The cosmic X-ray and gamma-ray background from dark matter annihilation

18 Pages·2011·0.59 MB·English

by  Mark Vogelsberger,,,

The cosmicX-ray and gamma-raybackground from dark matter annihilation
Jesu´sZavala,1,2,∗ MarkVogelsberger,3 TracyR. Slatyer,4 AbrahamLoeb,3 andVolker Springel5,6
1Max-Planck-Institutfu¨rAstrophysik,Karl-Schwarzschild-Straße1,85740GarchingbeiMu¨nchen, Germany
2Department ofPhysicsandAstronomy, UniversityofWaterloo, Waterloo, Ontario,N2L3G1, Canada†
3Harvard-Smithsonian CenterforAstrophysics, 60Garden Street, Cambridge, MA02138, USA
4School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
5HeidelbergInstituteforTheoreticalStudies, Schloss-Wolfsbrunnenweg 35, 69118Heidelberg, Germany
6Zentrumfu¨rAstronomiederUniversita¨tHeidelberg,AstronomischesRecheninstitut,Mo¨nchhofstr. 12-14,69120Heidelberg,Germany
The extragalactic background light (EBL) observed at multiple wavelengths is a promising tool to probe
thenatureofdarkmatter. Thisradiationmightcontainasignificantcontributionfromgamma-raysproduced
1 promptlybydarkmatterparticleannihilationinthemanyhalosandsubhaloswithinourpast-lightcone. Addi-
1 tionally,theelectronsandpositronsproducedintheannihilationgiveenergytothecosmicmicrowavephotonsto
0 populatetheEBLwithX-raysandgamma-rays.Tostudythesesignals,wecreatefull-skymapsoftheexpected
2 radiationfrombothofthesecontributionsusingthehigh-resolutionMillennium-IIsimulationofcosmicstruc-
n tureformation.OurmethodalsoaccountsforapossibleenhancementoftheannihilationratebyaSommerfeld
u mechanismduetoaYukawainteractionbetweenthedarkmatterparticlespriortoannihilation. Weuseupper
J limitsonthecontributionsofunknownsourcestotheEBLtoconstraintheintrinsicpropertiesofdarkmatter
usingamodel-independentapproachthatcanbeemployedasatemplatetotestdifferentparticlephysicsmod-
2
els.Theseupperlimitsarebasedonobservationalmeasurementsspanningeightordersofmagnitudeinenergy

(fromsoftX-raysmeasuredbytheCHANDRAsatellitetogamma-raysmeasuredbytheFermisatellite),and
]
O onexpectationsforthecontributionsfromnon-blazaractivegalacticnuclei,blazarsandstarforminggalaxies.
Toexemplifythisapproach,weanalyzeasetofbenchmarkSommerfeld-enhancedmodelsthatgivethecorrect
C
abundance ofdark matter, satisfyconstraintsfromthecosmicmicrowave background, andfitthecosmicray
.
h spectrameasuredbyPAMELAandFermiwithoutanycontributionfromlocalsubstructure. Wefindthatthese
p modelsareinconflictwiththeEBLconstraintsunlessthecontributionofunresolvedsubstructureissmalland
- thedarkmatterannihilationsignaldominatestheEBL.Weconcludethatprovidedthecollisionlesscolddark
o
matterparadigmisaccurate, evenforconservativeestimatesofthecontributionfromunresolvedsubstructure
r
t andastrophysicalbackgrounds,theEBLisatleastassensitiveaprobeofthesetypesofscenariosasthecosmic
s microwavebackground. Moregenerally,ourresultsdisfavoranexplanationofthepositronexcessmeasuredby
a
thePAMELAsatellitebasedonlyondarkmatterannihilationinthesmoothGalacticdarkmatterhalo.
[

2 PACSnumbers:95.35.+d,95.85.Nv,95.85.Pw
v
6
7 I. INTRODUCTION Theexcessofpositronsincosmicraysabove10GeVreported
7 bythePAMELAexperiment[3]isoneoftheseobservations,
0
andalthoughotherastrophysicalsources, suchaspulsars[4]
. A broad class of particles known as Weakly Interacting
3 andsupernovaremnants[5],couldexplainthesignal,thepos-
MassiveParticles(WIMPs),arethebeststudiedandarguably
0 sibilityofdarkmatterannihilationremainsattractiveandhas
1 the most favoredcandidatesto be the primarycomponentof motivated a significant number of papers on the topic. It is
1 cosmicdarkmatter.Themostprominentexampleofsuchpar-
howevernecessarytoinvokelargeannihilationratesandspe-
: ticles is the neutralino that arises naturally in supersymme-
v cificannihilationchannelstoexplaintheanomalieswithdark
try(SUSY);forrecentreviewsonneutralinodarkmattersee
i matterannihilationalone[6].Theseratesareordersofmagni-
X [1, 2]. WIMPs can explain the observed abundance of dark
tudelargerthantheonesobtainedassumingthestandardval-
r matterinanaturalwayandbecausetheybehaveascolddark
a uesfortheannihilationcrosssectionthatgivethecorrectrelic
matter(CDM)theyarealsofavoredbytheprevailingΛCDM
densityofdarkmatter. Duetotheirhigherdensities,substruc-
cosmology, which is the most successful model of structure
turesinthelocaldarkmatterdistributioncanboosttheanni-
formationtodate. Furthermore,mostWIMPsareparticularly
hilationrates,butnottotherequiredlevel[7]. Thenon-linear
appealingbecausetheyofferarelativelyhighchanceofdetec-
collapseofcollisionlessdarkmatterhalosleadstotheforma-
tioninthenearfuture,through:i)directdetectionexperiments
tionofcaustics,whichduetotheirhighdensitycouldsignifi-
onEarthlookingforthe recoilofordinarymatter byscatter-
cantlyincreasetheannihilationrate.However,thisincreaseis
ingofWIMPs,andii)indirectsearchesthatlookforstandard
actuallymuchlesssignificantthanpreviouslythought[8,9].
modelparticlesproducedintheannihilationofWIMPs.
Intheinnerpartsofhalos,itisessentiallynegligibleandcan
Anumberofobservationsinrecentyearshavehighlighted notbeinvokedtoexplainthehighannihilationratesrequired
anomalies that might be caused by dark matter annihilation. toexplainthePAMELAmeasurements.
