Lifetime constraints for late dark matter decay

Nicole F. Bell, Ahmad J. Galea, and Kalliopi Petraki

School of Physics, The University of Melbourne, Victoria 3010, Australia(Received 9 April 2010; published 16 July 2010)

We consider a class of late-decaying dark matter models, in which a dark matter particle decays to a

heavy stable daughter of approximately the same mass, together with one or more relativistic particles

which carry away only a small fraction of the parent rest mass. Such decays can affect galactic halo

structure and evolution, and have been invoked as a remedy to some of the small-scale structure formation

problems of cold dark matter. There are existing stringent limits on the dark matter lifetime if the decays

produce photons. By considering examples in which the relativistic decay products instead consist of

neutrinos or electron-position pairs, we derive stringent limits on these scenarios for a wide range of dark

matter masses. We thus eliminate a sizable portion of the parameter space for these late-decay models if

the dominant decay channel involves standard model final states.

DOI: 10.1103/PhysRevD.82.023514 PACS numbers: 95.35.+d, 95.85.Pw, 98.62.Gq, 98.70.Vc

I. INTRODUCTION

Despite the ample gravitational evidence for the exis-tence of dark matter (DM), its nature remains unknown. Apromising strategy for identifying the dark matter particleis to search for a signature produced by DM decay orannihilation, inside the Galactic halo or at cosmologicaldistances. Although such a signature has not yet beenidentified, the observed radiation backgrounds have servedto constrain various decay and annihilation channels [1–13]. In many cases, this has resulted in stringent limits onspecific DM candidates.

In this paper we derive constraints for a class of modelsin which DM decays dominantly via the process

� ! �0 þ l; (1)

where �0 is a massive stable particle, and l denotes one ormore light (or possibly massless) particles. Such decaymodes have been discussed in Refs. [14–19]. We focuson the case where the mass splitting between the unstableparent � and the daughter particle �0 is small,

�m ¼ m� �m�0 � "m�; (2)

where " � 1. Since the decay replaces an � DM particlewith an �0 DM particle of approximately the same mass, itdoes not change the DM energy density significantly and itis thus not constrained to occur at time scales much largerthan the age of the Universe. This is quite different fromdecaying DM scenarios in which the entire energy of theDM is converted into relativistic species during the decay,such as the model of Ref. [20]. The latter scenario isstringently constrained by the cosmic microwave back-ground anisotropies, which set a lower limit on DM life-time of 123 Gyr at 1� confidence level [21]. The DMdecay models we consider in this paper, in which most ofthe energy is retained in the form of nonrelativistic matter,may be most effectively constrained by comparing the flux

of the light particle(s) l emitted in the decay against theobserved radiation backgrounds.An interesting feature of this class of models is that they

offer a possible solution to some of the discrepanciesbetween galactic observations and small-scale structurepredictions of cold dark matter (CDM) simulations. InCDM simulations, clustering proceeds hierarchically, in a‘‘bottom-up’’ fashion, resulting in rich structure at smallscales. Several disparities with galactic observations havebeen identified [22–35]. For the disagreement to be re-solved within the CDM paradigm, some mechanism thatmodifies the standard CDM structure-formation picture isnecessary. If dark matter decays according to Eq. (1) attime scales comparable to the age of the Universe, thekinetic energy acquired by the heavy daughter, �0, willeffectively heat up the dark matter haloes, and cause themto expand. This can alleviate two of the most glaring CDMproblems, the cuspy density profiles of dark matter haloesand the over-prediction of satellite galaxies [14–19].Yuksel and Kistler [1] obtained bounds for the decay

channel � ! �0 þ �, using observations by SPI,COMPTEL, and EGRET. Since photons are the mosteasily detectable particles of the standard model (SM),these bounds constrain late-decaying DM scenarios in themost stringent fashion. However, the branching ratio fordirect photon production may be tiny in many models. Inthis paper we investigate DM decay modes to other pos-sible SM particles. We focus on two cases: when the lightdaughter particles l produced in the decay are a pair of thelightest charged leptons of the SM

� ! �0 þ e� þ eþ; (3)

or a pair of the least detectable stable SM particles

� ! �0 þ �þ ��: (4)

Given that photons provide the most stringent bounds, andneutrinos the least stringent, our results, together withthose of [1], span the full range of constraints for all SM

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final states. Moreover, constraining the decay into neutri-nos places a general lower bound on the DM lifetime fordecay to any SM particle.

The production of electrons and positrons by dark matterdecay can be constrained both by the observed positronflux and by the observed photon background. Electrons andpositrons propagate in the Galaxy and produce photons in avariety of ways. We consider a comprehensive list ofeffects which result in photon emission associated withthe dark matter decay into e� pairs, including inverseCompton scattering, bremsstrahlung, synchrotron radia-tion, in-flight and at-rest annihilations, and internal brems-strahlung. We compare the expected signals with data fromFermi LAT, COMPTEL, EGRET, and INTEGRAL. Thedecay into � �� can be constrained by the atmospheric-neutrino background, and by data from the Super-Kamiokande search for the diffuse supernova neutrinobackground.

In Sec. II we derive the constraints on the decay chan-nels (3) and (4). We present our results in Fig. 1, togetherwith the constraints derived in Ref. [1] for decay to aphoton. In Sec. III, we investigate whether these limitspermit DM decay to have a significant effect on the for-mation of galactic structure. We focus on late-decay sce-narios, at times* 0:1 Gyr, in accordance with the work ofRefs. [16–19]. For the decay modes described in Eq. (3)and (4), our constraints eliminate a sizable band of the � vsm� parameter space. However, significant parameter space

for which DM decay may influence halo structure remainsavailable. In Sec. IV we turn our attention to models whichare not necessarily motivated by the structure formationdebate, nevertheless they involve decay of a relic popula-tion of particles into states of similar mass. In such models,the decaying particles may amount only to a small fraction

of the total DM density of the Universe. We explore theapplicability of our constraints on these scenarios andcomment on specific particle-physics models.

II. CONSTRAINTS ON DARK MATTER DECAY

The energy of the light particles produced in the decaysdepends on the mass splitting between the parent and theheavy daughter particle, �m. Their flux is inversely pro-portional to the lifetime, �, and the mass, m�, of the parent

particle. The latter determines the number density of theDM particles. Observations thus set a lower limit on theproduct m�� as a function of �m.

