Multi-frequency analysis of neutralino dark matter annihilations in the Coma cluster

27 Pages·2006·0.93 MB·English

A&A455,21–43(2006) Astronomy
DOI:10.1051/0004-6361:20053887 &
(cid:1)c ESO2006 Astrophysics
Multi-frequency analysis of neutralino dark
(cid:1)
matter annihilations in the Coma cluster
S.Colafrancesco1,2,S.Profumo3,andP.Ullio4,5
1 INAF–OsservatorioAstronomicodiRoma,viaFrascati33,00040Monteporzio,Italy
2 IstitutoNazionalediFisicaNucleare,SezionediRoma2,00133Roma,Italy
e-mail:[email protected]
3 DepartmentofPhysics,FloridaStateUniversity,505KeenBldg.,Tallahassee,FL32306,USA
e-mail:[email protected]
4 ScuolaInternazionaleSuperiorediStudiAvanzati,viaBeirut2-4,34014Trieste,Italy
5 IstitutoNazionalediFisicaNucleare,SezionediTrieste,34014Trieste,Italy
e-mail:[email protected]
Received22July2005/Accepted10April2006
ABSTRACT
Westudytheastrophysicalimplicationsofneutralinodarkmatterannihilationsingalaxyclusters,withaspecificapplicationtothe
Comacluster. Wefirstaddress thedeterminationof thedarkhalo modelsfor Coma, startingfromstructureformation modelsand
observationaldata,andwediscussindetailtheroleofsub-halos.Wethenperformathoroughanalysisofthetransportanddiffusion
propertiesofneutralinoannihilationproducts,andinvestigatetheresultingmulti-frequencysignals,fromradiotogamma-rayfrequen-
cies.Wealsostudyother relevantastrophysical effectsofneutralinoannihilations,liketheDM-inducedSunyaev-Zel’dovich effect
andtheintraclustergasheating.Asfortheparticlephysicssetup,weadoptatwo-foldapproach,resortingbothtomodel-independent
bottom-upscenariosandtobenchmark,GUT-motivatedframeworks.WeshowthattheComaradio-halodata(thespectrumandthe
surfacebrightness)canbenicelyfittedbytheneutralino-inducedsignalforpeculiarparticlephysicsmodelsandformagneticfield
values,whichweoutlineindetail.Fittingtheradiodataandmovingtohigherfrequencies,wefindthatthemulti-frequencyspectral
energydistributionsaretypicallydimatEUVandX-rayfrequencies(withrespecttothedata),butshowanon-negligiblegamma-ray
emission,dependingontheamplitudeoftheComamagneticfield.Asimultaneousfittotheradio,EUVandHXRdataisnotpossible
withoutviolatingthegamma-rayEGRETupperlimit.Thebest-fitparticlephysicsmodelsyieldssubstantialheatingoftheintracluster
gas,butnotsufficientenergyinjectionastoexplainthequenching ofcoolingflowsintheinnermostregionofclusters.Duetothe
specificmulti-frequencyfeaturesoftheDM-inducedspectralenergydistributioninComa,wefindthatsupersymmetricmodelscan
besignificantlyandoptimallyconstrainedeitherinthegamma-raysoratradioandmicrowavefrequencies.
Keywords.cosmology:theory–galaxies:clusters–cosmology:darkmatter–galaxies:clusters:general
1. Introduction is Ω = 0.0233/h2, with the onlyother significantextra matter
b
termincolddarkmatterΩ =Ω −Ω ,thatourUniversehas
Mmaotsttero,fwthhoesempartetesernccoenitsenintdoicfattheedbuynisveevreserailsasintrofpohrmysiocfaldeavrik- aflatgeometryandacosmCoDloMgicalcmonstabntΛ,i.e.ΩΛ =1−Ωm,
and, finally,thatthe primordialpowerspectrumis scale invari-
dences(e.g.,gravitationallensing,galaxyrotationcurves,galaxy antandisnormalizedtothevalueσ = 0.897.Thischoicesets
clustersmasses)butwhosenatureisstillelusive.Onthecosmo- 8
ourframework,butitisnotactuallycrucialforanyoftheresults
logicalside,themostrecentresultsofobservationalcosmology,
presentedinthepaper,whichcanbeeasilyrescaledincaseofa
i.e. WMAP vs. distant SN Ia, indicate that the matter content
re-assessmentofbest-fitvaluesofthecosmologicalparameters,
oΩfbthh2e=un0i.v0e2r2s4e±is0Ω.0m0h029=(S0p.e1r3g5el+−00e..t0000a89l.w2i0th03a).bTahryeocnomdebnisniatytioonf acondsminolpoagritcicaullmaroodfelt,hwehvialeluweiodfeΩlyCuDsMed(,thheasprbeeseennatlcsoonccroitridcaiznecde
oftheavailabledataonlargescalestructures(Ly-αforestanal- and questionedin the lightofstill unexploredsystematics, see,
ysis of the SDSS, the SDSS galaxy clustering) with the latest e.g., Myersetal.2003; Sadatetal.2005; Lieu&Mittaz2005;
SNe and with the 1st year WMAP CMB anisotropies can im- Copietal.2004)
provethedeterminationofthecosmologicalparameters(Seljak
The Coma cluster has been the first astrophysical labora-
etal.2004)andhenceallowustosetaconcordancecosmologi-
tory for dark matter (DM) since the analysis of F. Zwicky
calmodel.
(Zwicky1933).Inthisrespect,wecanconsidertheComacluster
Werefer,inthispaper,toaflatΛCDMcosmologywithpa-
asanastrophysicalbenchmarkcase-studyforDM.Modernob-
rameters chosen according to the global best fitting results de-
servationshaveledto anincreasinglysophisticatedexploration
rivedinSeljaketal.(2004)(seetheirTable1,thirdcolumn):we
of the DM distribution in the universe, now confirmed to be
assume, in fact, thatthe presentmatter energydensityis Ω =
m a dominant component (relative to the baryonic material) over
0.281, that the Hubble constant in units of 100 kms−1Mpc−1
scalesrangingfromthoseofgalaxyhalostothatoftheparticle
is h = 0.71, that the present mean energy density in baryons
horizon.