Alternatively, an elegant solution may lie in an enhance-
mentoftheannihilationcrosssectionbyaSommerfeldmech-
anism produced by the mutual interaction between WIMPs
†Currentaffiliation
∗CITANationalFellow;Electronicaddress:[email protected] prior to their annihilation [10–12]. This enhancementcould
2
easilybelargeenoughtoexplaintheanomalousexcessofcos- Byusingthisapproachwearealsoabletoeasilyincludea
micraypositrons. velocity-dependentannihilation cross section via a Sommer-
The annihilation rate can however not be arbitrarily large feldmechanism. Typically,theenhancementisinverselypro-
eitherasitisconstrainedbydifferentobservables. Forexam- portionaltothelocalvelocitydispersionofdarkmatterparti-
ple, darkmatter annihilationcan ionizeandheatthe photon- cles. Sinceourmethodisbasedonaveragevaluesoftheanni-
baryonplasmaatrecombination,creatingperturbationsinthe hilationrateinsidehalosandtheirsubhalos,theSommerfeld
CosmicMicrowaveBackground(CMB)angularpowerspec- enhancementissimplygivenbythemeanvelocitydispersion
trum [13–15]. It can also alter the relic abundance of dark in each halo (subhalo), which is available in the simulation
matter significantly [16, 17], and produceimportantµ and andcanbemeasuredaccurately.
y typedistortionsoftheCMB[17,18].Alltheseobser−vables The paper is organized as follows. In Section II, we out-
he−nceconstrainthedegreeto whichthe Sommerfeldmecha- line the formalism to calculate the extragalactic background
nismcanenhancethecrosssection. Nevertheless,itispossi- radiation coming from in situ and up-scattered photons. A
bletosatisfyalltheseconstraintsandatthesametimeexplain description of the Sommerfeld enhancementmodel we used
thepositronexcessmeasuredbyPAMELA[19]. and its implementationis given in Section III. The observa-
tionalupperlimitsandmainresultsofourworkonthecosmic
Anadditionalsetofobservationswiththepotentialtocon-
backgroundradiationarepresentedinSectionIV. Finallywe
straintheannihilationcrosssectioncancomefromtheanaly-
presentasummaryandconclusionsinSectionV.
sisoftheextragalacticbackgroundradiationatmultiplewave-
lengths. The annihilation of WIMPs can manifestitself as a
cosmicbackgroundradiationwithgamma-rayphotonsbeing
II. ANNIHILATIONRADIATIONFORMALISM
producedpromptlyinallextragalacticsourceswithhighdark
matter density [20–28]. This gamma-ray radiation is com-
plemented towards lower energies by a diffuse extragalactic Our goal is to analyze the cosmic dark matter annihila-
backgroundinphotonsthatwerenotproduceddirectlyinthe tion background (CDMAB), or more specifically, the radia-
annihilationbutgainedenergyviainverseComptonscattering tionproducedbydarkmatterannihilationinallextragalactic
offtheenergeticelectronsandpositronsproducedduringthe sourcesintegratedoverallredshiftsalongtheline-of-sightof
annihilation[29–31]. a fiducialobserver,locatedat z = 0, foralldirectionson its
two-dimensionalfullsky.Tothisend,wefirstdefinethelocal
The data collected by several telescopes over the last
photonemissivity:
decadeshavegivenusameasurementoftheextragalacticra-
diationbackgroundfromsoftX-raystohardgamma-rays(e.g.
f dN σv
[32]). Inthisbroadenergyrange,mostoftheradiationisex- = WIMPEρ (~x)2S(σ (~x)), f = h i0
E 2 χ vel WIMP dE m2
pectedtobeproducedbyastrophysicalmechanismsdifferent χ
(1)
from dark matter annihilation. This has been partially con-
wherem andρ arethemassanddensityofWIMPs, σv
firmedbyaccountingfortheradiationofknownsourcesand χ χ h i0
isthethermallyaveragedproductoftheconstants-waveanni-
by estimating the contribution of an expected population of
hilationcrosssectionandthevelocityintheabsenceofSom-
sourcesyettobeobserved.Thiscombinedsetofobservations
merfeldenhancement,and dN/dE is the differentialphoton
andexpectationsputsstrongconstraintsonthecontributionof
yieldperannihilation.Thevelocitydispersiondependentfac-
darkmatterannihilation,beingspeciallystringentinthesoft-
tor S(σ ) booststhe value of σv througha Sommerfeld
X-ray regime where 90% of the emission comes from X- vel h i0
∼ mechanism(seesectionIII). Wenotethatforconsistency,the
raypointsources,mostlyActiveGalacticNuclei(AGN)[33],
value of σv should also give the correctdark matter relic
andonthegamma-rayregimewhereblazarsandstar-forming h i0
density.
galaxies are expectedto contribute significantly to the back-
TheCDMABisgivenbythespecificintensity,the energy
groundradiation,atthelevelof 70%[34,35].
∼ ofphotonsreceivedperunitarea,time,solidangleandenergy
Thehypotheticalbackgroundradiationcomingfrom(orbe-
range:
ing up-scattered in) all dark matter halos and their subhalos
withinourpastlightconehasbeenstudiedbydifferentauthors 1 dr
in the pastusing analyticapproachesto modelcosmicstruc- I = 4π E(E0(1+z),z)(1+z)4e−τ(E0,z), (2)
tureformation.Anapproachbaseddirectlyonhigh-resolution Z
numericalsimulationsishoweverdesirablesince it moreac- wheretheintegralisoverthewholelineofsight,r istheco-
curately capturesthe non-linearphase of the evolution, even movingdistanceandE isthephotonenergymeasuredbythe
0
though the simulation imposes a resolution limit for small- observeratz =0. Notethat isevaluatedattheblue-shifted
E
est structure. It is then possible to construct simulated sky- energy(1+z)E alongtheline-of-sighttocompensateforthe
0
mapsof the backgroundradiationthatgive a more complete cosmologicalredshifting.Theexponentialtermwithaneffec-
description of the signal. Such an approach was developed tive opticaldepth τ(E ,z) takes into accountthe absorption
0
in[36]toanalyzetheextragalacticgamma-rayradiationpro- ofphotonsbythematterandradiationfieldalongtheline-of-
ducedinsitubyannihilationusingthestate-of-the-artMillen- sight. The relevantprocessesof photonabsorptionand their
niumIIsimulation[37].Inthispaper,weextendthisapproach treatmentaredescribedinAppendixB.
to include the contribution from CMB photons scattered by Inthiswork,wefocusontwodifferentcontributionstothe
theelectronsandpositronsproducedduringannihilation. differential photon yield per annihilation event dN/dE. In
3
thefollowing,wedescribethesecontributionsthatwereferto Thisis becausespatialdiffusionis onlyrelevantatrelatively
asinsituandup-scatteredphotons. smallscales,withinafewkpcofthecenterofdarkmatterha-
los[38]. However,weareinterestedinacosmologicalback-
groundradiationwhere most of the signal in a given area in
A. Insituphotons theskycomesfromunresolvedsourcesfaraway,wherespa-
tialdiffusionisclearlyirrelevant.Inthiscase,thesteady-state
The in-situ photons are directly created due to the anni- solutiontoEq.(3)canbeapproximatedby:
hilation process. They are in the gamma-ray energy range
dn 1 mχ
andare producedbythreemechanisms: (i)continuumemis- e(E ,z) dE′ Q (E ,~x)
dE e ≈ b (E ,z) e e e
sionfollowingthedecayofneutralpionsproducedduringthe e e ZEe
hadronization of the primary annihilation products; (ii) mo- σv ρ (~x) 2 dn˜
noenergetic lines for WIMP annihilation in two-body final = h i0 χ S(σvel(~x)) e(Ee,z). (4)
2 m dE
statescontainingphotons;(iii)internalbremsstrahlungwhen (cid:18) χ (cid:19)
the final productsof annihilation are charged, leading to the Theenergylossrate,b (E ,z),forelectronsandpositrons
e e
emission of an additional photon in the final state. Process receivescontributionsfromdifferentinteractionprocesses:IC
(i) is dominant at most gamma-ray energies, but processes scatteringwithambientphotons,synchrotronradiationinthe
(ii) and (iii) produce distinctive spectral features intrinsic to ambient magnetic field, Coulomb scattering with free elec-
theannihilationphenomenon. Thisinsitucontributiontothe trons,ionizationofatomsandbremsstrahlungradiationinin-
CDMAB has been studied in detail before by different au- teractions with the ambient matter field. As we explain in
thors. Inthisworkwefollowtheanalysisof[36], extending AppendixA,amongallthesecoolingprocessesweonlycon-
theirresultstolowerenergiesasdescribedbelow. siderthefirstonesinceitdominatesthephotonenergyrange
weareultimatelyinterestedin.TheenergylossterminEq.(4)
ishencegivenbyEq.(A1).