However, if � is comparable to the age of the Universe,�0 � 13 Gyr, an essential element in the proposed late-decaying DM resolutions of the small-scale structure prob-lems of CDM [16–19], the abundance of � particles mayhave changed significantly due to decay. Observational

bounds should then be cast in the form m��e�0=� �

F ð�mÞ, reflecting the exponential suppression of theparent-particle number density today. (The observationalbound,F ð�mÞ, is presented in Fig. 1 and will be discussedfurther below.) Note that for sufficiently short lifetime,� � �0, all the � particles decay to the stable �0 statevery early on, thus leaving no observational signal in theUniverse today, while for sufficiently long lifetimes, � ��0, the current decay probability will be very small. Thelifetimes of interest to us, i.e., those that we can probe viaindirect detection of the DM decay products, lie betweenthese two extremes.In our analysis below, wewill assume that the � particles

originally constitute all of the DM, with no significantadmixture of the similar-mass �0 particles. Comparableinitial abundances of � and �0 might be anticipated incanonical thermal-relic realizations of such a DM model.However, our limits can easily be rescaled to account formulticomponent DM, and a down-scaling of our con-straints by a factor of �2 would not qualitatively affectour conclusions.The massive daughter particle �0 will be produced non-

relativistically, provided that �m � m�. Then, to a good

approximation, the light particles share the energy �mreleased in the decay, such that E1 þ E2 ’ �m, whereE1, E2 are the energies of the two light leptons. For the3-body decay channels described in Eq. (3) and (4), theenergy spectra of the daughter particles is in generalmodel-dependent. However, we find that the constraintson the dark matter decay are rather insensitive to the actualspectrum. We illustrate this by presenting results for twolimiting cases of the daughter energy spectra:(i) Monoenergetic, where the energy spectrum per de-

cay for each light daughter species is

dN=dE ¼ �ðE��m=2Þ: (5)

(ii) Flat, where is spectrum is uniform over the allowedenergy range such that

10 4 10 2 1 102 1041019

1021

1023

1025

1027

1029

1031

m GeV

m0

GeV

s

FIG. 1. Constraints on the dark matter decay channels:(i) � ! �0 þ �, using the �-ray line emission limits from theGalactic center region and the isotropic diffuse photon back-ground (irregular and smooth solid lines, respectively) [1];(ii) � ! �0 þ e� þ eþ (dashed line); (iii) � ! �0 þ �þ �� (dot-ted line). The regions below the lines are excluded.

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dN=dE ¼ 1=ð�m� 2mlÞ; (6)

whereml ¼ me for the decay into e� andml ¼ 0 for

the decay into neutrinos.For each of the decay modes analyzed, we pick the mostconservative limits to further continue our discussion.1

The constraints on DM decay depend on the DM halodensity profile. Although there are uncertainties in theprofile, particularly near the Galactic center (GC), we shallsee below that these uncertainties are subdominant. Forconsistency with the results of other analyses which weshall employ in our calculations, we use the Navarro-Frenk-White (NFW) [36] profile throughout. The standardparametrization is

�ðrÞ ¼ �0

ðr=rsÞ�½1þ ðr=rsÞ��ð��Þ=� ; (7)

where ð�;; �Þ ¼ ð1; 3; 1Þ. For the Milky Way rs ¼20 kpc, and the normalization �0 is fixed such that at theSolar-circle distance Rsc ¼ 8:5 kpc, the density is�ðRscÞ ¼ 0:3 GeV cm�3.

A. Decay into electrons and positrons

Electrons and positrons injected in the Galaxy willproduce photon fluxes spanning a wide range of energies,by a variety of different mechanisms. The injection rate canthus be constrained by the resulting contribution to theGalactic photon backgrounds. We will also derive con-straints based upon direct detection of positrons. Below,we describe the methods used to estimate the signal pro-duced by each effect. We present the resulting constraintsin Fig. 2.

1. Internal bremsstrahlung

Electromagnetic radiative corrections to the DM decaychannel to charged particles inevitably produce real � rays.If the process � ! �0 þ e� takes place, then � ! �0 þe� þ � must also take place since either one of the finalstate charged particles can radiate a photon. This process iscalled internal bremsstrahlung (IB). Note that the photon inthe IB process arises directly in the Feynman diagram forthe decay, and is unrelated to regular bremsstrahlung whichresults from interaction in a medium. Importantly, the IBspectrum and normalization is approximately model-independent. In addition, as the IB photon flux does notdepend on the propagation of charged decay products inthe Galaxy, it is free of uncertainties associated with theastrophysical environment. IB is thus a very clean andreliable technique for obtaining limits on DM decay to

charged particles, with the only source of uncertaintyresiding in the DM profile adopted.The IB differential decay width is [37]

d�IB

dE¼ � hðEÞ; (8)

where E is the energy of the photon emitted, � ¼ ��1 is therate of lowest order decay process � ! �0 þ e�, and hðEÞis the photon spectrum per � ! �0 þ e� decay. This spec-trum is independent of the new physics which mediates thedecay, and is given by

hðEÞ ’ �

1

Eln

�s0

m2e

��1þ

�s0

s

�2�; (9)

where s0 ¼ sð1� E=EmaxÞ, with s being the energy re-leased in the decay, and Emax the maximum energy thatcan be imparted to the radiated photon.2 In the 3-bodydecay we consider here, s ¼ ð�m� 2meÞ2, andEmax ¼

ffiffiffis

p ð1� "=2Þ ’ ffiffiffis

p.3

The shape of the IB photon spectrum is determined bythe 1=E factor in Eq. (9), while the logarithmic factorprovides a sharp cutoff at E ’ Emax. Since the Galacticphoton background varies approximately as E�2, the high-

IB

IA511 keV

Synch

ICe flux

10 3 10 2 10 1 1 101 102 103 1041020

1022

1024

1026

1028

1030

m GeV

m0

GeV

s

FIG. 2 (color online). Limits on dark matter decay process� ! �0 þ e� þ eþ, obtained from internal bremsstrahlung (IB,cyan), in-flight annihilation (IA, blue), annihilation at rest(511 keV, purple), inverse Compton scattering and bremsstrah-lung emission (IC, green), synchrotron radiation (Synch, or-ange), and positron flux (eþ flux, red). Solid lines correspondto a monoenergetic spectrum of electrons and positrons, whiledashed lines assume a flat injection spectrum over the allowedenergy range. The regions below the lines are excluded.

1As a consistency check, we verify that for a specific �m, thelimit obtained on m��e

�0=� when an extended injection distri-bution is used, does not exceed the maximum value of this limitobtained for monoenergetic injection for mass splittings up to2�m, i.e. ½m��e

�0=�ð�mÞ�ext max�<2�m½m��e�0=�ð�Þ�mono.

