E2I GeV cm2 s sr

10 2 10 1 100 101 102E GeV10 14

10 1310 1210 1110 1010 9

E2I GeV cm2 s sr

W W

bb

FIG. 4 (color online). Left: The mean intensity for the focus point model, averaged over all directions c > 18� away from thegalactic center. The plot format is the same as for Fig. 1. Right: The same calculation for a 150 GeV dark matter particle thatannihilates purely to WþW�, b �b, or �þ��.

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photon-emitting product is at lower energy. At 150 GeVdark matter annihilation, the photons from annihilation toWþW� are indistinguishable from annihilation to b �b.These pure branching ratio intensities can be used to con-struct the intensity profile for any theory that annihilates tothese states, with known branching ratios. For larger darkmatter masses, the W and b signals become more distin-guishable from one another.

IV. NEUTRINO RESULTS

Because the models discussed in the previous section

also contribute a neutrino annihilation spectrum dN�

dE�, it is

interesting to consider this component of the signal as well.Because the neutrino is electrically neutral and weaklyinteracting, it also propagates relatively freely throughthe cosmos, and the annihilation signal will have bothgalactic and extragalactic contributions. This calculationis completely analogous to that for the gamma-ray signal.We neglect any cosmic opacity for the neutrinos in thesample calculations that follow.

In Fig. 5, we show the galactic, extragalactic, and netintensity of cosmic neutrinos from annihilations of thesame 150 GeV focus point neutralino dark matter consid-ered in the previous section. In the galactic signal, weclearly see the peaks from primarily and secondarilyproduced neutrinos from the W decays. However, thosefeatures are washed out in the redshift-modulatedextragalactic signal. We note how both components con-tribute significantly to the total signal in all of the shown

lines of sight. Again, reasonable adjustments of dark mat-ter distribution parameters and consideration of halo sub-structures could significantly alter this balance in eitherdirection.Although the neutrino signal still suffers from uncer-

tainties in the galactic core density profile, it does notsuffer from the same astrophysical contamination as dogamma-rays. Therefore, there is no reason to exclude thegalactic center in these experiments. In fact, if a neutrinodetector with high angular resolution can be developed, itis a good strategy to focus on the galactic center.Figure 6 shows the neutrino signal for the focus point

model averaged over the whole sky, directions away fromthe core, and directions focused on the core, respectively.We see how the galactic signal is seen to dominate thesignal at the galactic core if we assume the NFW profileholds to the center, and we neglect extragalactic sub-structures. The same dominance of the galactic core occurswith annihilation gamma-rays, but it is very difficult to seethose photons from the noisy center of the galaxy. Furtherwork, with more realistic distributions, should better elu-cidate the situation at the galactic core, and provide anunderstanding of the information about the dark matterdistribution uncertainties that may be available in anobserved neutrino signal.It is common in the literature to express neutrino signals

as binned detector event rates per detector mass. If hI�i�is the mean annihilation intensity in a solid angle �of observation, the event rate for a neutrino �f of flavor

f ¼ e, �, or � in an energy bin Ei < E� < Eiþ1 is

10 2 10 1 100 101 102E GeV

10 11

10 10

10 9

10 8E2I GeV cm2 s sr

10 2 10 1 100 101 102E GeV

10 11

10 10

10 9

10 8E2I GeV cm2 s sr

10 2 10 1 100 101 102E GeV

10 11

10 10

10 9

10 8E2I GeV cm2 s sr

FIG. 5 (color online). The neutrino signal from annihilating dark matter in the directions c ¼ 30�, 90�, and 150� from the galacticcenter, for the same particle model and plot format as in Fig. 1.

10 2 10 1 100 101 102E GeV

10 1110 1010 910 810 710 6

E2I GeV cm2 s sr

10 2 10 1 100 101 102E GeV

10 1110 1010 910 810 710 6

E2I GeV cm2 s sr

10 2 10 1 100 101 102E GeV

10 1110 1010 910 810 710 6

E2I GeV cm2 s sr

FIG. 6 (color online). The mean neutrino intensity for the focus point model. Left: The all-sky intensity, 0� < c < 180�. Middle:The anticore intensity, c > 18�. Right: The core intensity, c < 5�. The plot format is the same as in Fig. 1.