(cid:2) AppendixAisonlyavailableinelectronicformat The nature of DM is not yet known and several de-
http://www.edpsciences.org tection techniques have been used so far. Obviously, direct
Article published by EDP Sciences and available at http://www.edpsciences.org/aaor http://dx.doi.org/10.1051/0004-6361:20053887
22 S.Colafrancescoetal.:DMannihilationsinComa
detectionisthecleanestandmostdecisivediscriminant(seee.g. finalstateproductsofneutralinopairannihilations,andthemass
Munoz2003, fora review).However,it wouldbeinterestingif of the neutralino itself. We will indicate this first strategy as a
astronomicaltechniquesweretorevealsomeofthefundamental bottom-upapproach(seeSect.4.1fordetails).
propertiesofDMparticles.Infact,ifDMissupposedtoconsist Sincemostphenomenologicalstudieshavebeensofarbased
offundamentalparticleswhichareweaklyinteracting,thentheir on GUT-motivated models, and a wealth of results on accel-
owninteractionwillleadtoanumberofastrophysicalsignatures erator physics, direct and indirect detection has accumulated
(e.g.,high-energygamma-rays,electronsandpositrons,protons withintheseframeworks,wedecidedtoworkouthere,aswell,
andneutrinosandhencebytheiremission/interactionproperties) the astrophysicalconsequences,for the system underconsider-
indicativeoftheirnatureandcomposition. ation, of a few benchmark models. The latter have been cho-
These facts provide the basic motivations for our study, sen among the minimal supergravity (mSUGRA) models indi-
whichisaimedto:i)describethemulti-wavelengthsignalsofthe catedinBattagliaetal.(2004)withthecriterionofexemplifying
presenceofDMthroughtheemissionfeaturesofthesecondary thewidestrangeofpossibilitieswithinthatparticulartheoretical
products of neutralino annihilation. These signals are of non- setup(seeSect.3.2fordetails).
thermal nature and coverthe entire electro-magneticspectrum,
from radio to gamma-ray frequencies;ii) indicate the best fre-
quencywindowswhereitwillbepossibletocatchphysicalindi- 1.2.Theastrophysicalframework
cationsforthenatureofDM;iii)applythisanalysistothelargest
To make our study quantitative, we will compare the predic-
boundcontainersofDMintheuniverse,i.e.galaxyclusters.We
tionsoftheabovementionedneutralinomodelswiththeobser-
shall focushere on the case of the Coma cluster, a particularly
vationalset-upoftheComacluster,whichrepresentsthelargest
richandsuitablelaboratoryforwhichanextendedobservational
available observational database for a galaxy cluster. The to-
databaseisathand. tal mass of Coma found within 10 h−1 Mpc from its center is
M<10h−1 Mpc ≈ 1.65×1015 M(cid:3) (Gelleretal.1999).Theassump-
1.1.Thefundamentalphysicsframework tion of hydrostatic equilibrium of the thermal intra-cluster gas
in Coma provides a complementary estimate of its total mass
SeveralcandidateshavebeenproposedasfundamentalDMcon- enclosed in the radius r. A value M ≈ 1.85×1015 M(cid:3) within
stituents,rangingfromaxionstolight,MeVDM,fromKKpar- 5h−1MpcfromtheclustercenterhasbeenobtainedfromX-ray
ticles, branons, primordial BHs, mirror matter to supersym- 50
data(Hughes1989).
metric WIMPs (see, e.g., Baltz2004; Bertoneetal.2004; and
X-rayobservationsofComaalsoyielddetailedinformation
Bergstrom2000, for recent reviews). In this paper we will as-
about the thermal electrons population. We know that the hot
sume thatthe mainDM constituentis thelightestneutralinoof thermal electrons are at a temperature k T = 8.2 ± 0.4 keV
the minimal supersymmetric extension of the Standard Model (Arnaudetal.2001) and have a central dBenesity n = (3.42 ±
(MSSM).Althoughnoexperimentalevidenceinfavorofsuper- 0
0.047)h1/2×10−3cm−3,withaspatialdistributionfittedbyaβ-
symmetryhasshownuptodate,severaltheoreticalmotivations 70
indicatethattheMSSM isoneofthe bestbetsfornewphysics model,n(r)=n0(1+r2/rc2)−3β/2,withparametersrc =10.5(cid:4)±0.6(cid:4)
beyondthe Standard Model. Intriguinglyenough,and contrary andβ ≈ 0.75(Brieletal.1992).Assumingsphericalsymmetry
tothemajorityofotherparticlephysicscandidatesforDM,su- andthepreviousparametervalues,theopticaldepthofthether-
persymmetrycanunambiguouslymanifestitselfinfutureaccel- mal gas in Coma is τth (cid:5) 5.54×10−3 and the pressure due to
eratorexperiments.Furthermore,providedneutralinosarestable thethermalelectronpopulationis Pth (cid:5) 2.80×10−2 keVcm−3.
andarethelightestsupersymmetricparticles,nextgenerationdi- The hot intra-cluster gas produces also a thermal SZ effect
rectdetectionexperimentsfeaturegoodchancestoexploremost (Sunyaev&Zel’dovich1972; Sunyaev&Zel’dovich1980; see
oftheneutralino-protonscatteringcrosssectionrangepredicted Birkinshaw1999, for a general review) which has been ob-
bysupersymmetry. served over a wide frequencyrange, from 32 to 245 GHz (see
A long standing issue in phenomenologicalstudies of low- DePetrisetal.2002,andreferencestherein).