2These general definitions of s, s0 reproduce the correctexpressions for the case of annihilation into two charged parti-cles [37] and for muon decay [38].

3Note that the phase space available to the photon when one ofthe decay products is emitted nonrelativistically (i.e. when " �1) is larger than when all decay products are relativistic (e.g.muon decay, where " ’ 1). This is because a heavy daughterparticle can absorb recoil momentum, without carrying away asignificant amount of energy.

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energy region of the IB photon spectrum will provide themost stringent constraints.

The flux of photons due to IB will be proportional to thedark matter density integrated over the line of sight. At anangle c from the Galactic center, the line-of-sight integralis

Jðc Þ ¼Z lmax

0dl��ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2sc � 2lRsc cosc þ l2

qÞ; (10)

where lmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2MW � R2

scsin2c

qþ Rsc sinc . The average

of J over a cone of half-angle c centered on the GC is

J ��ðc Þ ¼ 2

��

Z c

0Jðc 0Þ sinc 0dc 0; (11)

where �� ¼ 2ð1� cosc Þ.The expected photon flux, per unit solid angle, due to IB

is then

d�IB

dE¼ 1

4

1

m��e�0=�

J ��hðEÞ: (12)

COMPTEL observations show that the �-ray back-ground remains approximately constant within jlj< 30�.Thus, along the Galactic plane, the signal-to-backgroundratio for IB is maximized at the GC. Here, we will averageover the region c < 5�, which exceeds the angular reso-lution with which both COMPTEL and EGRET data havebeen reported [39,40]. The IB limits are shown in Fig. 2.

2. In-flight and at-rest annihilations

Positrons produced by DM decay will propagate in theinterstellar medium, lose energy, and annihilate with elec-trons. Most positrons will survive until they become non-relativistic, and annihilate at rest to produce a 511 keVannihilation line (together with a three-photon continuum).However, a portion of the positrons annihilate in flightwhile still relativistic. These in-flight annihilations (IA)can produce a significant flux of � rays [41].

Reference [41] considered the injection of positrons atthe Galactic center, and related the flux of IA � rays to theflux of 511 keV photons. This was used to derive a con-straint on the injection energy of positrons that contributeto the observed Galactic 511 keV line.

We will follow a similar approach to obtain limits onDM decay into e�. In our case, however, the overallnormalization of the photon flux is not fixed by the ob-served 511 keV line intensity, but is instead determined bythe DM decay rate. The continuum �-ray flux due to IA,and the 511 keV photons from annihilation at rest, thusyield thus two separate sets of constraints. The positroninjection, deceleration, and annihilation budget is de-scribed by the equation [42]

@n

@tþ @

@�

�d�

dtn

�¼ �nH�ð�Þnð�; tÞ þ �inj

dN

d�; (13)

where nð�; tÞd� is the density of positrons with energy

�me, at time t, �inj ¼ ��e��0=�=m�� is the positron-

density injection rate due to DM decay, and dN=d� isthe positron spectrum per decay. The positron annihilationcross section �ð�Þ on electrons at rest is [43]

� ¼ r2e�þ 1

��2 þ 4�þ 1

�2 � 1lnð�þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

qÞ � �þ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2 � 1p

�;

(14)

where re is the classical electron radius.At energies lower than �1 GeV, ionization and

Coulomb losses (in a neutral or an ionized medium, re-spectively) dominate over energy loss due to synchrotron,bremsstrahlung, and inverse Compton effects, while athigher energies synchrotron losses are more important.The energy loss rates due to ionization, Coulomb scatter-ing, and synchrotron emission are [42]

��������d�

dt

��������i’ 4:4 � 10�15 s�1

�nH

0:1 cm�3

�ln�þ 6:8

; (15)

��������d�

dt

��������C’ 1:5 � 10�15 s�1

�nH

0:1 cm�3

�ln�þ 75:9

; (16)

��������d�

dt

��������s’ 5:4 � 10�24 s�1

�UB

0:2 eVcm3

�ð�2 � 1Þ; (17)

where nH is the hydrogen number density of the medium,and UB ¼ B2=8 is the energy density of the magneticfield. Equations (15) and (16) apply to a fully neutral andfully ionized medium, respectively. As a conservativechoice, we set the ionization fraction of the interstellargas to be xi 0:51. Compared with a completely neutralmedium, energy losses in an ionized medium are larger andthus IA constraints weaker, by a factor of a few [44]As evident from Eqs. (15) and (16), it takes approxi-

mately 107 yr for relativistic positrons to slow down andannihilate. If positron injection (DM decay) has been tak-ing place over a much larger time period, � > 0:1 Gyr, thepositrons reach a steady-state distribution. The time-independent solution of Eq. (13) is

nð�Þ ¼ ��

m��e�0=�

1

j d�dt jZ 1

�d�0 dN

d�0 P�0!�; (18)

where

P�0!� � exp

��nH

Z �0

�

�ð�00Þ00

j d�00dt j

d�00�

(19)

is the survival probability, introduced in [41], of positronsinjected at energy �0me to have not annihilated by the timethey reach energy �me.

NICOLE F. BELL, AHMAD J. GALEA, AND KALLIOPI PETRAKI PHYSICAL REVIEW D 82, 023514 (2010)

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Here, we have ignored the effects of spatial diffusion ofpositrons, which would require detailed modeling. Thisapproximation is sufficient provided that the positronsremain trapped within the region we consider.4 In whatfollows, we will calculate the expected photon flux due toannihilation in a cylindrical region of radius rmax ¼ 3 kpcand half-height zmax ¼ 0:5 kpc, centered at the GC. This iswell inside what is taken to be the diffusion area in mostmodels simulating cosmic-ray propagation in the Galaxy.Because of the Galactic magnetic field, charged particlesare largely expected to follow the magnetic lines andconverge toward the GC, as long as their Larmor radius,rL ¼ �m=qB, is much smaller than the scale of variationof the magnetic field. In the Milky Way, B * few�G, andðd lnB=drÞ�1 � few kpc [5]. Positrons are thus expected tofollow the magnetic lines, even for energies much higherthan the ones considered in this analysis. Any influx ofpositrons in the region under consideration would only leadto more stringent constraints than the ones derived here.(The possibility of transport of the positrons from the diskto the GC has been discussed in detail in [45].)