B. Up-scatteredphotons We further assume that the electrons and positrons pro-
ducedintheannihilationprocessloseenergyandreachequi-
Theup-scatteredphotonsoriginateindifferentlyproduced libriuminstantaneously(inacosmologicaltimeframe),scat-
backgroundphotonsthatgainenergyduetotheirinteractions teringtheCMBphotonsatthesameredshiftatwhichthean-
withparticlesproducedintheannihilationofdarkmatter. We nihilationtakesplace(e.g. [39]). Theseup-scatteredphotons
concentrateexclusivelyonInverseCompton(IC)scatteringas haveadifferentialphotonspectrumgivenby:
themechanismcontributingtotheup-scatteringofthesepho-
dN dn˜
tons,andontheCMBasthemainphotonbackground.There IC(E,z)= dE e(E ,z)P˜ (E,E ,z), (5)
e e IC e
dE dE
areadditionalbackgrounds,likestellarandinfraredlight,that Z
aredominantclosetogalacticdiscsinthecenterofrelatively wheretheICpowerperscatteredphotonenergyis:
massive halos. However, most of the CDMAB comes from
theintegratedeffectoflowmasshalosandsubhalos(seeAp- P˜ (E,E ,z)=c dE˜ n (E˜,z)σ (E,E ,E˜). (6)
IC e CMB KN e
pendix C). In these places, the stellar component is rather
Z
small and the mean number density of starlight and infrared
Here,n (E˜,z)dE˜ isthenumberdensityofCMBphotons
photonsismuchlowerthanthatoftheCMB. CMB
intheenergyrange(E˜,E˜+dE˜)atredshiftz:
Electronsandpositronsaretheannihilationbyproductspar-
ticipatinginthescattering.Theseparticlesarequiteenergetic 8π E˜2dE˜
andhavethereforeusuallyalargeγ =1/ 1−(v/c)2factor. nCMB(E˜,z)dE˜ = (hc)3exp[E˜/(k T (1+z))] 1, (7)
This implies that they can up-scatter low energy photons to B 0 −
p
significantly higher energies, because of the γ2-dependence where T = 2.725 K is the CMB temperature today, which
0
of the peak energy of up-scattered photons. In this process, increases with redshift like (1 + z)T . Finally, σ is the
0 KN
CMB photons increase their energy into the X-ray and low differentialKlein-Nishinacross-sectionforICscattering
gamma-rayregimes[29].
Thedifferentialelectron(andpositron)yieldthatisrelevant 3σ m c2 2
σ (E,E ,E˜)= T e G(q,Γ ), (8)
fortheICup-scatteringoftheCMBphotonsisfoundbysolv- KN e 4E˜ (cid:18) Ee (cid:19) e
ing a diffusionequationthat takes into accountthe diffusion
whereσ istheThomsoncross-section,m theelectronmass,
andenergylossesoftheseparticles: T e
and
∂ dn dn ∂ dn
∂tdEee =∇(cid:20)De∇dEee(cid:21)+ ∂Ee (cid:20)bedEee(cid:21)+Qe, (3) G(q,Γe)= 2qlnq+(1+2q)(1 q)+ (Γeq)2(1−q) ,
" − 2(1+Γeq) #
where dn /dE is the equilibriumelectron spectrum, D =
e e e (9)
D (E ,~x) is the diffusion coefficient, b = b (E ,~x) is the
e e e e e with
energy loss term and Q = Q (E ,~x) = /E [77] is the
e e e e e
sourcefunction. SpatialdiffusionduetoscaEtteringonthein- 4E˜E E
Γ = e , q = . (10)
homogeneitiesoftheambientmagneticfieldcanbeneglected. e (m c2)2 Γ (E E)
e e e
−
4
Note thattheKlein-Nishinacross-sectiondependsonthe in- caseschosenabove,anduseEq.(11)togettheaverageanni-
creasedenergyoftheup-scatteredphotonE,theenergyofthe hilation boost S(σ ) for each halo. Since we can estimate
vel
originalCMBphotonE˜,andontheenergyE oftheelectron the change on the values of S and σ for a differ-
e max vel,max
that does the IC scatter. The limits for the various integrals ent set of parameters, the results we obtain later using these
abovearegivenbythekinematicconstraintoftheICscatter- representative cases serve us to analyze the whole range of
ingrequiring1/[4(E /(m c2))2]<q <1. possibilities that are expected for a Sommerfeld mechanism
e e
producedbyaYukawapotential.
The Sommerfeld enhancement alters the relic density of
III. SOMMERFELDENHANCEMENT dark matter [16, 17]. During freeze-out, while the Sommer-
feld enhancement is generally (1), it is not negligible and
O
Weincludeinouranalysisascenariowheretheannihilation canconsequentlyhavean (1)effectontherelicdensity(re-
O
isenhancedbytheSommerfeldmechanism(e.g.[10–12,40]), quiring a reduction of the underlyingannihilation cross sec-
restrictingittothecasewheretheinteractionbetweenWIMPs tion to compensate). After kinetic decoupling of the dark
prior to annihilation is mediated by a scalar boson of mass matter from the radiation bath, the typical velocities of the
m through a Yukawa potential with coupling constant α darkmatterparticlesdecreaserapidly: evenfornon-resonant
φ c
(e.g. [41]). This case encompassesthe largemajority of the (butunsaturated)enhancement,theenhancedannihilationrate
modelsthataretypicallyusedintheliteraturetoaccountfor keepspacewiththeuniverse’sexpansion,andforresonanten-
theSommerfeldenhancement,includingthemostcommonof hancementthedarkmatterannihilationscanactuallyrecouple
thesewhereS 1/σ (theso-called“1/v”boost). Evenin (dependingontherelativetemperaturesoffreeze-out,kinetic
vel
modelswithne∝arly-degenerateinteractingstatesand/ormulti- decouplingandsaturationoftheenhancement),greatlyreduc-
pleforcecarriers,whilethedetailsoftheenhancementdiffer, ingtherelicdensity. Oncetheenhancementsaturates,thean-
the general features remain similar. The enhancement satu- nihilationrate no longerkeepspace with the expansionrate,
rates at sufficiently low velocities due to the finite range of andthecomovingdensityofdarkmatterremainsfixed. Inall
the Yukawa interaction. For certain combinationsof α and cases there is a significant effect on the relic density, and in
c
m ,resonancesassociatedwithzero-energyboundstatesap- ordertoproducethecorrectabundancetoday,thevalueofthe
φ
pear [78]. Close to these resonances, the enhancement gets annihilationcross-sectionbeforetheonsetoftheenhancement
significantlylargerforlowvelocitiesandscalesas1/σ2 . In needstobesmallerthanforthecasewithoutSommerfelden-
vel
thiscase,theenhancementalsosaturateseventuallyduetothe hancement.