energy supersymmetry is traced to the parameterization of the Beyond the presence of DM and thermal gas, Coma also
supersymmetrybreakingterms(seeChungetal.2003,forare- showshintsforthepresenceofrelativisticparticlesinitsatmo-
centreview).Inthisrespect,twosomehowcomplementaryatti- sphere. The main evidence for the presence of a non-thermal
tudeshavebeenpursued.On theonehand,onecanappealtoa population of relativistic electrons comes from the observation
(setof)underlyinghighenergyprinciplestoconstraintheform of the diffuse radio halo at frequencies νr ∼ 30 MHz−5 GHz
ofsupersymmetrybreakingterm,possiblyatsomehighenergy (Deissetal.1997;Thierbachetal. 2003).Theradiohalospec-
(often at a grand unification) scale (see e.g. Baeretal.2000). trum can be fitted by a power-law spectrum Jν ∼ ν−1.35 in the
The low energy setup is then derived throughthe renormaliza- range 30 MHz-1.4 GHz with a further steepening of the spec-
tion group evolution of the supersymmetry breaking parame- trum at higher radio frequencies. The radio halo of Coma has
ters down to the electroweak scale. Alternatively, one can di- anextensionofRh ≈ 0.9h−701 Mpc,anditssurfacebrightnessis
rectly face the most general possible low energy realization quite flat in the inner 20 arcmin with a radial decline at larger
of the MSSM, and try to figure out whether general proper- angulardistances(e.g.,Colafrancescoetal.2005).
ties ofsupersymmetryphenomenologycan bederived(see e.g. Otherdiffusenon-thermalemissionshavebeenreportedfor
Profumo&Yaguna2004). Coma (as well as for a few other clusters) in the extreme UV
In this paper we will resort to both approaches. We will (EUV) and in the hard X-ray (HXR) energybands. The Coma
showthatthefinalstateproductsofneutralinopairannihilations fluxobservedinthe65−245eVband(Lieuetal.1996)is∼36%
show relatively few spectral patterns, and that any supersym- abovetheexpectedfluxfromthethermalbremsstrahlungemis-
metric configuration can be thoughtas an interpolation among sion of the k T ≈ 8.2 keV IC gas (Ensslin&Biermann1998)
B
the extreme cases we shall consider here. The huge number of and it can be modeled with a power-law spectrum with an ap-
free parameters of the general MSSM are therefore effectively proximately constant slope ≈1.75, in different spatial regions
decoupled, and the only relevant physical properties are the (Lieuetal.1999; Ensslinetal.1999; Bowyer et al. 2004). The
S.Colafrancescoetal.:DMannihilationsinComa 23
EUV excess in Coma has been unambiguously detected and discussion) and in this case the EUV emission should be pro-
it does not depend much on the data analysis procedure. The ducedeitherbyadifferentrelativisticelectronpopulationorby
integrated flux in the energy band 0.13−0.18 keV is F ≈ a warmthermalpopulationbothconcentratedtowardstheclus-
EUV
(4.1±0.4)×10−12ergcm−2s−1 (Bowyeretal.2004).According ter center. Lastly, models in which the EUV and HXR emis-
to the most recent analysis of the EUVE data (Bowyer et al. sion can be reproduced by synchrotron emission from the in-
2004), the EUV excess seems to be spatially concentrated in teraction of Ultra High Energy cosmic rays and/or photons
the inner region (θ <∼ 15−20 arcmin) of Coma (see also (Timokhinetal.2004; Inoueetal.2005) have also been pre-
Bonamenteetal.2003). The nature of this excess is not defi- sented.Thesituationisfarfrombeingcompletelyclearandsev-
nitely determined since both thermal and non-thermal models eralproblemsstillstandonboththeobservationalandtheoretical
areabletoreproducetheobservedEUVflux.However,theanal- sidesoftheissue.
ysisofBowyeretal.(2004)seemstofavouranon-thermalori- SinceDMisabundantlypresentinComaandrelativisticpar-
ginoftheEUVexcessinComageneratedbyanadditionalpop- ticlesareamongthemainannihilationproducts,weexplorehere
ulation of secondary electrons. A soft X-ray (SXR) excess (in theeffectofDMannihilationonthemulti-frequencyspectralen-
theenergyrange≈0.1−0.245keV)hasbeenalsodetectedinthe ergydistribution(SED)ofComa.Theplanofthepaperisthefol-
outer region(20(cid:4) < θ < 90(cid:4)) of Coma (Bonamenteetal.2003; lowing.WediscussinSect.2theDMhalomodelsfortheComa
Finoguenovetal.2003). The spectral features of this SXR ex- cluster,thesetofbestfittingparametersfortheDMdistribution,
cessseemtobemoreconsistentwithathermalnatureoftheSXR andthe roleofsub-halos.The annihilationfeaturesof neutrali-
emission,whileanon-thermalmodelisnotabletoreproduceac- nosandthemainannihilationproductsarediscussedinSect.3.
curately the SXR data (e.g., Bonamenteetal.2003). The SXR The multi-frequency signals of DM annihilation are presented
emission from the outskirts of Coma has been fitted in a sce- and discussed in details in Sects. 4 and 5, while the details of
narioinwhichthethermalgasatk T ∼ 0.2keVwith∼0.1so- thetransportanddiffusionpropertiesofthesecondaryparticles
B e
larabundance(see,e.g.Finoguenovetal.2003whoidentifythe aredescribedintheAppendixA,togetherwiththederivationof
warmgaswithaWHIMcomponent)residesinthelow-density the equilibrium spectrum of relativistic particles in Coma. The
filamentspredictedtoformaroundclustersasaresultoftheevo- conclusionsofouranalysisandthe outlineforforthcomingas-
lution of the large-scale web-like structure of the universe(see trophysicalsearches for DM signals in galaxyclusters are pre-
Bonamenteetal.2003). It has been noticed, however, that the sentedinthefinalSect.6.