The diffuse �-ray background of the Galactic plane hasbeen measured by COMPTEL in the energy range 1–30 MeV for the region jlj< 30�, jbj< 5� (Galactic longi-tude and latitude, respectively) and by EGRET, in theenergy range 30 MeV–30 GeV and the region jbj< 10�[39,40]. The cylindrical volume defined above is coveredby these observations, and encompassed within a conicalpatch of solid angle �� ¼ 4lmax sinbmax ’ 0:13 sr, withlmax ¼ arcsinðrmax=RscÞ ’ 20� and bmax ¼arctan½zmax=ðRsc � rmaxÞ� ’ 5�. The comparison of thesignal from the cylindrical region, with the backgroundradiation emanating from the larger conical region, keepsour constraints conservative.

The flux of photons (per unit solid angle) due to IA is

d�IA

dk¼ 1

��

1

4R2sc

ZdV

Zd�nð�ÞnH d�ðk; �Þ

dk;

where the spatial integration is over the cylindrical region

considered. The differential cross section d�ðk;�Þdk for a posi-

tron of energy �me to produce a photon of energy E ¼ kme

is [41,46]

d�ðk; �Þdk

¼ r2e�2 � 1

�� 3þ�1þ� þ 3þ�

k � 1k2

ð1� k1þ�Þ2

� 2

�; (20)

while the minimum positron energy �minme required toproduce a photon of energy kme is given by

�minðkÞ ¼ k� 1

2þ 1

2ð2k� 1Þ : (21)

For the steady-state positron distribution nð�Þ, given in Eq.(18), the expected �-ray signal from IA is

d�IA

dk¼ 1

m��e�0=�

RdV��

4R2sc��

Z 1

�minðkÞd�

dN

d�Qðk; �Þ;

(22)

where

Q ðk; �Þ � nHZ �

�minðkÞd�0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�02 � 1

p�0

d�ðk; �0Þdk

P�!�0

j d�0dt j

:

(23)

The IA flux of Eq. (22) is compared to the �-ray back-ground reported by COMPTEL and EGRET [39,40], andcompiled in Ref. [12]. The resulting lower limit on

m��e�0=� is presented in Fig. 2 and increases with increas-

ing �m, since positrons injected at higher energy spendmore time as relativistic, and are more likely to annihilatein flight. However, the annihilation probability diminishesat high energies, which is manifest by the plateau in thelifetime bound for energies * 3 GeV.The majority of positrons survive until they become

nonrelativistic. Then, they either annihilate directly withelectrons to produce 511 keV photons, or they form apositronium bound state. Positronium subsequently anni-hilates, with probability 25% into two 511 keV photons, orwith probability 75% into a three-photon continuum. Thepositronium fraction in the Galaxy is fixed by the relativeintensities of the �-ray continuum below 511 keV, and the511 keV line. In our Galaxy f ¼ 0:967� 0:022 [47].If DM decays into e� pairs, the flux of 511 keV photons

produced is determined, in the steady-state regime, by therate at which positrons arrive at rest. For each nonrelativ-istic positron, there will be 2ð1� fÞ þ 2f=4 ¼2ð1� 3f=4Þ photons contributing to the line. The expectedflux of 511 keV photons is

�511 ¼ 2ð1� 3f=4Þm��e

�0=�

RdV��

4R2sc��

Z 1

1d�

dN

d�P�!1: (24)

The �-ray observations by INTEGRAL have revealed a511 keV line emission of intensity 0:9410�3 ph cm�2 s�1 within an integration region jlj< 20�,jbj< 5� [48]. The average flux per steradian is then0:0077 ph cm�2 s�1 sr�1. The limits on DM decay arisingfrom annihilation at rest are presented in Fig. 2 . Theyexhibit a very slight negative slope toward increasing �m,which accounts for the (small) portion of positrons anni-hilating in-flight when injected at high energies.

4Along with spatial diffusion, we also ignore the related effectof reacceleration. This becomes more important with increasingenergy, since the diffusion coefficient in momentum space isDpp / p2�� with � < 1 [5,6,42]. However, the probability toannihilate drops sharply at high energies, and thus pumpingmore energy in already energetic positrons does not affect theIA constraints significantly.

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The IA constraints are stronger than those for annihila-tion at rest for average positron injection energies hEi ¼�m=2 * 80 MeV. This is considerably higher than thecorresponding value of �3 MeV found by an appropriateanalysis in Ref. [41]. The latter limit was determined byrequiring that the IA � rays do not exceed 30% of theobserved background, instead of the more conservative100% adopted here. However, the more important factorin reconciling these two analyses is the different signalregions considered. The analysis in Ref. [41] consideredthe conical region within a �0:37 kpc radius of the GC,which contains the peak of the 511 keVannihilation signal.This is to be compared to our larger cylindrical volume(rmax ¼ 3 kpc, zmax ¼ 0:5 kpc) outlined above. The effectof choosing the larger volume is to make the annihilation-at-rest constraints stronger, with respect to those from IA,than for the choice of a smaller region at the Galacticcenter. The Galactic center is not the optimal observationregion to use in setting an annihilation-at-rest limit, be-cause that is where the observed 511 keV background ishighest. In addition to resulting in more sensitiveannihilation-at-rest limits, the larger volume also rendersus insensitive to possible effects of diffusion.

Notice also that the photon fluxes, Eqs. (22) and (24), areindependent of the medium number density nH, as long asionization and Coulomb losses dominate, for �m &fewGeV. Since the probability to annihilate at even higherenergies is insignificant, the results are rather insensitive tonH, as well as UB, throughout the energy spectrum.

3. Inverse Compton scattering, bremsstrahlung emission,synchrotron radiation

At high energies IA become rather rare, and electronsand positrons propagating in the Galaxy produce radiationmore efficiently via other mechanisms. Gamma rays areproduced by bremsstrahlung and inverse Compton (IC)scattering of low-energy photons, with the latter effectyielding the dominant contribution. Synchrotron emissionin the Galactic magnetic field gives rise to a radio wavesignal.

Zhang et al. [5,6] calculated the photon spectrum ex-pected if electrons and positrons are injected in the Galaxyby DM decay. They assumed a monoenergetic injectionspectrum, and modeled the propagation of e� in the inter-stellar medium in detail, including the effects of spatialdiffusion, convection, energy loss, and reacceleration.They compared the resulting spectrum at Earth with ob-served backgrounds, and encoded their calculations inresponse functions which can be utilized to obtain con-straints when convoluted with the DM decay spectrum of aparticular model. Here, we use the response functionsderived in Ref. [5] for synchrotron radiation, and inRef. [6] for IC and bremsstrahlung emission, to obtainconstraints for the � ! �0 þ e� decay process.