finite lifetime of the states. Regardless of the values of the This result is relevant because it implies that any particle
parameters, the boost to the cross-section disappears for ve- physicsmodelwithoutenhancementchosentosatisfytheob-
locities comparableto the speed of light. This argumenthas servationalboundsontheabundanceofdarkmatterneedsto
often beeninvokedto infer thatthe darkmatter relic density berevisedoncetheenhancementisincludedtotestwhetheror
isunaffectedbytheSommerfeldenhancement,butithasbeen notitstillgivesthecorrectrelicdensity. Thefullyconsistent
shownrecentlythatthisassumptionisnotcorrect[17]. way to do so is to incorporatethe Sommerfeldenhancement
A detailed description of the Sommerfeld model studied intoaBoltzmanncodeandre-sampletheparameterspaceof
here has been presented elsewhere (e.g. [12, 41]). For the that particular model to find allowed regions. Here, we fol-
purposesof thiswork, we follow the descriptionof[17] and lowasimplerapproach.Accordingto[17],thevalueof σv
0
h i
mentionthattheenhancementS(σ )tothes-wavecontribu- shouldbelowerbyafactorbetween1and10comparedtothe
vel
tiontotheannihilationrateisgivenby: casewithoutenhancementinordertogetthecorrectrelicden-
sity.Theprecisereductionfactor,f ,dependsontheintensity
Ω
σv = σv S(σ ), oftheenhancement:f 0.1nearresonancesandf 0.5
h i h i0 vel Ω ∼ Ω ∼
S(σ )= 1 1S(β)β2e−β2/4σv2el dβ , (11) ospffo-nredsionngafncef.aTcthoerr,ewfoerer,obuyghmlyulttaipkleyiinngtohσacvcio0ubnyttthheeceoffrerec-t
vel 2σ3 √π Ω
(cid:18) vel Z0 (cid:19) on the relic density. In this way, a model without enhance-
mentthatgivesthe correctrelic density with σv willalso
whereβ = vrel/cistherelativevelocitybetweentheannihi- givetherightrelicabundancewithSommerfeldhenih0ancement,
latingpair[79].
provideditsannihilationcrosssectionintheearlyUniverseis
Fordefiniteness,wechoosetwosetsofparametersthatfall
chosentobef σv .
within currentlyfavoredregionsof the parameterspace (e.g. Ωh i0
[41]): case i) off-resonance: m /m = 5 10−4, α =
φ χ c
3 10−2andcaseii)near-resonance:m /m ×=2.98 10−4,
φ χ
α×= 3 10−2. Theformerisrepresentativeofthes×tandard IV. EXTRAGALACTICCDMAB
c
×
“1/v”boostwithamaximumenhancementS 2000for
max
σ 10−5. Thelatterisatypicalresonance∼case with The procedure we follow to construct the simulated sky
vel,max
S 1/σ∼ atintermediatevelocitiesandS 1/σ2 atlow maps of the contribution of dark matter annihilation to the
vel∝ocitiesveulptoasaturationS 106 for∝σ vel 6 X-ray and gamma-ray extragalactic background radiation is
max vel,max
10−7. ∼ ∼ × essentially an extension of the one discussed in [36]. For a
By solving the Schro¨dinger equation for s-wave annihila- detailed description of the map-making technique we used,
tion in the non-relativistic limit, we obtain S(β) for the two wehencereferthereadertosection5.1ofthatpaper.
5
AccordingtoEq.(1),thelocalannihilationratedependson
thesquareofthelocaldensityofdarkmatter. Forthecompu-
tationofthecosmologicalbackgroundfromanN-bodysimu-
lation,itismorereliabletouseanalyticallyintegratedquanti-
tiesoverwholedarkmatterhalos(basedonscalinglawstested
withextremelyhigh-resolutionsimulationsofMW-likehalos
[42]) instead of trying to use individual simulation particles
directly, which are subject to stronger resolution effects and
numericalnoise[36].Usingthismethod,eachpixelinoursky
mapsreceivescontributionsofallinterveningresolvedhalos
and subhalos along its correspondingpast light cone. Addi-
tionally,weaddtheexpectedcontributionofunresolvedstruc-
tures down to the dampingscale limit of WIMPs (10−6M⊙,
seesections5.2-5.4of[36]). Sinceweareexploringthecase
with Sommerfeld enhancement in this paper, the formulae
givenin [36] needto be alteredaccordingly. In AppendixC
wedescribehowweaccomplishthis.
The signal depends of course on the value of the photon
yield dN/dE as well, which contains contributions from in
situ and up-scattered photons. These are determined by the
intrinsicpropertiesofWIMPs. Asanexample,wetakeaneu-
tralinowithamainannihilationchannelintob¯b. Inparticular, FIG.1: Photon yieldfor dark matter annihilation intheX-rayand
gamma-ray energy range for a∼ 185 GeV neutralino annihilating
we use a benchmark point within the minimal supergravity ¯
intobb. Thecontributionsfromin-situandup-scatteredCMBpho-
(mSUGRA) framework (model L in Table I of [36]). This
tonsareshownwithsolidredandbluelines, respectively. Forref-
benchmarkpointhasmχ 185GeVwithannihilationintob¯b erence,thein-situandequilibriumelectronyieldsfromannihilation
witha99%branchingrati∼o,and σv 6.2 10−27cm3s−1. areshownwithdashedredandgreenlinesrespectively(theequilib-
Itbelongstotheso-called“bulkrhegioin∼”with×inthemSUGRA riumspectrumasdefinedinEq.(4)wasscaledbyafactorof10−16
5-dimensionalparameterspacethatisconsistentwithcurrent toshowitinthesamefigure). Alsoshowninthefigurewithablack
constraints on the relic density of neutralinos (if neutralinos dottedlineisthetotalphotonyieldfrombenchmarkmodel1ofTable
makeupforalltheobservationallyinferreddarkmatterden- 1,seesectionIVC.
sity). We obtain the photon, electronand positronyields for
thismodelusingthenumericalcodeDarkSUSY[43,44]with
theinterfaceISAJET[45]. mumandminimumvaluesoftheextrapolationforunresolved
InFig.1,weshowthefinalphotonyieldspectrumfordark subhalos, which encompassesthe astrophysicaluncertainties
matterannihilationintheX-rayandgamma-rayenergyrange in the contributionby low-mass subhalosthat can not be re-
fortheexamplejustdescribed.Thecontributionsfrominsitu solvedbytheMillenniumIIsimulation(seeAppendixC).The
andup-scatteredCMBphotonsareshownwithsolidredand medium-grayanddark-grayshadedregionsare forthe cases
blue lines, respectively. For reference, the in situ and equi- with Sommerfeld enhancement, off- and near-resonance, re-
libriumelectron(positron)yieldsfromannihilationareshown spectively.
withdashedredandgreenlines,respectively[80]. Themain AccordingtoFig.2,anymodelwithaphotonyieldsimilar
bumpandsecondarypeakthatareclearlyshownfortheinsitu to the one we used as an example (see Fig. 1), and with a
photonscorrespondtothetwomainmechanismsmentionedin mass 200GeV and σv 6 10−27cm3s−1, couldbe
0
∼ h i ∼ ×
section IIA, neutralpiondecayand internalbremsstrahlung, ruled out, depending on how relevant the contribution from
respectively. The figure shows clearly that although in this unresolvedhalosandsubhalosis.AnysignificantSommerfeld
case the largest photon yield is in the gamma-ray regime, enhancementisclearlyruledoutbyabsolutemeasurementsof
there is a significantamountof X-ray radiation producedby thebackgroundinthiscase.