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R�f;i ¼NA�

nm

Z Eiþ1

Ei

dE� ��fNðE�Þ hI�i�ðE�Þ;

where NA is Avogadro’s number, nm is the molar mass ofthe detector material, and ��fN is the neutrino-nucleon

charged current scattering cross section [34]. Note thatNA=nm is simply the nucleon number per detector mass.To ease conversion of the results for different detectormaterials, we show the results for nm ¼ 1 g=mol.

Figure 7 shows the neutrino event rates for annihilationinto W bosons, b quarks, or � leptons. The logarithmicGeV energy bin size used is

� ¼ log10

�Eiþ1

Ei

�¼ 0:04:

At these neutrino energies, the electron and muon neutri-nos have indistinguishable nucleon-scattering cross sec-tions, which are larger than that for the tau neutrinos.Hence, the tau neutrino event rates are a little smaller.

Since � leptons always decay to a primary neutrino,while W bosons only decay directly to leptons some ofthe time, the � production from �’s is more intense. The bquarks do not produce primary neutrinos, and only have alower energy neutrino spectrum from secondary chains.Thus, the flux of neutrinos from annihilations breaks thedegeneracy between annihilation into WþW� and b �b thatoccurred in the gamma-ray signal.

Another class of models that results in interesting phe-nomenology for dark-matter-annihilation neutrinos is theUð1ÞB�L extension of the MSSM, described in Sec. II B, inthe parameter spaces where the sneutrino ~N is the darkmatter particle. The neutrino detector rates for one ex-ample model are shown in Fig. 8. Here, the 150 GeV ~Nannihilates into 120 GeV right-handed neutrinos Nc thatdecay to a standard model neutrino and Higgs boson, thelatter of which decays mostly to b’s and �’s. The ~Nannihilation in this model does have a slight p-wave

component, and the s-wave cross section is �v ¼ 1:1�10�26 cm3=s, giving the correct thermal dark matter relicdensity.The secondary neutrinos produced from the Higgs decay

result in a broad, soft spectrum, whereas the neutrinosproduced directly from Nc decays produce a narrowerpeak at lower energies on the order of the mass differencebetween the Nc and the higgs. Because of the Higgsdecays, there is also a gamma-ray component to the signal.In the case where the ~N dark matter is heavier (larger

than twice the Higgs mass), and the Nc mass still slightlysmaller than it, then the physical neutrino peak occurscloser to the dark matter mass energy. This will producea hard spectrum with narrow peak from the primary neu-trinos, and broad low-energy tail produced by the Higgsdecays.Another intriguing scenario occurs when the dark matter

annihilates solely to two light neutrinos �. In the context ofthe B� L model previously described, this corresponds tothe limit where the Higgs mass is small, negligible com-pared to the ~N mass, and the mass difference between ~Nand Nc is also very small. Then the spectrum of theproduced light neutrinos is at the energy of the ~N, andthe width of the spectrum is very small. This simplescenario results in a prominent neutrino line feature withno corresponding gamma-ray observations. At this energyscale of neutrino energies, the dominant astrophysicalsource is atmospheric neutrinos. The solid lines in Fig. 9show the detector rates for annihilation of 150 GeV darkmatter particles into prompt neutrinos, to each flavorequally, with a cross section of �v ¼ 1:1� 10�26 cm3=swith our modeled dark matter distribution. The upper lineshows the electron flavor rates and muon flavor rates. Thelower line is the tau flavor detection rate. Shown is themean all-sky signal (0� < c < 180�), an anticore signal(c > 18�), a core signal (c < 5�), and an inner core signal(c < 1�). The width of the spectral line feature is due tothe velocity distribution of dark matter in the galactic halo,which is negligible compared to the energy resolution ofviable detectors. Therefore, it is completely contained in

1005020 30 15070E GeV

10 8

10 7

R kton 1 yr 1

FIG. 7 (color online). All-sky neutrino plus antineutrino de-tection rates for 150 GeV dark matter annihilation. The thicklines are for electron or muon flavor, and the thin lines show thetau flavor rate. At 70 Gev, the top two lines (green online) are forannihilation to �þ�� leptons, the middle two lines (blue online)show annihilation to WþW� bosons, and the bottom two lines(red online) are for annihilation to b �b quarks.