The main source of uncertainties in deriving the re-sponse functions is poor knowledge of the various astro-physical parameters which determine the propagation ofe� in the Galaxy. The most significant contribution comesfrom the height of the diffusion zone. The Galactic mag-netic field, although also quite poorly known, contributessubdominantly to the total uncertainty. This is true even forthe synchrotron response functions, since although themagnetic field may be rather uncertain in the Galacticbulge, the directions which optimize the signal-to-background ratio point away from the GC. Differentchoices for the DM halo profile have a small influence onthe response functions, since the flux is only proportionalto the density (as opposed to the �2 dependence applicablefor DM annihilation) [5,6].The IC and bremsstrahlung emission is constrained by

using the Fermi LAT �-ray maps derived in [49], for theenergy range (0.5–300) GeV. Because of reacceleration,this permits limits for e� injection energies in a muchwider range, ð0:01–104Þ GeV. The strength of the limits,of course, sharply diminishes in the low-energy part of thisrange. At energies above 50 GeV, the Fermi data maysuffer from significant background contamination, whilepoint sources have not been subtracted. These effects leadto more conservative constraints. There are also variousastrophysical contributions to the �-ray flux, such asnucleus-nucleus photoproduction via 0 decay, and ICand bremsstrahlung radiation from cosmic-ray electronsand positrons. In the spirit of setting conservative limitson DM decay, we will use response functions where theseforegrounds have not been subtracted. The patch of skychosen to compare the expected signal to observations isjlj< 20� and �18� < b<�10� [6].The synchrotron response functions derived in Ref. [5]

correspond to 408 MHz, 1.42 GHz, and 23 GHz sky maps,and yield constraints for e� injection energies in the range,ð0:1–104Þ GeV. The directions that optimize the signal-to-background ratio are different for each of the sky mapsused, and are located close to, but not at, the GC. (Similarsensitivity may be achieved with the isotropic diffuse fluxmeasured by Fermi; see Ref.[50] for relevant limits on theprocess � ! e�.)The response functions reported all assume a NFW halo

profile. The expected signal from particles produced byDM decay is compared to observed photon flux including a2� error. The dependence on the propagation model, forboth the synchrotron and the IC plus bremsstrahlung emis-sion, becomes more significant at e� injection energieslower than about 10 GeV. This is because the limits in thatrange rely solely on the effect of reacceleration, whichdepends strongly on the model adopted, while no actuallow-energy background data are used. For the synchrotronconstraints, we adopt the ‘‘DR’’ model [5], which yieldsmoderately conservative constraints along the entire en-ergy range. For the IC plus bremsstrahlung constraints we

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adopt the ‘‘L1’’ model, whose response functions are mostcomprehensively reported in [6].

The limits on DM decay from IC plus bremsstrahlungradiation and synchrotron emission are shown in Fig. 2.

4. Positron flux

The DM decay into e� pairs can be constrained by directobservations of the positron flux on Earth. Utilizing thesame propagation models as the ones used for the synchro-tron emission, Zhang et al. [5] constructed response func-tions, comparing the expected positron flux to PAMELAdata [51]. The positron fraction reported by PAMELAwasconverted into positron flux using the e� total flux ob-served by the Fermi telescope [52].

The positron-flux response functions are based on ob-servations of positrons in seven energy bands in the range(10.17–82.55) GeV. Depending on the model used, theycan provide meaningful constraints for positron injectionenergies within the range 0.1 GeV–10 TeV. These con-straints appear to prevail over the IB, IA, IC, and synchro-tron constraints at the high-energy part of the spectrum. Inthe same spirit as before, we adopt the DR model, andobtain fairly conservative constraints for the entire energyspectrum. The results are presented in Fig. 2.

Much like the photon-emission response functions, thepositron-flux response functions are sensitive to theadopted propagation model, which determines the diffu-sion and reacceleration of the injected positrons. However,they are insensitive to the DM halo profile, since most ofthe positrons come from the local region,�1 kpc from theSun [5].5

B. Decay into neutrinos

Neutrinos are the least detectable stable SM particles,hence constraints on DM decay (or annihilation) to neu-trinos can be used to set conservative but robust lowerlimits on the DM lifetime (or upper limit on the annihila-tion cross section) to any SM final state [7]. Palomarez-Ruiz [2,3] obtained limits on the decay channel � ! �þ��. Here we will adapt the analysis of Refs. [2–4], toconstrain the decay channel � ! �0 þ �þ ��. This willset the most conservative limit on DM decay modes ofthe form � ! �0 þ l, where l is any SM final state.

The neutrino flux on Earth from DM decay in theGalactic halo is

d��

dE�

¼ 1

4

1

m��e�0=�

J ��

2

3

dN

dE�

: (25)

The multiplicity of 2 accounts for the sum of � and ��produced in the decays. As in Refs. [2,3], we assume equaldecay width to all three neutrino flavors, which accounts

for the factor of 1=3 in Eq. (25). This is a reasonableapproximation, since neutrino flavor oscillations betweenthe production and detection points will considerablyweaken any preference toward a particular flavor. Forsimplicity, and in order to minimize the uncertainty arisingfrom a particular choice of halo profile, we average theexpected neutrino signal over the whole sky, c ¼ 180�.Directional information, whenever available, is in generalexpected to lead to more stringent limits [2,3,8].For energies E� * 50 MeV, the neutrino flux on Earth is

dominated by atmospheric neutrinos. The atmosphericneutrino flux has been well measured by a number ofexperiments [53–58] and is in good agreement with theo-retical predictions. We utilize the results of the FLUKA[59,60] calculation of the atmospheric �� þ ��� back-

ground, over the energy range 50 MeV to 10 TeV. Wedetermine a DM decay limit by requiring the expectedneutrino signal from DM decay not to exceed the atmos-pheric neutrino flux, integrated over energy bins of width�log10E� � 0:3. This choice of bin size is in accordancewith that adopted in Refs. [2,3], and encompasses theexperimental resolution of the neutrino detectors. (Wenote that future measurement by IceCube-DeepCore willlead to sensitive limits at high energy [61,62].)At low energies E� & 100 MeV, the relevant data come

from the diffuse supernova neutrino background (DSNB)search performed by the SK experiment [63]. SK searchedfor positrons produced by incident ��e on free protons in-side the detector, via the inverse -decay reaction ��e þp ! eþ þ n. Incoming �e and ��e interact also with boundnucleons, producing electrons and positrons. The mainsources of background for these observations are atmos-pheric �e and ��e, and the Michel electrons and positronsfrom decays of subthreshold muons. A DSNB signal wasnot detected. In Refs. [2–4] a �2 analysis of these data wasperformed, and used to place limits, at 90% confidence

10 1 1 101 102 103

1019

1021

1023

1025

1027

1029

m GeV

m0

GeV

s

FIG. 3. Limits on dark matter decay � ! �0 þ �þ ��. As inFig. 2, the solid (dashed) lines correspond to monoenergetic(flat) injection distribution of �, ��. The regions below the linesare excluded.