WHIM componentcannotreproduce,by itself, the Coma SXR
excess because it would producea SXR emission by far lower 2. AΛCDM modelfortheComacluster
(seeMittazetal.2004),andthusoneisforcedtoassumealarge
amountofwarmgasintheoutskirtsofComa.Thus,itseemsthat To describe the DM halo profile of the Coma cluster we refer,
theavailableEUVandSXRdataindicate(atleast)twodifferent asageneralsetup,totheΛCDMmodelforstructureformation,
electronpopulations:anon-thermalone,likelyyieldingthecen- implementingresultsofgalaxyclusterformationobtainedfrom
trallyconcentratedEUVexcessandathermal(likelywarm)one, N-bodysimulations.Freeparametersarefittedagainsttheavail-
providingthe periphericallylocated SXR excess. In this paper, abledynamicalinformationandarecomparedtothepredictions
wewillcompareourmodelswiththeEUVexcessonly,whichis of this scheme. Substructures will play a major role when we
intimatelyrelatedtoComabeingspatiallyconcentratedtowards will discuss the predictionsfor signals of DM annihilations.In
theinnerregionofthecluster. this respect, the picture derived from simulations is less clean
There is also evidence of a hard X-ray (HXR) and,hence,wewilldescribeindetailsoursetofassumptions.
emission observed towards the direction of Coma
with the BeppoSAX-PDS (Fusco-Femianoetal.1999;
2.1.Thedarkmatterhaloprofile
Fusco-Femianoetal.2004) and with the ROSSI-XTE
experiments (Rephaelietal.1999). Both these measure- TodescribetheDMhaloprofileoftheComaclusterweconsider
ments indicate an excess over the thermal emission which the limit in which the mean DM distribution in Coma can be
amounts to F(20−80)keV = (1.5 ± 0.5) × 10−11 ergs−1cm−2 regardedassphericallysymmetricandrepresentedbythepara-
(Fusco-Femianoetal.2004). It must be mentioned, for the metricradialdensityprofile:
sake of completeness, that the HXR excess of Coma is still ρ(r)=ρ(cid:4)g(r/a). (1)
controversial(see RossettiandMolendi2004), but, for the aim
of our discussion, it could at worst be regarded as an upper Twoschemesareadoptedtochoosethefunctiong(x)introduced
limit.ThenatureoftheHXRemissionofComaisnotyetfully here. In the first one, we assume that g(x) can be directly in-
understood. ferred as the function setting the universal shape of DM halos
Finally,agamma-rayupperlimitof F(>100MeV) ≈ 3.2× found in numerical N-body simulationsof hierarchicalcluster-
10−8 phocm−2 hasbeenderivedforComafromEGRETobser- ing. We are assuming,hence,thatthe DM profileis essentially
vations(Sreekumaretal.1996;Reimeretal.2004). unalteredfromthestageprecedingthebaryoncollapse,whichis
Thecurrentevidencefortheradio-haloemissionfeaturesof –strictlyspeaking–thepictureprovidedbythesimulationsfor
Comahasbeeninterpretedassynchrotronemissionfromapopu- thepresent-dayclustermorphology.Afewformsfortheuniver-
lationofprimaryrelativisticelectronswhicharesubjecttoacon- sal DM profile have been proposedin the literature:we imple-
tinuousre-accelerationprocesssupposedlytriggeredbymerging mentherethenon-singularform(whichwelabelasN04profile)
shock and/or intracluster turbulence (e.g., Brunettietal.2004). extrapolatedbyNavarroetal.(2004):
TheEUVandHXRemissionexcessesarecurrentlyinterpreted g (x)=exp[−2/α(xα−1)] with α(cid:5)0.17, (2)
N04
as Inverse Compton scattering (ICS) emission from either pri-
andtheshapewithamildsingularitytowardsitscenterproposed
maryorsecondaryelectrons.Alternativemodelinghasbeenpro-
byDiemandetal.(2005)(labeledhereasD05profile):
posed in terms of suprathermal electron bremsstrahlung emis-
sion for the HXR emission of Coma (see Ensslinetal.1999; 1
g (x)= with γ(cid:5)1.2. (3)
Kempner&Sarazin2000, see also Petrosian2001for a critical D05 xγ(1+x)3−γ
24 S.Colafrancescoetal.:DMannihilationsinComa
The other extreme scheme is a picture in which the baryon in- as defined implicitly by σ(M (z)) = δ (z), with δ being
(cid:2) sc sc
fall inducesa largetransferof angularmomentumbetweenthe the critical overdensity required for collapse in the spherical
luminousandthedarkcomponentsofthecosmicstructure,with model and σ(M) being the present-day rms density fluctuation
significantmodificationoftheshapeoftheDMprofileinitsin- inspherescontainingameanmass M (see,e.g.,Peebles1980).
ner region. According to a recent model (El-Zantetal.2001), An expression for δ is given, e.g., in Ekeetal.(1996). The
sc
baryons might sink in the central part of DM halos after get- rms fluctuation σ(M) is related to the fluctuation power spec-
ting clumped into dense gas clouds, with the halo density pro- trumP(k)(seee.g.Peebles1993)by
file inthefinalconfigurationfoundtobedescribedbyaprofile (cid:3)
(labeled here as B profile) with a large core radius (see, e.g.,
σ2(M)≡ d3kW˜2(kR)P(k), (8)
Burkert1995):
g (x)= 1 · (4) where W˜ is the top-hat window function on the scale R3 =
B (1+x)(1+x2) 3M/4πρ¯ with ρ¯ the mean (proper) matter density, i.e. ρ¯ =
Ω ρ with ρ the critical density. The power spectrum P(k)
Once the shape of the DM profile is chosen, the radial den- m c c
is parametrized as P(k) ∝ knT2(k) in terms of the primordial
sity profile in Eq. (1) is fully specified by two parameters: the
length-scale a and the normalization parameter ρ(cid:4). It is, how- power-spectrumshape∝knandofthetransferfunctionT2(k)as-
sociatedtothespecificDMscenario.Wefixtheprimordialspec-
ever, useful to describe the density profile model by other two
tral index n = 1 and we take the transfer function T2(k) given
parameters, i.e., its virial mass M and concentration param-
vir byBardeenetal.(1986)foran adiabaticCDM model,withthe
eter c . For the latter parameter, we adopt here the definition
vir shapeparametermodifiedtoincludebaryonicmatteraccording
byBullocketal.(2001).WeintroducethevirialradiusR ofa
vir totheprescriptionin,e.g.Peacock(1999)(seetheirEqs.(15.84)
halo of mass M as the radius within which the mean density
ofthehaloiseqvuiraltothevirialoverdensity∆ timesthemean and(15.85))andintroducingamultiplicativeexponentialcutoff
backgrounddensityρ¯ =Ω ρ : vir atlargek correspondingtothefree-streamingscaleforWIMPs
m c (Hofmann et al. 2001; Chen et al. 2001; Green et al. 2005;
4π Diemand et al. 2005). The spectrum P(k) is normalized to the
Mvir ≡ 3 ∆virρ¯R3vir. (5) valueσ8 =0.897aswasquotedabove.