IC scatter of the CMB photons. The contribution from up- ItisimportanttonotethattheSommerfeldmechanismisof
scatteredCMBphotonsisthedominantfeatureforotherpar- relevanceforneutralinosonlyformasses&TeVinthecaseof
ticlephysicsmodels. AsanexampleofthisweshowinFig.1 aminimalSUSYmodellikemSUGRA(e.g.[12,40]). Inthis
the total photon yield for one of a set of benchmarkmodels case, the force carriers responsible for the enhancement are
thatweuselaterinsectionIVC.Theshapeandnormalization the W and Z gauge bosons [81]. Therefore, boosts of order
forthisbenchmarkmodel1(seeTable1)arerepresentativeof 1000or evenlargerare onlypossible forneutralinoswith
allthebenchmarkmodelswewilluse. m∼uchhighermassesthanthemodelwehavechosenasanex-
Fig.2showsthecontributionfromdarkmatterannihilation ampleinFigs. 1and2. Theneteffectofahigherneutralino
to the X-rayandgamma-rayextragalacticbackgroundradia- mass in the inputphoton and positron (electron)spectra is a
tionfortheparticularSUSYmodeldescribedabove.Thecase shiftoftheX-rayandgamma-raypeaksshowninthesefigures
withoutSommerfeldenhancementisshownwithin thelight- towardshigherenergies. Nevertheless,thedN/dE spectrum
gray shaded region. This region is bracketed by the maxi- shown in Fig. 1 is generic for any model with a WIMP an-
6
matesoftheextragalacticemissionattheseenergieshavenot
beenpossible[33,53]. Intherange1keV E 200keV,
0
≤ ≤
theextragalacticX-raybackgroundhasbeenstudiedindetail
by satellites such as CHANDRA, SWIFT and INTEGRAL.
We take the absolute measurementsobtainedusingthe latter
twosatellitesaccordingtotheanalysisof[46](redsymbolsin
Fig.2),[47](orangesymbols)and[32](yellowsymbols). At
intermediateenergies, 300 keV E 30 MeV, the mea-
0
≤ ≤
surements come from the Solar Maximum Mission (SMM)
[49] and COMPTEL [50]. These measurements are shown
with green and light blue points, respectively. Finally, ob-
servationsbasedonEGRET [51] andrecentlyonFermi[52]
have estimated the cosmic background in gamma-rays from
40 MeV to 100 GeV. These estimates are shown with dark
blueandpurplesymbols,respectively.
The observational data we have described above give a
measurementofthetotalextragalacticX-rayandgamma-ray
background radiation. Over the full energy range, most of
thesignalisexpectedtocomefromphotonsproducedbydif-
ferentastrophysicalsourcesinmechanismsthatareunrelated
FIG.2: CDMABspectrumincludingin-situandup-scatteredCMB to dark matter annihilation. The contribution from the latter
photons (gamma-rays from annihilation and CMB photons up- is likely to be a subdominantcomponent of the total signal,
scattered to X-ray energies by electrons and positrons of the an- whichisespeciallytrueatlowerenergies. Upperlimitstothe
nihilation) for the cases of no-enhancement (light-gray), enhance- emissionthathavenotbeenaccountedforbyknownsources
ment away from a resonance (medium-gray) and near a resonance forE <8keV,havebeenfoundusingCHANDRAdata(red
0
(dark-gray). The upper and lower limitsof each stripebracket the arrows)[33]. Approximatelylessthan10%oftheintegrated
uncertainty in the extrapolation of unresolved subhalos in thesim-
specificintensityisunresolvedbetween1keV<E <8keV.
ulation. All cases are for a model with annihilation mainly into 0
b¯b, mχ ∼ 185 GeV and hσvi0 ∼ 6.2 × 10−27cm3s−1. They sFioornhiasrdexXp-ercatyesd(t1o0ckoemVe≤froEm0C≤o2m0p0tokne-Vth),inmAoscttiovfethGeaelamctiisc-
allgiveapproximately thecorrect relicdensity. Observations from
Nuclei (AGN). We use the model presented in [48] to put a
soft X-rays to gamma-rays are marked with red to violet, follow-
ing approximately a rainbow color pattern: red symbols [46], red conservativeupperlimitontheunresolvedcomponentofthe
arrows (Chandra, [33]), orange symbols (INTEGRAL, [47]), yel- emissionattheseenergies(yellowarrows)[82]. Themodeling
low symbols (SWIFT BAT, [32]), yellow arrows [48], green area intheMeVrangeismoreuncertain.Accordingtosomeanal-
(SMM,[49]),lightblue(COMPTEL,[50]),blue(EGRET,[51]),vi- yses,blazarsarethoughttocontributesignificantlytotheradi-
olet(Fermi-LAT,[52]),violetarrows[34,35]. Thepointswitherror ation[54],butothersarguefornon-blazarAGNsasthemain
bars are absolute measurements with2σ or 1σ errors. The arrows contributorsto the MeV radiation [55]. We will not attempt
pointingdownwardsarebestestimateupperlimitsoftheunresolved
tomodelthecontributionofthesesourcesduetothiscontro-
componentofthesignal,thatis,thesignalthatcannotbeaccounted
versy, butwe note thatthe constraintson the contributionof
forbyalreadyknownorexpectedsources.
darkmatterannihilationtotheMeVbackgroundareexpected
to besignificantlylower thanthoseseen in Fig. 2. Recently,
the Fermi-LATcollaborationhas estimated the blazar contri-
nihilatingmainlyintob¯b. Ifsuchagenericmodelallowsthe
bution to the gamma-ray backgroundin the 0.1 100 GeV
inclusion of a new scalar boson responsible of the Sommer- −
energyrange. Its totalspecific intensity in this energyrange
feldenhancement,thentheformalismdescribedinSectionIII
(i.e. its integrated flux between these energies) down to the
isapplicableandFig.2showstheexpectedlevelofenhance-
minimumdetectedsourcefluxis 16%ofthederivedvalue
mentoftheCDMABduetothismechanism. ∼
for the cosmic gamma-raybackground[34]. We use the en-
The symbols shown in Fig. 2 represent inferences for
ergy spectrum given by these authors (see their Table 6 and
theextragalacticX-rayandgamma-raybackgroundradiation
Fig.20)toaccountfortheblazarcontributionnotingthatthis
basedonobservationaldataasdescribedinthefollowing.