5This is consistent with our assumption in Sec. II A 2, that thee� do not escape from the inner part of the Galaxy.

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level, on the contribution to the signal from DM annihila-tion, �� ! � ��, or decay, � ! � ��. We shall employ theDM decay limits of Refs. [2,3]. A simple rescaling sufficesto convert these limits into constraints on the decay modeof interest here, � ! �0� ��, under the assumption that � ��are emitted monoenergetically. The resulting constraintsare shown in Fig. 3.

III. LATE-DECAYING DARK MATTER AND THESMALL-SCALE GALACTIC STRUCTURE

The standard �CDM cosmological model successfullyreproduces the observed structure of the Universe at largescales. However, at small scales, there appear to be severalinconsistencies between observations and CDM simula-tions of galaxy formation. The negligible primordial ve-locity dispersion of CDM particles allows gravitationalclustering to occur down to very small scales. Structureforms hierarchically, with small scales collapsing first andforming dense clumps. The resulting phase packing in theinner regions of galaxies leads to cuspy central densityprofiles [24], which are currently disfavored by the rota-tional curves of dwarf spheroidal galaxies [22,23,25–28].Dense lumps of matter also survive the mergers and be-come satellite galaxies. The number of satellite galaxiespredicted by CDM simulations greatly exceeds the numberobserved around the Milky Way [24,30,32]. A number ofother disparities between observations and CDM predic-tions have been reported. These include the overestimationof the number of halos in low-density voids [33,35], thenonprediction of pure-disk or disk-dominated galaxies[29], the large angular-momentum loss by condensinggas [34], and the ‘‘bottom-up’’ hierarchical formation [31].

It is not yet known whether the solution to these incon-sistencies lies outside the dark matter sector. Indeed, indi-vidual astrophysical solutions to some of these problemshave been suggested. However, it is possible that theapparent discrepancies between CDM and observed galac-tic structure point toward a modification of the standardCDM scenario. The significance of these hints for deci-phering the nature of DM is underscored by the fact thatgravitational effects remain the only confirmed evidencefor the existence of DM.

A possible alternative to the CDM model is warm darkmatter (WDM), whose thermal free-streaming propertiesresult in a suppression of structure on small scales.Although well-motivated particle-physics candidates forWDM exist, WDM lacks the ‘‘naturalness’’ of the standardCDM scenario in which the observed relic DM abundancearises from thermal freeze-out (CDM candidates producedby other mechanisms also exist). It is possible, though, toincorporate some WDM-like structure-formation featureswithin the CDM paradigm, if we discard the notion thatCDM particles are completely stable.

It has been suggested that if the heavy relic particlesdecay according to Eq. (1), the energy released in the decay

will change the standard picture of structure formation. Ifthe decays occur at early times,�105 s–108 s, before gravi-tational collapse, the massive decay products acquire non-negligible velocities, which results in a suppression of theamplitude of density perturbations at small scales (still inthe linear regime). The daughter DM particles behaveeffectively as WDM. Existing limits on the free-streamingand phase-packing properties of WDM can be directlytranslated to determine the interesting and the excludedregions of the CDM decay parameter space [14,15].Another interesting possibility, investigated in

Refs. [16–19], arises if the decays take place during thenonlinear stages of gravitational collapse. In this scenariothe energy acquired by the decay products causes the haloto expand and, as a result, softens the steep central cuspspredicted by CDM [16]. It also results in a decrease in thehalo circular velocity, which can account for the deficit ofsatellite galaxies with velocity 10–20 km/s, with respect tostandard CDM predictions [17].The efficacy and viability of the late DM decay mecha-

nism to alleviate the CDM small-scale structure problemsdepend on two parameters: the parent particle lifetime, �,and the ratio " � �m=m� which gives the recoil velocity

of the massive daughter particle, v�0 ’ ".6 Qualitatively,

one expects � to be comparable to the age of the Universe,so that the decays occur during galaxy formation. In thelimit of very large �, the standard CDM scenario is recov-ered, while small � corresponds to the early decay sce-nario, examined in [14,15]. The velocity imparted to theheavy daughter particles should be sufficiently large—ofthe order of the virial velocity of the galaxy—in order tohave an effect on the formation of the halo. It should notexceed, though, the escape velocity from the halo, whichwould be catastrophic for the galaxy, unless, of course, � isvery large.Abdelqader and Melia [17] used a semianalytical ap-

proach that allowed them to incorporate the effect of DMdecay in halo evolution. They showed that DM decaypreferentially heats smaller haloes, causing them to ex-pand, and reduces their present-day circular velocity. Theyargued that if "� ð5–7Þ 10�5 and �� ðfew� 30Þ Gyr,dark matter decay can well account for the deficit in theobserved number of galaxies with circular velocities in therange 10–20 km/s.Peter et al. [18,19] performed a more extensive inves-

tigation of the �-" parameter space, by means of bothsemianalytical calculations and simulations. They demon-strated the way in which DM decay affects the halo mass-concentration relation and mass function, and used mea-surements of these quantities to constrain the DM decayparameter space. They located the allowed and potentially

6This becomes a strict equality in the case of a 2-body decay. Ifmore than one relativistic particle is produced, as in the casesexamined in this paper, the correction in v�0 is insignificant.

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interesting parameter space roughly within the range

" ¼ 3:4 10�6–3:4 10�4; (26)

� ¼ ð0:1–100Þ Gyr: (27)

The corresponding recoil velocity is v�0 ¼ ð1–100Þ km=s.

The actual limits on " and � are, of course, correlated.In Sec. II, we derived constraints on the lifetime for late

decay of DM into a massive daughter and either e� or a � ��pair. For a given value of ", the bounds of Sec. II determinewhether these decays can take place in time to affect haloevolution.