The toy model of Bullock et al. (2001) prescribes a one to
Weassumeherethatthevirialoverdensitycanbeapproximated
one correspondencebetween the density of the Universeat the
by the expression (see Bryan&Norman1998), appropriatefor
collapseredshiftz ofthe DMhaloanda characteristicdensity
c
aflatcosmology,
ofthehaloattheredshiftz;itfollowsthat,onaverage,thecon-
(cid:1) (cid:2)
centrationparameterisgivenby
18π2+82x−39x2
∆vir (cid:5) 1−x , (6) c (M,z)= K1+zc = cvir(M,z=0), (9)
with x ≡ Ω (z)−1.Inourcosmologicalsetupwefindatz = 0, vir 1+z (1+z)
m
∆ (cid:5) 343 (we refer to Colafrancescoetal.1994, 1997 for a
vir with K being a constant (i.e. independent of M and cosmol-
general derivation of the virial overdensity in different cosmo-
ogy) to be fitted to the results of the N-body simulations. We
logicalmodels).Theconcentrationparameteristhendefinedas
plot in Fig. 1 the dependence of c on the halo mass M, at
vir
R R z = 0, according to the toy model of Bullock et al. (2001) as
cvir = r−v2ir ≡ x−v2ira, (7) eloxstrmapaodleatoedf WdoIwMnPtso,tih.ee.farreoeu-sntdre1a0m−i6nMgm(cid:3) a(sssesecHaloeffmoarnDnMethaal-.
with r−2 the radius at which the effective logarithmic slope of 2001;Chenetal.2001;Greenetal.2005;Diemandetal.2005).
the profileis −2. We find that x−2 = 1 forthe N04 profile(see The predictions are compared to the results of a few sets of
Eq.(2)),x−2 =2−γforD05profile(seeEq.(3)),andx−2 (cid:5)1.52 N-bodysimulations:weuse“data”pointsandrelativeerrorbars
fortheBurkertprofile(seeEq.(4)). fromBullocketal.(2001)(representingabinninginmassofre-
Sincethefirstnumericalresultswithlargestatisticsbecame sultsforalargesampleofsimulatedhalos;ineachmassbin,the
available (Navarro et al. 1996, 1997), it has been realized that, marker and the error bars correspond,respectively,to the peak
at any given redshift, there is a strong correlation between c andthe 68%width inthe c distribution)todeterminethe pa-
vir vir
and M ,withlargerconcentrationsfoundinlighterhalos.This rameter K. The same value will be used to infer the mean c
vir vir
trendmaybeintuitivelyexplainedbythefactthatmeanoverden- predictedinourcosmologicalsetup.Other“datasets”referactu-
sities in halos should be correlated with the mean background allytodifferentvaluesofσ anddifferentredshiftsz(z=26for
8
densitiesatthetimeofcollapse,andinthehierarchicalstructure thetwominihalosfittedinFig.2ofDiemandetal.2005andfor
formationmodelsmallobjectsformfirst,whentheUniversewas the upper bound in the range up to 10M(cid:3) quoted in the same
indeeddenser.Thecorrelationbetweenc and M isrelevant paper; z = 3 for the sample from Colin et al. 2004) and have
vir vir
inourcontextattwolevels:i)whendiscussingthemeandensity been extrapolated,consistentlywith ourprescriptions,to z = 0
profile of Coma and; ii) when including substructures. Hence, andσ8 = 1.Sincesmallobjectstendtocollapseallatthesame
wewillreviewthisrelevantissuehereandwewillapplyittothe redshift, the dependenceon mass of the concentrationparame-
presentcaseofComa.Bullocketal.(2001)proposedamodelto tersflattensatsmallmasses;themeanasymptoticvaluewefind
describethiscorrelation,improvingonthetoymodeloriginally isslightlylargerthanthetypicalvaluesfoundinDiemandetal.
outlined in Navarro et al. (1996, 1997). A collapse redshift z (2005),butitisstillconsistentwiththatanalysis.
c
is assigned, on average,to each halo of mass M at the epoch z Analternativetoy-modeltodescribetherelationbetweenc
vir
throughthe relation M (z ) ≡ FM. Here it is postulatedthat a and M has been discussed by Eke, Navarro & Steinmetz (Eke
(cid:2) c
fixedfractionFofM(followingWechsleretal.2001wechoose et al. 2001, hereafter ENS model). The relation they propose
F = 0.015) corresponds to the typical collapsing mass M , has a similar scaling in z, but with a different definition of the
(cid:2)
S.Colafrancescoetal.:DMannihilationsinComa 25
cvir z = 0 cvir
102 σ8 = 1
Colin et al.
Diemand et al.
10
10
Bullock et al.
1
Bullock et al. N04 profile
ENS Tasitsiomi et al.
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
-7 -5 -3 -1 3 5 7 9 11 13 15
10 10 10 10 10 10 10 10M10 [ h10-1 M10 ❍⋅1 ]0 Mvir [ 1015 M ❍⋅ h-1 ]
Fig.1.Thedependence ofc onthehalomass M,atz = 0,asinthe cvir
vir
Bullocketal.toymodel(solidline)andintheENStoymodel(dashed
line); predictions are compared to a few sets of simulation results in
differentmassranges.Aflat,vacuum-dominatedcosmologywithΩ =
M
0.3,ΩΛ =0.7,h=0.7andσ8=1isassumedhere.