is a conservativeestimate since undetected sources certainly
contribute to the signal, see below. Star forming galaxies
are also expected to be a significant source of gamma-rays
A. Observations in this energy range. We use the model by [35] to include
thiscontribution(seetheirFig.1),whichaccountsfor 53%
∼
We are interested in measurements of the cosmic back- of the total specific intensity. We note that the energy spec-
ground radiation in an energy range going from soft X-rays trum of the contribution of star forming galaxies to the cos-
to gamma-rays: 0.1 keV E . 100 GeV. Because for mic gamma-ray background (as plotted in Fig. 2, i.e. E I)
0 0
≤
E < 1 keV the signal is completely dominated by galactic peaks at E 0.3 GeV, dominating over the blazar spec-
0 0
∼
and local emission that varies with time and position, esti- trum, and drops more steeply towards higher energies than
7
thetotalbackground.AtE &10GeVblazarsdominateover Eq.(2)canbewrittenas:
0
starforminggalaxieswithaspectrumshallowerthantheob- c
I(E )= E f (E (1+z∗))
served background. In this way, both populationscombined 0 0 WIMP 0
8π
taecncsoiutyntatfoEr ∼ 860%.3 G(∼eV46(E%) of 7th0eGmeeVa)s.uOrevdersptheceifiwchoinle- ρ2χ(~x,z)S(σvel(~x,z))e−τ(E0,z)dz, (12)
0 ∼ 0 ∼ (1+z)3 H(z)
0.1 100GeVenergyrange,theyaccountfor 69%ofthe Z
− ∼
totalintegratedflux. Based on this, the correspondingupper whereH(z)istheHubbleparameter[83]. Ingeneral,wedo
limits on the contributionfromadditionalsourcesare shown not know the value of z∗, it is model dependent. Neverthe-
withvioletarrowsinFig.2. less,wecansafelyapproximatetheupperlimitoftheintegral
inEq.(12)byz = 4inthecaseofthelowestX-rayenergies
We shouldcommentonthe uncertaintiesassociated to the
andbyz = 1forthehighergamma-rayenergies(andvalues
contributionofblazarsandstarforminggalaxies. Forthelat-
inbetweenforintermediateenergies).Thisisbecause 90%
ter,theseareconnectedtothegamma-rayluminosityfunction ∼
ofthesignalisproducedforz <4(z <1)intheformer(lat-
ofgalaxieswhichisultimatelyrelatedto thetime-dependent
ter) case. The relevantredshiftrangeis significantly smaller
global star formation rate density. The possible behaviors
at higher energies where photon absorption plays an impor-
of the gamma-rayluminosity functionand the most relevant
tantrole. Thisapproximationis goodenoughforany model
sources of uncertainty in the model (more importantly the
with a photon yield spectrum dN/dE similar to the one de-
cosmic star formation rate and the normalization given by
pictedinFig.1. Moregenerally,foranymodelwithaphoton
the inferred gamma-ray luminosity of the Milky-Way) have
yield thatis monotonicallydecreasingwith energy,the large
been considered by [56] using a similar modeling to that of
majorityof the signalat any givenenergywould come from
[35]. Theauthorsfindthatstar forminggalaxiesaccountfor
relatively low redshifts, since the contribution from higher
between 10% to 90% of the EBL measured by FERMI at
redshifts would correspondto higher-energy(and hence less
E 0.3GeV(seeFig.1of[56])withaspectralshapevery
0 ∼ abundant)initialphotons.Thisistruebecausetheastrophyis-
similar to the model we have chosen here. Since the contri-
calpartof the specific intensity thatgoesin the integrandof
butionfromstarforminggalaxiesisquiteuncertainandsince
Eq. (12), excluding the absorption factor, is essentially flat
the fiducial model we use lies closer to the upper value of
withredshift(see forexampleFig. 1 of[52]). Thus, we can
this contribution, we explore below the effects that a lower
usetheobservedupperlimitsontheunaccountedcontribution
contribution has in our results (see section IVC). If the ob-
toI(E ),andthevaluesofthissamequantitypredictedbya
servedcountdistributionofblazarsisextrapolatedtozeroflux 0
reference particle physics model to estimate upper limits on
then their contribution to the total observed signal between
f (E (1+z∗)):
0.1 GeV and 100 GeV is 23( 9)% (including statistical WIMP 0
andsystematicuncertaintie∼s)[34]±. Aswe mentionedbefore, IOBS(E )
f (E (1+z∗)) fREF (E (1+zREF)) 0 ,
thispercentagedropsto 16( 9)%whenonlysourcesdown WIMP 0 ≤ WIMP 0 IREF(E )
∼ ± 0
to the minimum flux are considered. At E 0.3 GeV the (13)
0
∼
minimumcontributionfromblazarsto the measuredspecific where the values associated with the reference model are
intensityis 7%(includinguncertainties). Takingthelower given with a superscript REF. The values of z∗ and zREF
limits of al∼l these uncertainties into account, star forming are in the interval (0,4) for E 10−5 GeV and in the in-
0
∼
galaxies and blazars would contribute minimally by 17% terval (0,1) for E 10 GeV. By choosing z = 0, so
atE 0.3GeV,afactorof5lowerthantheestimate∼weuse fREF is evaluate0d∼at the measured energy raRthEeFr than the
0 ∼ WIMP
here. (higher) effective energy of injection, we obtain a conserva-
tive constraint, since this function is monotonically decreas-
ing with respect to energy. Taking a higher value for z
REF
will strengthen the bound. By choosing z∗ = 0, we evalu-
ate f at the lowest possible energy for the purpose of
WIMP
comparing to the limit: this is not conservative for models
B. Constraintsonparticlephysicsmodels where f is a monotonically falling function of energy,
WIMP
in the sense that taking a larger z∗ leads to a weaker limit,
butsincethe signalis dominatedby the lowestredshifts, the
With the procedure we have previously outlined, we can
resulting uncertainty is quite small. We denote the upper
compare the prediction of any given particle physics model
bound on f (E) obtained by setting z∗ = z = 0
with the observational upper limits shown in Fig. 2. The WIMP REF
byfMAX (E).
modelgivesa photonand a positron (electron)inputspectra WIMP
We take as a reference model the example we have used
from the annihilation, and our map-makingcode producesa
throughoutthetext,andcomputefMAX forthecaseswithout
simulatedmapforaprescribedenergy.Afullspectrumcanbe WIMP
enhancement,andwithSommerfeldboost,off-resonanceand
thenproducedoncemapsatdifferentenergiesareconstructed.
near-resonance. The exclusion regions we obtain are shown
It is possible however to present robust limits on the part in Fig. 3 with the light-gray,medium-grayand dark-grayre-
of the signal that only dependson the intrinsic propertiesof gions, respectively, for these three cases, assuming a min-
WIMPs,namelyf inEq.(1).Thiscanbedonebynoting imum extrapolation for the contribution of unresolved sub-
WIMP
thatthereexistsaredshiftz∗alongtheline-of-sightforwhich structurestothesimulatedmaps. Thedashedlinesshowhow
8
FIG. 3: Limits on the value of fWIMP for dark matter annihila- FIG.4: Constraintsonthelocalvalueofthethermallyaveragedan-
tionaccordingtoobservationsofthecosmicX-rayandgamma-ray nihilation cross section (assuming a MB velocity distribution with
background radiation. The light-gray, medium-gray and dark-gray σvel = 150kms−1) as afunction of WIMP massfor annihilation
areas markexclusion regions for the casewithno Sommerfeld en- intoµ+µ−finalstates.Theseconstraintscomefromobservationsof
hancement, off- and near-resonance enhancement, respectively, in thecosmicX-rayandgamma-raybackground radiation. Theviolet
the case where the contribution of unresolved substructures to the contourshowsthe2σ bestfitregionofthismodeltothePAMELA
signalisminimal. Thedashedlinesshowhowtheseregionsareex- positrondataaspresentedin[52]. Theother linestylesandcolors
tended if thiscontribution ismaximal. The area between the limit areasinFig.3.
of each shaded region and its corresponding dashed line encom-
passes the uncertainty on the contribution of unresolved subhalos.