1005020 30 15070E GeV

10 11

10 10

10 9

10 8

10 7

R kton 1 yr 1

FIG. 8. All-sky neutrino plus antineutrino event rates for150 GeV sneutrino dark matter that annihilates to two135 GeV right-handed neutrinos (each flavor equally repre-sented), each of which decays to a light neutrino and 120 GeVstandard model Higgs particle.

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the energy bin at the dark matter mass. The diffuse com-ponent is due to the redshifted extragalactic neutrinos.The dotted lines in the figure are the predicted meanatmospheric neutrino rates, as would be seen at theKamioka site during minimum solar activity [35]. Theupper line shows the �� þ ��� rates, and the lower line

shows the �e þ ��e rates.By comparing them with the previous neutrino rate

plots, we see that the typical diffuse signals are well belowthe current measured atmospheric neutrino rates. Again,the situation likely improves with the consideration of halosubstructure, and the background can also be reduced withrespect to the signal by focusing on a nearby dark-matter-dense region of space, as we shall discuss for the promptneutrino production example.

The prominence of the peak at the galactic core showshow a neutrino detector with high angular resolution mayextract a spectral line feature by focusing on a dense regionof space. Although the signal to background ratio improveswith small solid angles of observation, the detection ratesbecome forbiddingly small. With better energy resolution,an experiment can also gain a stronger signal in the spectralline scenario. Thinner energy bins have a higher spectralline height. The energy bin width at E� ¼ m requiredfor the bin height to be at the corresponding atmo-spheric neutrino rate when observing in solid angle � isapproximately

�Eðm;�Þ � 1

�

Gðm;�ÞIatmðmÞ � IEGðmÞ ;

where Gðm;�Þ is the flux of galactic annihilation neu-trinos of energy at the dark matter particle mass m origi-nating within the solid angle �, IatmðmÞ is the meanintensity of atmospheric neutrinos of energy m, andIEGðmÞ is the mean extragalactic annihilation neutrinointensity. The corresponding required logarithmic binwidth is (assuming �E m)

� � �E

m ln10:

This approximate logarithmic energy bin size is shownin Fig. 10 for ranges of the dark matter mass, and fordifferent solid angles centered on the galactic center. Forcomparison, Fig. 9 used � ¼ 0:04. We see the energyscales where the spectral line is most hindered by theatmospheric neutrinos. At high dark matter mass, the elec-tron neutrinos are much easier to see, since the electronatmospheric neutrinos are much less abundant than themuon atmospheric neutrinos.The neutrino annihilation signal is also complementary

to the detection of neutrinos from dark matter annihilationsin the Sun. While the galactic and extragalactic signalsdepend on the dark matter’s self-annihilation cross section,the solar annihilation signal is primarily dependent on the

1005020 30 15070E GeV

10 14

10 11

10 8

10 5

0.01

R kton 1 yr 1

1005020 30 15070E GeV

10 14

10 11

10 8

10 5

0.01

R kton 1 yr 1

1005020 30 15070E GeV

10 16

10 13

10 10

10 7

10 4

R kton 1 yr 1

1005020 30 15070E GeV

10 18

10 15

10 12

10 9

10 6

R kton 1 yr 1

FIG. 9. Neutrino plus antineutrino event rates for 150 GeV dark matter annihilating to 2 prompt neutrinos � with cross section�v ¼ 1:1� 10�26 cm3=s (solid lines), shown with the mean atmospheric neutrino plus antineutrino rates at the Kamioka site duringlow solar activity (dotted lines). For the atmospheric neutrinos, the upper line is the muon flavor, and the lower line is the electronflavor. For the annihilation neutrinos, the upper line shows the rate for electron flavor, as well as the rate for muon flavor. The lower lineshows the rates for �� þ ���. Upper left: the mean neutrino rates from the whole sky. Upper right: rates when excluding the galacticcore, c > 18�. Lower left: rates when focused on the galactic core, c < 5�. Lower right: rates when focused on the inner galacticcore, c < 1�.