10
collapseredshiftz andamilderdependenceofc onM.Inour
c vir
notation,theydefinez throughtheequation
c
1
D(zc)σeff(Mp)= Cσ (10) 1 D05 profile
whereD(z)representsthelineartheorygrowthfactor,andσeff is 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
an“effective”amplitudeofthepowerspectrumonscaleM:
(cid:4) (cid:5) Mvir [ 1015 M ❍⋅ h-1 ]
σeff(M)=σ(M) −ddllnn((Mσ))(M) =−ddMσM (11) Fvaigri.a2b.leWienEshqo.w(1t9h)e,f1oσr,th2eσNaanvdar3roσectoanl.tohuarlsoapsrodfielreiv,eEdq.fr(o2m),(tuhpepχe2rr
panel) and for the Diemand et al. halo profile, Eq. (3) (lower panel).
whichmodulatesσ(M)andmakesz dependentonboththeam- Alsoshown aremeanvaluesfor thecorrelationbetween Mvir andcvir
c
asinthetoymodelsofBullocketal.(2001)(solidline)andthatofEke
plitude and on the shape of the power spectrum, rather than
etal.(2001)(dashedline).
just on the amplitude, as in the toy model of Bullock et al.
(2001).Finally,inEq.(10),M isassumedtobethemassofthe
p
halo contained within the radius at which the circular velocity 2.2.FittingthehaloparametersofComa
reachesitsmaximum,whileC isafreeparameter(independent
σ
Foragivenshapeofthehaloprofilewemakeafitoftheparam-
of M and cosmology)which we will fit again to the “data” set
etersM andc againsttheavailabledynamicalconstraintsfor
in Bullocketal.(2001). With such a definition of z it follows vir vir
c
Coma.We considertwoboundsonthetotalmassofthecluster
that,onaverage,c canbeexpressedas:
vir
at large radii, as inferred with techniques largely insensitive to
(cid:4) (cid:5) thedetailsofthemassprofileinitsinnerregion.InGelleretal.
∆ (z )Ω (z) 1/3 1+z
c (M,z)= vir c M c· (12) (1999),atotalmass
vir ∆ (z)Ω (z ) 1+z
vir M c M(r <10h−1Mpc)=(1.65±0.41) ×1015 h−1M(cid:3) (13)
As shown in Fig. 1, the dependence of c on M given by
vir isderivedmappingthecausticsinredshiftspaceofgalaxiesin-
Eq.(12)aboveisweakerthanthatobtainedintheBullocketal.
fallinginComaonnearlyradialorbits.Severalauthorsderived
(2001)toy-model,withasignificantmismatchintheextrapola-
massbudgetsforComausingopticaldataandapplyingthevirial
tionalreadywithrespecttothesamplefromColinetal.(2004)
theorem,orusingX-raydataandassuminghydrostaticequilib-
andanevenlargermismatchinthelowmassend.Moreover,the
rium.We considertheboundderivedbyHughes(1989), cross-
extrapolation breaks down when the logarithmic derivative of
correlatingsuchtechniques:
theσ(M)becomesverysmall,intheregimewhenP(k)scalesas
k−3.Notealsothatpredictionsinthismodelarerathersensitive M(r <5h−501Mpc)=(1.85±0.25) ×1015 h−501M(cid:3), (14)
to the specific spectrum P(k) assumed (in particular the form
in the public release of the ENS numerical code gives slightly whereh istheHubbleconstantinunitsof50kms−1Mpc−1.
50
largervaluesofc initslowmassend,aroundavaluec ≈40 Inourdiscussionsomeinformationontheinnershapeofthe
vir vir
(wecheckedthatimplementingourfittingfunctionforthepower massprofileinComaisalsoimportant:weimplementherethe
spectrum,werecoverourtrend). constraintthatcanderivedbystudyingthevelocitymomentsof
26 S.Colafrancescoetal.:DMannihilationsinComa
agiventracerpopulationinthecluster.Asthemostreliableob- vir
c
servablequantityonecanconsidertheprojectionalongtheline
ofsightoftheradialvelocitydispersionofthepopulation;under
theassumptionofsphericalsymmetryandwithoutbulkrotation,
this is related to the total mass profile M(r) by the expression
10
(Binney&Mamon1982):
(cid:3)
∞
2G
σ2 (R) = dr(cid:4)ν(r(cid:4))M(r(cid:4))(r(cid:4))2β−2
los I(R)
(cid:3) R (cid:4) (cid:5)
r(cid:4) R2 r−2β+1
× dr 1−β √ , (15)
R r2 r2−R2
1
Burkert profile
whereν(r)isthedensityprofileofthetracerpopulationandI(R)
represents its surface density at the projected radius R. In the
derivationofEq.(15),aconstant-over-radiusanisotropyparam- 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
eterβdefinedas Mvir [ 1015 M ❍⋅ h-1 ]
σ2(r) Fig.3.Weshowherethe1σ,2σand3σcontoursasderivedfromtheχ2
β≡1− θ , (16) variableintroducedinthetext,fortheBurkertprofile,Eq.(4). r
σ2(r)
r
hasbeenassumedwithσ2 andσ2 being,respectively,theradial Nonetheless, we have derived in Fig. 2 the 1σ, 2σ and 3σ
r θ
and tangential velocity dispersion (β = 1 denotes the case of contoursinthe(M , c )planefortheNavarroetal.halopro-
vir vir
purely radial orbits, β = 0 that of system with isotropic veloc- file (Eq. (2) and for the Diemand et al. halo profile (Eq. (3)).
itydispersion,whileβ → −∞labelscircularorbits).Following In Fig. 3 we show the analogouscontoursfor the Burkert pro-
Lokas&Mamon(2003),wetakeastracerpopulationthatofthe file (Eq. (4)). In all these cases we have performed the fit of
E-S0galaxies,whoseline ofsightvelocitydispersionhasbeen the line-of-sightradial velocitydispersion of E-S0 galaxiesas-
mapped,accordingto Gaussian distribution,in nine radialbins suming that this system has an isotropic velocity dispersion,
from 4(cid:4) outto 190(cid:4) (see Fig. 3 in Lokas& Mamon 2003), and i.e. we have taken β = 0. Best fitting values are found at
whosedensityprofilecanbedescribedbythefittingfunction: Mvir (cid:5)0.9×1015M(cid:3) h−1andcvir (cid:5)10(thatweconsider,hence,
as reference values in the following analysis), not too far from
1
ν(r)∝ , (17) themeanvalueexpectedfrommodelssketchingthecorrelation
(r/rS)(1+r/rS)2 betweenthesetwoparametersintheΛCDMpicture.Weshowin
with r = 7.(cid:4)05. Constraints to the DM profile are obtained Figs.2and3thepredictionsofsuchcorrelationinthemodelsof
S Bullocketal.(solidline)andofEkeetal.(2001)(dashedline).