Thesymbolswitherrorbarsinthebottomshowthetheoreticalun-
has a very similar shape to the black dotted line shown in
certaintyontheconstructionofthisfigurefromFig.2; seetextfor
Fig.1,whichisbythewaysharedbyallthebenchmarkmod-
details.ThethickmagentalineisfortheSUSYmodelwehaveused
as an example: a ∼ 185 GeV neutralino annihilating into b¯b with elswedescribeinsectionIVC. Forthiscase,andformχ >
hσvi0=6×10−27cm3s−1. 100 GeV, the up-scattered photons contribute dominantly to
the background radiation. Instead of showing constraints
on σv we show in Fig. 4 the constraints on σv =
0 H
these exclusionlimits are extendedif a maximumextrapola- S(σhvel =i 150kms−1) σv 0,whichisathermalaverhageiovera
h i
tionistaken. Theright-anddown-wardserrorbarsinthefig- Maxwell-Boltzmann(MB)velocitydistributionwithaveloc-
uremarktheuncertaintyinthevaluesofz∗andzREF,respec- itydispersionof150kms−1(5 10−4 c). Thisroughlycor-
×
tively. Asmentionedbefore,theamplitudeoftheuncertainty respondsto the estimated local dark matter one-dimensional
dependsonthevalueoftheobservedenergy,beinglowerfor velocitydispersion. For both cases of Sommerfeldenhance-
higher energies. The thick magenta line shows the value of mentweareconsidering:S(σvel =150kms−1) 230.With
∼
f forourreferencecase. thischoice,wecancomparetheconstraintscomingfromthe
WIMP
The validity of a givenmodelcan be tested directlyusing extragalactic background radiation with the local values of
Fig. 3 without the need of computing the CDMAB for this σv that better fit the PAMELA data for an explanation of
h i
model. KeepinmindthatforthecaseswithSommerfelden- the positronexcessbased solely on darkmatter annihilation.
hancement,thevalueof σv inf isthevalueofthes- The violet contourin Fig. 4 shows the 2σ best fit region ac-
0 WIMP
waveannihilationcrossshectiionwithoutSommerfeldenhance- cording to [52]. For this particular model with annihilation
ment. intoµ+µ−withabranchingratioof100%,theconstraintswe
Onceaspecificmodelischosen,dN/dE iscalculatedand find donotfavor an explanationof the PAMELA data based
Fig.3canbeusedtoproduceconstraintson σv 0 asafunc- onlyondarkmatterannihilationformχ >260GeV(asimilar
tion of WIMP mass. As an example, we takhe aimodel with conclusionwasfoundin[29,52]).AlargesaturatedSommer-
annihilationintoleptons,specificallyµ+µ− withabranching feld enhancement (Smax > 2000) essentially rules out this
ratio of 100%. Modelssuch as this are typicallyused in the possibility.
literaturetoexplaintheanomalousabundanceofpositronsin WenotethatanymodeltestedusingFig.3alsoneedstobe
cosmic rays above 10 GeV reported by the PAMELA satel- checked for consistency with the correct relic density. Con-
lite[6].WeuseDarkSUSY[43]tocomputetheinsitupositron trary to Fig. 2 that was used to exemplify a case where a
(electron)andphotonspectra.Theresultingtotalphotonyield specific model gives the correctdark matter abundance with
9
andwithoutenhancement(recallthatfortheformerwemul-
tipliedthevalueof σv withoutenhancementbyafactorf
Ω
h i
toaccomplishthis, seeendofsectionIII), theupperlimiton
f inEq.(13)deliberatelydoesnottakethisintoaccount.
WIMP
In this way, the upper limits on the cases with and without
SommerfeldenhancementinFigs.3-4arenotrelatedtoeach
otherthroughtheirvaluesof σv .
0
h i
C. Benchmarkmodelsfittingthecosmicrayexcesses
In addition to the reference µ+µ− model, we investigate
benchmarkmodelsrecentlypresentedin[19],whichproduce
the correct thermalrelic density, fit the cosmic ray (CR) ex-
cesses measured by PAMELA and Fermi, and are currently
allowedbyboundsonS fromthecosmicmicrowaveback-
max
ground. Whereasourpreviousreferencemodelsdemonstrate
theeffectofSommerfeldenhancementontheEBLconstraints
in a broad class of scenarios, these benchmarks allow us to
test specific proposed models, and compare our bounds to
those from the cosmic microwave background. The param-
FIG.5:RatiooftheobservedboundonfWIMPtothevaluepredicted
eterscharacterizingthese modelsare summarizedinTable I; by the model, for each of the six benchmarks in Table I: 1=red,
see[19]forfurtherdetails[84]. 2=green, 3=blue, 4=yellow, 5=violet, 6=cyan. Energies where the
The benchmarks feature dark matter masses in the 1-1.7 ratioislessthan1areruledout. Forthepurposesofthisfigure,we
TeVrange,withnearly-degenerateexcitedstatesδ 0.1 1 assumeminimalcontributionfromunresolvedsubstructure.
∼ −
MeVabovethegroundstate. BothSommerfeldenhancement
andannihilationtoStandardModelfinalstatesoccurviavec-
tormediatorswithmassesm rangingfrom200 900MeV. factorof 7 10. Fig. 5displaystheratiofOBS /fMOD
Thesemodelswere chosentoφfitthe CR datawit−h nocontri- forthese∼mode−ls. WithalargercontributionfroWmIMuPnresWolIvMePd
butionfromlocalsubstructure,sotheyarenotperfectlycon- substructure,allthebenchmarkscanberuledoutindependent
sistentwiththeassumptionsofthiswork. However,theSom- ofastrophysicalcontributionstotheEBL.
merfeld enhancement in these models saturates at relatively Thereareseveraleffectsthatcouldamelioratethisconflict,
high velocities in order to evade constraints from the CMB, inadditiontothesmallsubstructurecorrectionmentionedal-
andthusweexpectthesubstructureboosttothelocallymea- ready.Theuncertaintyontheestimationoff ,asshown
WIMP
sured CR signals to be only a factor of 1.5 5, based on in Fig. 3, can alleviate the tension slightly (by less than a
∼ −
the results of [57]. We neglect this effect in the following factor of 2); a potentially larger effect is the uncertainty in
discussion; if the local substructure boost is substantial then the subtraction of astrophysical contributions to the EBL. If
these benchmarkswouldalso significantlyoverpredicte+e− ourbestestimateforthecontributionofastrophysicalsources
cosmicrays,andarelessinterestingfordirectcomparisonsto is too high (by a factor of up to 5, as discussed previ-
∼
data. ously), then in the case of minimal contribution from unre-
Sincethese benchmarkmodelshavelowervaluesofS solved(sub)halosthetensiondiminishessignificantly. Inthis
max
thantheoff-resonancecaseweconsideredinFig.3, wesim- casehowever,darkmatterannihilationwouldneedtobedom-
ply scale-downthe upperlimits of the latter to the appropri- inantlyresponsiblefortheEBLintheenergyrangeobserved
ate value of each benchmark model. We note that although byFermi,unlessothereffectsreducethedarkmattersignal.