The flux of DM decay products in the Galaxy today is

proportional to 1=ðm�e�0=�Þ, and the limits in Fig. 1 areexpressed in terms of this quantity. Translating these limitsto bounds on � alone, we see that the constraints may besatisfied for either (i) very long lifetimes, � � �0, forwhich the current DM decay rate is very small; or(ii) very short lifetimes, � � �0, for which the abundanceof the unstable parent in the Universe today is very small(as most decays have already taken place). Between theseextremes limits lies a band of excluded lifetimes. Thesebands are shown in Figs. 4–6 for three values of " spanningthe range given in Eq. (26), " ¼ 10�5, 5 10�5 and 310�4, corresponding to recoil velocities v�0 ’ 5 km=s,

15 km/s, and 90 km/s, respectively.

For the values of " considered, the excluded band spansseveral orders of magnitude in lifetime. These excludedbands become smaller as " is decreased, such that forsufficiently small epsilon there is no constraint. Note thatwhile the upper limit of the excluded band may be im-proved with more sensitive flux observations, the same isnot true for the lower limit, at �� 1 Gyr, since for these

lifetimes the factor e��0=� dominates. For this reason, thelower limit of the excluded regions for the photon and e�decay modes are very similar.We now compare the bounds on the DM lifetime � with

the interesting range of values specified in Eq. (27).Figures 4–6 show that these bounds significantly constrain,but do not rule out, unstable dark matter models whose

3 10 4

CDM like

ruled out structure formation

e

10 4 10 2 1 102 10410 2

1

102

104

106

108

10 1 101 103 105 107

m GeV

Gyr

m GeV

FIG. 6 (color online). Same as in Fig. 4, but for " ¼ 3 10�4,or recoil velocity v�0 ’ 90 km=s.

5 10 5

CDM like

ruled out structure formation

e

10 4 10 2 1 102 10410 2

1

102

104

106

108

1 102 104 106 108

m GeV

Gyr

m GeV

FIG. 5 (color online). Same as in Fig. 4, but for " ¼ 5 10�5,or recoil velocity v�0 ’ 15 km=s, suggested in [17] as suitable

for resolving the missing-satellite problem. This would require adark matter lifetime within the narrower interval 1 Gyr< �<30 Gyr shown (dashed lines).

10 5

CDM like

ruled out structure formation

e

10 4 10 2 1 102 10410 2

1

102

104

106

108

101 103 105 107 109

m GeV

Gyr

m GeV

FIG. 4 (color online). Constraints on DM lifetime � vs masssplitting �m and mass m�, for " ¼ 10�5 (recoil velocity v�0 ¼5 km=s), for the decay channels � ! �0 þ � , � ! �0 þ e� þeþ, and � ! �0 þ �þ ��. Color-shaded regions are excluded bythe observed radiation backgrounds (Ref. [1] and this work). Thedecays may have an observable effect on galactic halo structureif they occur at times 0:1 Gyr< �< 100 Gyr (interval betweensolid lines). Longer lifetimes � * 100 Gyr correspond to thestandard CDM scenario, while decays occurring at � & 0:1 Gyr(but after the matter-radiation equality) disrupt the galaxy for-mation significantly and are excluded [18,19].

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dominant decay channel produces the SM particles con-sidered. For the � or e� decay modes, lifetimes greaterthan �1 Gyr are eliminated, and we thus identify ��ð0:1–1Þ Gyr as the range of allowed lifetimes for whichdecays may affect structure formation. For decay modes toneutrinos, however, the allowed lifetimes span the entireinterval of Eq. (27) (at least for some masses). Of particularinterest, perhaps, is the case of weakly interacting massiveparticle (WIMP) dark matter of mass m� � 102 GeV, de-

caying via the � �� channel, with � and " in accordance withthe values suggested in Ref. [17] for resolving the missingsatellite problem. These decay parameters are indicated bythe dashed lines in Fig. 5.

IV. LIMITS ON MODELS WITH LATE DARKMATTER DECAY

A number of particle-physics models have been pro-posed which predict or invoke a DM decay mode describedby Eq. (1) [64–67]. The motivation for these models varieswidely, thus the parameter space of interest does not al-ways overlap with that relevant for structure formation. Wenow describe, in general terms, when and how the boundsderived in Sec. II can constrain decay modes of the type inEq. (1). We then apply these considerations to particularmodels found in recent literature.

In Secs. II and III we assumed that all of the DM was inthe form of the heavy, unstable, � particles prior to decay.However, this need not be the case. In fact, many modelspredict or require that the relic � abundance be only afraction of the total CDM density. This may be true even ifthe decay time is larger than the age of the Universe,provided that either (i) other processes (e.g. inelastic scat-tering) efficiently eliminate the heavy particles (�) in favorof the light ones (�0) [65], or (ii) the �-�0 sector makesonly a subdominant contribution to the total DM density.The relic � abundance is, of course, critical in derivinglimits on specific models.

Allowing the � abundance to be a free parameter, thefour variables which determine the relevance of constraintsinferred from the present radiation backgrounds are (i) themass splitting,�m; (ii) the mass of the DM particle,m�, or

equivalently " ¼ �m=m�; (iii) the decay lifetime of the

heavy DM state, �; and (iv) the relic fraction of DM in theform of � particles before decay occurs, f� � Y�=YCDM.

In terms of these parameters, the constraints of Sec. II canbe expressed as

�e�0=� � ð" � f�ÞF ð�mÞ=�m; (28)

whereF ð�mÞ can be read off Fig. 1 for the decay channelsconsidered.

A. Fixed " � f�Consider constraints on the lifetime for models in which

the parameters " and f� are specified. Similar to the

analysis of Sec. III, Eq. (28) implies an excluded bandfor �, provided that the fraction of energy released perdecay (here quantified by ") and/or the fraction of decayingdark matter, f�, are sufficiently large. More specifically, in

order for Eq. (28) to translate into constraints on �, theparameters ", f� must satisfy

" � f� � �0e

F ð�mÞ=�m (29)