throughitscontributionto M(r),inwhichweincludetheterms
due to spiral and E-S0 galaxies (each one with the appropriate
density profile normalized to the observed luminosity through 2.3.SubstructuresintheComacluster
an appropriate mass-to-light ratio), and the gas component (as
Since the astrophysical signals produced by WIMP pair anni-
inferred from the X-ray surface brightness distribution) whose
hilation scale with the square of the WIMP density, any local
numberdensityprofilecanbedescribedbythefittingfunction:
overdensitydoesplayarole(seee.g.Bergstrometal.1999,and
⎡ (cid:4) (cid:5) ⎤
n(r)=n ⎢⎢⎢⎢⎢⎣1+ r 2⎥⎥⎥⎥⎥⎦−1.5b, (18) raenfaelroegnocuessltyhteoretihne).gTeonedriasclupsicstsuurbesitnrutrcotduurecsedinatbhoevCeofmoraDclMushtear-,
0 rc los,welabelasubhalothroughitsvirialmassMsanditsconcen-
trationparameterc (orequivalentlyatypicaldensityandlength
with n0 = 3.42 × 10−3 cm−3, rc = 10.(cid:4)5 and b = 0.75 scale,ρ(cid:4)sandas).Thsesubhaloprofileshapeisconsideredhereto
(Brieletal.1992). besphericalandofthesameformasfortheparenthalo.Finally,
Tocompareamodelwithsuchdatasets,webuildareduced as for the mean DM density profile, the distribution of subha-
χ2-likevariableoftheform: los in Coma is taken to be spherically symmetric.The subhalo
(cid:12) number density probabilitydistribution can then be fully spec-
χ2 = 1 1(cid:13)2 (M(r<ri)−Mi)2 ified through Ms, cs and the radial coordinate for the subhalo
r 2 2 (∆M)2 position r. To our purposes,it is sufficientto consider the sim-
i=1 i
(cid:1) (cid:2) plifiedcasewhenthedependenceonthesethreeparameterscan
+1(cid:13)9 σlos(Rj)−σljos 2(cid:14) (19) befactorized,i.e.:
9 j=1 (∆σljos)2 dr3ddMns dc = ps(r)ddMns (Ms)Ps(cs). (20)
s s s
wheretheindexiinthefirstsumrunsovertheconstraintsgiven
Herewehaveintroducedasubhalomassfunction,independent
inEqs.(13)and(14),while,inthesecondsum,we includethe
ofradius,whichisassumedtobeoftheform:
nine radial bins over which the line of sight velocity disper- ⎡ (cid:4) (cid:5) ⎤
stiimonatoedf.EW-Se0igghatlfaaxciteosrsanhdavietsbseteanndaadrddeddetvoiagtiivoenthhaessabmeeensetas-- dns = A(Mvir)exp⎢⎢⎢⎢⎢⎣− Ms −2/3⎥⎥⎥⎥⎥⎦, (21)
dM M1.9 M
tistical weight to each of the two classes of constraints, see, s s cut
e.g., Dehnen&Binney(1998) where an analogous procedure Diemand et al. (2005) where M is the free streaming cutoff
cut
hasbeenadopted. mass(Hofmannetal.2001;Chenetal.2001;Greenetal.2005;
S.Colafrancescoetal.:DMannihilationsinComa 27
Diemandetal.2005),whilethenormalizationA(M )isderived distributiondNf/dE,andN (r)isthenumberdensityofneu-
vir i pairs
imposing that the total mass in subhalos is a fraction f of the tralino pairs at a given radius r (i.e., the number of DM par-
s
totalvirialmassM oftheparenthalo,i.e. ticles pairs per volume element squared). The particle physics
vir
(cid:3) frameworksets the quantity(cid:12)σv(cid:13) and the list of B . Since the
MvirdM dns M = f M . (22) neutralino is a Majorana fermion0light fermion finafl states are
sdM s s vir suppressed, while – dependingon mass and composition – the
Mcut s
dominantchannelsareeitherthosewithheavyfermionsorthose
AccordingtoDiemandetal.(2005), f isabout50%foraMilky
s with gauge and Higgs bosons. The spectral functions dNf/dE
Way size halo, and we will assume that the same holds for i
are inferred from the results of MonteCarlo codes, namely
Coma. The quantity P (c ) is a log-normaldistribution in con-
s s thePythia(Sjöstrand1994,1995;Sjöstrandetal.2003)6.154,
centrationparametersaroundameanvaluesetbythesubstruc-
as included in the DarkSUSY package (Gondoloetal.2004).