S(σ )doesnothavethesameshapeforthebenchmarkmod- Star forming galaxies dominate the astrophysicalgamma-
vel
els than for the Yukawa case, the previous approximationis ray background model we have used for 0.3GeV . E .
0
good enough because most of the signal comes from struc- 10GeV, whereas blazars dominate at higher energies. The
tures that are already in the saturated regime. In any case, contributionoftheformertothetotalobservedsignalispar-
thisapproximationactuallyunderestimatesthesignalfromthe ticularlyimportanttoconstraintheroleofdarkmatterannihi-
benchmarkmodelsbecauseS(σ )islargerintheintermedi- lation.Toillustratethis,wetranslatetheconstraintsonf
vel WIMP
atevelocitydispersionsthanintheYukawacaseforthesame givenin Fig.3toconstraintsonthevalueoftheannihilation
valueofS . cross section at saturation by taking the value of σv =
max sat
h i
We find that these benchmarks are in conflict with Fermi S σv asafreeparameterlimitedbytheFermimeasure-
max 0
h i
measurementsin the energyrange 0.3 20 GeV, even in mentsand the astrophysicalbackground. Fig. 6 shows these
the case of minimal contribution fr∼om un−resolved substruc- constraintsasafunctionoffFermi(E >0.1GeV),thecontri-
SF
ture,ifthecurrentbestestimatesofcontributionsfromblazars butionofstarforminggalaxiestotheobservedintegratedflux
andstar-forminggalaxiestotheEBLaresubtractedfromthe between0.1GeVand100GeV.Weshowtheconstraintsonly
data. TheconflictismaximalatE 300MeV,whereitisa for the case of minimalcontributionof unresolvedsubhalos.
∼
10
Benchmarkno. AnnihilationChannel mφ(MeV) mχ(TeV) αc δ(MeV) 3×1S0m−a2x6hcσmv3i0s−1
1 1:1:2e± :µ± :π± 900 1.68 0.04067 0.15 530
2 1:1:2e± :µ± :π± 900 1.52 0.03725 1.34 360
3 1:1:1e± :µ± :π± 580 1.55 0.03523 1.49 437
4 1:1:1e± :µ± :π± 580 1.20 0.03054 1.00 374
5 1:1e± :µ± 350 1.33 0.02643 1.10 339
6 e±only 200 1.00 0.01622 0.70 171
TABLEI:Particlephysicsparametersandsaturatedannihilationcrosssectionsforbenchmarkpoints.
Thesixbenchmarkmodelsappearwiththesamecolorsasin
Fig.5. Asareference,thevaluesof σv thatfitthecosmic
sat
h i
rayexcessesforthesemodelsaremarkedwitharrowsnextto
the verticalaxison the rightside. The other two modelswe
have used throughoutthe paper are also included in the fig-
ure: m = 185GeVannihilatingintob¯b(magentaline)and
χ
m = 1.5TeV annihilatingintoµ+µ− (blackline). Forthe
χ
benchmark models and for the µ+µ− model, the constraint
on σv decreasesrapidlywithfFermi(E > 0.1GeV),be-
h isat SF
cause these modelsare constrainedat E 0.3 GeV where
0
∼
the contribution from star forming galaxies peaks. Even as-
sumingonlya5%contributionofstarforminggalaxies(recall
thatwehaveused53%asafiducialvalue),theconstraintson
σv stillexcludethevaluesneededbythesemodelstofit
sat
htheciosmicrayexcesses.Theb¯bmodelismoreindependentof
the star formingcontributionsince this model is constrained
atE 10GeV,whereblazarsdominate.
0
∼
Wewouldliketomentionthatthemodelwehaveusedtoin-
cludethecontributionfromstarforminggalaxiesassumesthat
theMilky-Waygamma-rayspecificintensityhasapowerlaw
energyspectrumwithanexponentof 2.7forE 0.6GeV FIG. 6: Constraints to the annihilation cross section at saturation
− ≥
[35], which seems to be too steep at high energies accord- as a function of the contribution from star forming galaxies to the
ingtotherecentanalysisoftheFermi-LATcollaborationthat observedintegratedfluxbetween0.1GeVand100GeVasmeasured
points to an exponent close to 2.5 for E 10 GeV (see byFermi.Thevaluesabovethelinesareexcluded.Wehavetakenthe
− ≥
Table I and Fig. 3 of [58]). Assuming a shallower spectrum modelgivenin[35]togetthespectralshapeofthiscontribution.We
fortheMilky-Waygamma-rayspecificintensitywouldresult showthesixbenchmarkmodelsofTable1,settinghσvisatasafree
in a contribution of star forminggalaxies to the EBL with a parameter,withthesamecolorsasinFig.5. Thesmallarrowsnext
totheverticalaxisontherightsidemarkthecorresponding values
shallowerspectrumaswell,makingitmorerelevantathigher
energiesthatitisinthemodelwehaveusedinthiswork,and ofhσvisat forthesebenchmarksasgiveninTable1. Wealsoshow
theresultsfor amodel withannihilation intoµ+µ− andamassof
slightlystrengtheningthederivedconstraintsinFigs.2,3and
1.5TeV(blackline),andthemodelweusedasreferenceinFigs.1-
5athighenergies.Nevertheless,themostrelevantuncertainty 3: a 185 GeV neutralino annihilating into b¯b (magenta line). As
istheoverallcontributionofstarforminggalaxiesdiscussedin inFig.5wehaveassumedaminimalcontributionfromunresolved
thepreviousparagraphandwhoseeffectsareshowninFig.6. substructures. Blazarsareassumedtocontributebyafixedamount
We haveassumedalow-masscutoffof10−6 M⊙: kinetic (16%)intheenergyrangemeasuredbyFermi[34].
decoupling can occur quite late in models of this type [59],
leadingtoahighercutoffofupto0.1 1M⊙[60]. However,
we estimate that a changeof five ord−ersof magnitudein the ence between those profiles modifies the gamma-ray signal
low-masscutoffwillaffectthefinalresultbyafactorofonly by roughly an order of magnitude. The presence of signif-
2 6,andthehighendofthisrangewillonlybeattainedfor icant dark matter self-interactions and nearly-degenerateex-
no−n-minimal contributions from unresolved (sub)halos (i.e., cited states in models of this type can lead to disruption of
scenarios that are presently in conflict with even the unsub- low-masshalosandthedepletionofcentraldensitycusps(see
stracted data, for these benchmark models). If the slope we e.g.[61,62]andreferencestherein);whiletheseeffectscould
have assumed for the central density profile of the halos is potentiallyreducethetensionwiththeEBLdata,theirinclu-
steeperthanreality,thiscouldalsoaffectourlimitsbyafac- sionisbeyondthescopeofourcurrentanalysis.
torofafew:theNFWprofilewehavechosenliesbetweenthe WethereforeseethatinaCDMscenario,inthecontextof
MooreandBurkertprofilesconsideredby[29],andthediffer- currentstructureformationmodels,theEBLcanrobustlyact