(since �e�0=� � �0e, for any �).In Fig. 7 we sketch the regions defined by Eq. (29), for

the three decay channels under consideration. Modelswhich lie above (below) the curves are constrained (notconstrained) by the bounds derived in Sec. II using theobserved radiation backgrounds. For models that lie on thecurves, the constraint (28) produces a zero-width exclusionband for the lifetime at � ¼ �0. Larger values of " � f�yield exclusion bands which span a range of lifetimesaround �0.The above discussion is relevant to a class of models

proposed in Refs. [64,65], in which DM possesses anexcited state. As a result of an approximate symmetry,the excited state is separated by a small mass splittingfrom the ground state, to which it decays with emissionof e� pairs. For a particular implementation of thesemodels, with �m 2 MeV and m� 500 GeV, the ki-

netic excitations and the subsequent decays of DM parti-cles inside the Galactic halo can produce the positronswhich give rise to the observed 511 keV annihilation linein the center of our Galaxy [64]. For the range of parame-ters favored by the 511 keV signal, f� & 10�3 and " ’10�7 � 10�6, the scenario lies comfortably below the e�curve in Fig. 7, and is thus not constrained by our obser-vational bounds. However, significantly larger values of " �

10 4 10 2 1 102 104

10 11

10 10

10 9

10 8

10 7

10 6

10 5

m GeV

f

FIG. 7. The observed radiation backgrounds can constrain thelifetime of the heavy DM state only in models which lie in theregions above the lines. Models which lie below the lines cannotbe constrained by the bounds presented in Sec. II. As in Fig. 1,the different curves correspond to the decay channels � ! �0 þ� (solid lines), � ! �0 þ e� þ eþ (dashed line), and � ! �0 þ�þ �� (dotted line).

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f� are possible within the same generic class of models, for

which our bounds would impose a nontrivial constraint.

B. Fixed �

Returning to the constraint of Eq. (28), we now considermodels in which � (rather than " � f�) is narrowly speci-

fied. Equation (28) then limits the energy release via decay,and relic abundance of the unstable state, as per

" � f� �e�0=�

F ð�mÞ=�m : (30)

This constraint is substantial only if

�e�0=� F ð�mÞ=�m (31)

(since by definition ", f� 1). We obtain a nontrivial

constraint provided that 1 Gyr & � & 106–1012 Gyr,where the exact range depends on the mass splitting andthe decay channel considered. Models with � & 1 Gyrtrivially satisfy the bounds placed by radiation back-grounds, as do models with very large �, in accordancewith the discussion in Sec. III.

The ‘‘degenerate gravitino’’ scenario of Ref. [67] falls inthe category of models whose energy release may be con-strained by observations. In this scenario, the next-to-lightest supersymmetric particle (NLSP) and LSP have asmall mass splitting, �m� 10�2 GeV, such that theslightly heavier NLSP can decay into the LSP with emis-sion of a photon, with a lifetime comparable to the age ofthe Universe. These parameters satisfy the inequality (31)and thus the observational bounds represent a nontrivialconstraint on the model. Specifically, the photon bounds[1] constrain the relic NLSP abundance before decay in amixed (NLSPþ LSP) DM scenario, as explicitly shown inRef. [67].

As an example of models which trivially evade theconstraint (30), we mention here the inelastic DM scenar-ios [66]. These models aim to reconcile the results ofDAMA and CDMS direct-detection experiments, by evok-ing small mass splittings �m� 100 keV. However, sincethe decays occur very early (before recombination) thisscenario cannot be constrained by the present radiationbackgrounds. Substituting the lifetime of the heavier state�� ð102–103Þ yr into Eq. (31) of course trivially verifiesthis fact.

V. CONCLUSIONS

The decay lifetime is a fundamental property of darkmatter. Knowledge of this parameter may provide an es-sential clue for unravelling the particle nature of darkmatter. In the standard CDM scenario, dark matter particleinteractions play no role in the evolution of the Universesubsequent to DM freeze-out, which limits our ability toprobe DM particle interactions via cosmological observa-tions. However, there is a large class of DM models in

which DM particle interactions are in fact invoked toexplain cosmological observations. One such hypothesisis that DM decay may provide a possible solution to thesmall-scale structure problems of standard CDM.We have considered a class of DM models in which DM

decays at late times to a nearly degenerate daughter plusrelativistic particles, � ! �0 þ l. In this scenario, the rela-tivistic daughters carry only a small fraction, " ¼ �m=m�,

of the parent DMmass and thus decays do not alter the DMenergy density. However, the energy acquired by thedaughter �0 effectively heats the halo, and has been pro-posed as a means of modifying halo structure in such a wayas to bring CDM predictions into better agreement withobservation.We focused on the scenario in which the relativistic

daughters consist of standard model particles, and derivedconstraints on the cases in which those daughter particlesare either e� or neutrinos. In Fig. 1 we summarize theseconstraints and compare them with the analogous photonbounds obtained in Ref. [1]. Since photons provide themost stringent bounds and neutrinos the least stringent,these results, taken together, delineate the range of lifetimebounds for these models.Our constraints on the e� decay mode lie between those

for photons and neutrinos, though we note that there aresome masses for which the e� constraints become compa-rable to those for photons. Note that many models thatallow the process � ! �0 ��� will also allow � ! �0eþe�,for which stronger bounds apply.For the e� (and �) decay modes, the lifetime must either

be much greater than the age of the Universe (for whichdecay cannot affect structure formation) or smaller than�1 Gyr. Though this eliminates some of the parameterspace of interest in structure evolution, a significant portionremains open. For the ��� decay modes, a much largerrange of lifetimes is permitted.Models which are not necessarily motivated by structure

formation considerations, but nonetheless entail decay be-tween particle states of similar mass, may also be con-strained by the bounds presented here. This is possible,provided that the relic density of the heavier state from theearly Universe is not too small, and/or the decay lifetime islarger than 1 Gyr. The constraints and the analysis pre-sented here can provide generic, useful guidelines forconstructing models which feature decay modes betweenclosely degenerate states.We note that our limits in Figs. 1–3 are strictly appli-

cable only for " � 1 (so that the approximation El ’ �mis valid). However, for " ¼ 1, our monoenergetic con-straints correspond to those for the decays � ! eþe�and � ! ��� (i.e., where there is no �0). For the ��� casethis simply reproduces the results of Ref. [2]. For the e�case, our bounds inferred using inverse Compton scatteringand bremsstrahlung emission, synchrotron radiation, andthe positron flux, reproduce the ones derived in Ref. [10],

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using the same response functions employed here.However, the monoenergetic limits of Fig. 2 inferredfrom the in-flight and at-rest annihilations represent strongnew constraints of general applicability for the decay mode� ! eþe�, in the interval 2 MeV & mDM & 1 GeV.

ACKNOWLEDGMENTS

N. F. B. and K. P. were supported, in part, by theAustralian Research Council, and A. J. G. by theCommonwealth of Australia. We thank F. Melia, J.Beacom, and H. Yuksel for useful discussions.

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