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elastic nucleon-scattering cross section, provided that theannihilation rate and capture rate of dark matter particle bythe Sun are in equilibrium. This fact means that it is stillpossible to probe models for which the elastic scatteringcross section is too low for annihilations to be observedfrom the Sun. For example, in the context of B� Lmodelswith ~N dark matter, this occurs when there is a small masssplitting between the real and imaginary parts of ~N, and thedark matter becomes the lighter of the two [17].

V. DISCUSSION

The possibility of indirect detection of dark matter an-nihilation through astrophysical observation of its productsis an idea that improves our understanding of the particlenature of dark matter. It can already provide constraints onthe s-wave part of its annihilation cross section for a rangeof particle masses that annihilate according to some speci-fied annihilation spectrum and assumptions about the darkmatter distribution. These dark matter constraints are notavailable from any other kind of experiment.

Observation of annihilation products would pro-vide valuable information about the dark matter self-interactions and distribution. The precise nature of theinformation that would be available to be extracted isdependent on the details of the dark matter properties,and on the nature of the data collected.

In this paper, we explored how the simultaneous obser-vation of annihilation gamma-rays and neutrinos allows formore constrained conclusions of the dark matter propertiesand distribution. We gave sample calculations of the galac-tic and extragalactic contributions to signals from differentparticle physics models using a simple smooth halo modelof dark matter distribution. It is interesting to considergamma-rays and neutrinos because the direction of theirsource is constrained, and their signal contains contribu-tions from both galactic and extragalactic sources.Observing these signals would improve the ability to re-construct the particles produced in the annihilations, andthe dark matter mass, by breaking degeneracies that existin one signal. For example, dark matter particles of mass150 GeV that annihilate directly intoWþW� bosons or b �bquarks were seen to produce very similar gamma-ray spec-tra, but those different cases could be distinguishable in

observations of cosmic neutrinos. For larger dark mattermass, the gamma-ray signals from annihilations intoWþW� will be more distinguishable from annihilationsinto b �b.It is also possible for theories to be dominant in one kind

of signal, and therefore undetectable in other channels. Wediscussed scenarios where annihilation could be neutrino-dominant. One intriguing case where this can happen is ifdark matter is made up of supersymmetric partners ofneutrinos, as we showed in the context of a low scaleB� L gauge theory, which can account for narrow spectralpeaks in the cosmic neutrinos.Once information about the dark matter particle mass is

obtained, this theoretically constrains the Jeans mass scaleresponsible for the dark matter halo minimum mass.According to the halo model, the extragalactic signal isproduced nearly democratically by all present scales ofhalo masses, when halo substructure is neglected. Sincesubstructure has the most significant effect on large halos,one would predict the extragalactic signal to be dominatedby the substructure of the most massive halos, which inturn is dependent on the scale of minimum halo mass.From this analysis, we find that the magnitude of the

annihilation products would provide information about theannihilation cross section, the concentration of subhaloswithin the most massive halos, and the density profile oftheir cores. The degeneracies that these quantities have onthe intensity could be broken by information in a signalextracted verifiably from galactic sources, if enough inde-pendent observations can sufficiently constrain the darkmatter distribution at those sources, perhaps via the angulardistribution of the galactic signal to extract informationabout a universal core density profile. Neutrinos may bemuch more suited than gamma-rays for observing a signalfrom the dense galactic core, since there are fewer astro-physical sources of neutrinos at the relevant energies orig-inating from that region.It is evident that interpretation of observations will

require precise, realistic theoretical predictions. It is pos-sible to efficiently explore the effects of various aspects oflarge-scale structure and particle physics with the develop-ment of semi-analytic descriptions of the structure.Through these methods, the important scales contributing

0.1 1 10 100 1000 104m GeV

2 10 5

5 10 51 10 42 10 4

5 10 4

0.1 1 10 100 1000 104m GeV

0.0200.0100.005

0.0020.001

0.1 1 10 100 1000 104m GeV

0.2000.1000.050

0.0200.0100.005

FIG. 10. The approximate logarithmic bin size required for the spectral line detector rate bin to reach the atmospheric neutrinodetection rate. The upper line is for �e þ ��e and the lower line is for �� þ ���. Left: c < 180�. Center: c < 5�. Right: c < 1�.

ALLAHVERDI, CAMPBELL, AND DUTTA PHYSICAL REVIEW D 85, 035004 (2012)

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to the predictions can be identified, and the robustness ofthe calculations against the uncertainties determined. It isthus that we may determine the constraints possible withmodern and future experiments searching for indirect sig-nals of dark matter annihilation.

ACKNOWLEDGMENTS

The authors thank Carsten Rott for helpful comments.The work of R. A. is supported by the University of NewMexico Office of Research. The work of S. C. and B.D. issupported in part by DOE Grant No. DE-FG02-95ER40917.

APPENDIX A: ALGORITHM FOR MEANANNIHILATION INTENSITIES ACROSS THE

CORE OF NFW PROFILES

While the density cusp at the center of dark matter halosin the NFW model causes the observed dark matter anni-hilation intensity to be infinite in the direction toward thecenter of the halo, mean intensities over solid anglesincluding the center are finite. Baryon cooling and, inlarger halos, the presence of a supermassive black hole

are some of the important effects that ultimately generate amore realistic core profile. To keep the dark matter distri-bution in this paper relatively simple, we do not attempt tomodel these effects, but assume the NFW profile through-out the halo.In this appendix, we share our method for accurate

calculation of annihilation intensity averages �IðcMÞ fromobservations over solid angles centered on the galacticcenter, with angular radius cM. Referring to Eqs. (5)–(7),we wish to evaluate

�IðE; cMÞ ¼ �v

8�m2

dN

dE

ðEÞ �JðcMÞ

with

�JðcMÞ ¼ 1

1� coscM

Z cM

0dc sinc Jðc Þ:

Let x be the distance from the solar system, in units of thegalactic halo scale radius rs;G, along a line of sight at anglec from the galactic center, and x� be distance of the solarsystem from the galactic center, also in units of rs;G. Then

1� coscM

�2s;Grs;G

�JðcMÞ ¼Z cM

0dc sinc

Z xmaxðc Þ

0

dx

ðx2 � 2x�x cosc þ x2�Þð1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � 2x�x cosc þ x2�

p Þ4;

where

xmaxðc Þ ¼ x� cosc þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2G � ðx� sinc Þ2

q

expresses the halo boundary and, as before, the halo con-centration is cG ¼ Rvir;G=rs;G. The integrand of �J in thesecoordinates is irregular in the neighborhood of c ¼ 0 andx ¼ x�, precisely where the modeled density diverges atthe halo center.

The accurate evaluation of this expression is more easilyattained when x is replaced in favor of �, as pictured inFig. 11

sin� ¼ x� sincs

¼ x� sincffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� x� cosc Þ2 þ x2�sin2cp

In these coordinates,

1� coscM

�2s;Grs;G

�JðcMÞ

¼ 1

x�

Z cM

0dc

Z ��c

�Mðc Þd�

�sin�

sin�þ x� sinc

�4;

where

�Mðc Þ ¼ sin�1

�x�cG

sinc

�:

The inner � integration is now well-defined and easyto numerically evaluate, except for when c ¼ 0,where the � path of integration becomes degenerate,initially at the Sun having the value of �, and instanta-neously becoming 0 when crossing the galactic center.

FIG. 11. Galactic coordinates used for calculating the meanintensity due to dark matter annihilation in the smooth compo-nent of the galactic halo.

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Since this degenerate point is an end of the c integration,it is sufficient for numerical evaluation to considerthe value of the inner integration in the limit as capproaches 0.

For c ! 0, we have �M ! x�c =cG ! 0, and the innerintegral approaches

Z ��c

�Mðc Þd�

�sin�

sin�þ x�c

�4 !

Z �

0d� ¼ �:

For c ¼ �, the � integration path is simply of zeromeasure with � ¼ 0 constant along the path. Therefore,the inner integration vanishes for this value of c .

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