ture mass; the trend linking the mean c to M is expected to
s s Finally,N (r)isobtainedbysummingthe contributionfrom
be analogous to that sketched above for parent halos with the pairs
the smooth DM component,which we write here as the differ-
Bullocketal.orENStoymodels,exceptthat,onaverage,sub-
encebetweenthecumulativeprofileandthetermthatatagiven
structurescollapsedinhigherdensityenvironmentsandsuffered
radiusisboundinsubhalos,andthecontributionsfromeachsub-
tidalstripping.Bothoftheseeffectsgointhedirectionofdriving
halo,inthelimitofunresolvedsubstructuresandinviewofthe
largerconcentrations,asobservedinthenumericalsimulationof
factthatweconsideronlysphericallyaveragedobservables:
Bullocketal.(2001),whereitisshownthat,onaverageandfor
(cid:12)
M ∼5×1011M(cid:3) objects,theconcentrationparameterinsubha- (ρ(cid:4)g(r/a)− f M p (r))2
losisfoundtobeafactorof≈1.5largerthanforhalos.Wemake N (r) = s vir s
pairs 2M2
herethesimplifiedansazt: (cid:3) χ (cid:3)
dn (cid:15) (cid:16)
(cid:12)cs(Ms)(cid:13)= Fs(cid:12)cvir(Mvir)(cid:13) with Ms = Mvir, (23) +ps(r) dMsdMs dcs(cid:4) Ps cs(cid:4)(Ms)
(cid:3) (cid:15) s (cid:16) (cid:14)
dwoheesren,oftordesipmenpdlicointy,Mwse. FasoslulomweinthgaatgthaeinenBhualnloccekmeetnatlf.a(c2t0o0r1F)s, × d3rs ρ(cid:4)sg2(rMs/2as) 2 · (27)
the1σdeviation∆(log c )aroundthemeaninthelog-normal χ
10 s
distribution Ps(cs), is assumed to be independentof Ms and of Thisquantitycanberewritteninthemorecompactform:
cosmology,andtobe,numerically,∆(log10cs)=0.14. (cid:12)
Finally, we have to specify the spatial distribution of sub- ρ¯2 (ρ(cid:4)g(r/a)− f ρ˜ g(r/a(cid:4)))2
N (r) = s s
structureswithinthecluster.Numericalsimulations,tracingtidal pairs 2M2 ρ¯2
stripping, find radial distributions which are significantly less χ (cid:14)
ρ˜ g(r/a(cid:4))
concentrated than that of the smooth DM component. This ra- +f ∆2 s , (28)
dialbiasisintroducedhereassumingthat: s ρ¯
p (r)∝g(r/a(cid:4)), (24) where we have normalized densities to the present-day mean
s
matter density in the Universe ρ¯, and we have defined the
withgbeingthesamefunctionalformintroducedaboveforthe quantity:
parenthalo,butwitha(cid:4)muchlargerthanthelengthscaleafound (cid:17)
for Coma. Following Nagai&Kravtsov(2005), we fix a(cid:4)/a (cid:5) dM dns M ∆2 (M )
7. Since the fraction fs of DM in subhalos refers to structures fs∆2 ≡ sdMMs s Ms s (29)
withinthevirialradius,thenormalizationof p (r)followsfrom (cid:17) vir
s
t4hπe(cid:3)reqRvuirirre2mpe(nr)t=1. (25) = fs dM(cid:17)sdddMMnsssMddMns∆ssM2Mss(Ms), (30)
s
0 with
(cid:3)
3. NeutralinoannihilationsinComa ∆2Ms(Ms)≡ ∆vi3r(z) dcs(cid:4) Ps(cid:15)cs(cid:4)(cid:16)[II12((ccs(cid:4)s(cid:4)xx−−22))]2(cs(cid:4)x−2)3 (31)
3.1.Statisticalproperties
and
Havingset thereferenceparticlephysicsframeworkandspeci- (cid:3)
x (cid:18) (cid:19)
fiedthedistributionofDMparticles,wecannowintroducethe I (x)= dyy2 g(y) n. (32)
n
sourcefunctionfromneutralinopairannihilations.Foranysta- 0
ble particle speciesi, generatedpromptlyin the annihilationor
Suchdefinitionsareusefulsince∆2 givestheaverageenhance-
produced in the decay and fragmentation processes of the an- Ms
nihilationyields, the source function Qi(r,E) givesthe number mentinthesourceduetoasubhaloofmass Ms,while∆2 isthe
ofparticlesperunittime,energyandvolumeelementproduced sumoverallsuchcontributionsweightedoverthesubhalomass
locallyinspace: functiontimesmass.Finally,inEq.(28)wehavealsointroduced
thequantity:
(cid:13)dNf
Qi(r,E)=(cid:12)σv(cid:13)0 f dEi (E)Bf Npairs(r), (26) ρ˜s ≡ 4π(a(cid:4))3MI1vi(rRvir/a(cid:4))· (33)
where (cid:12)σv(cid:13) is the neutralino annihilation rate at zero temper- In the limit in which the radial distribution of substructures
0
ature. The sum is over all kinematically allowed annihilation tracestheDMprofile,i.e.a(cid:4) = a,ρ˜ becomesequaltothehalo
s
final states f, each with a branching ratio B and a spectral normalizationparameterρ(cid:4).
f
28 S.Colafrancescoetal.:DMannihilationsinComa
Ms 108 comesinsteadfromlargeradii.Thismeansthattheenhancement
2∆ B. et al., Fs = 2 σ8 = 0.89 fromsubhaloslargelyinfluencestheresultswhentheneutralino
107 sourceisextended.Thisisthecaseofgalaxyclusters,andmore
specificallyoftheComaclusterwhichisourtargetinthispaper.
106
B. et al., F = 1 3.2.Sourcefunctionsspectralproperties:generalities
s
105 andsupersymmetricbenchmarks
104 ENS, Fs = 1 Ttiohnesspdeecpternadl pornolpye,rptireisoroftosedciffonudsiaornyapnroddeuncetsrgoyflDosMsesa,nnoinhitlhae-
N04 DM particle mass M and on the branchingratio BR(χχ → f)
χ
103 D05 forthefinalstate f intheDMpair-annihilation.TheDMparticle
Burkert physicsmodelfurthersetsthemagnitudeofthethermallyaver-
102 agedpairannihilationcrosssectiontimestherelativeDMparti-
10-710-510-310-1 10 103 105 107 109101110131015 clesvelocity,(cid:12)σv(cid:13) atT =0.
0
Ms [ M ❍⋅ ] The range of neutralino masses and pair annihilation
cross sections in the most general supersymmetric DM setup
Fig.4.Scalingoftheaverageenhancementinsourcefunctionsduetoa
is extremely wide. Neutralinos as light as few GeV (see
subhaloofmass M.Weshowresultimplementingthethreehalopro-
s Bottinoetal.2003) and as heavy as hundreds of TeV (see
filesintroduced,i.e.theN04,D05andBurkertprofile,thetwotoymod-
Profumo2005) can account for the observed CDM density
elsforthescalingofconcentrationparameterwithmass,i.e.theBullock
et al. and the ENS, and two sample values of the ratio between con- throughthermalproductionmechanisms,andessentiallynocon-
centrationparameterinsubhalostoconcentrationparameterinhalosat straintsapplyinthecaseofnon-thermallyproducedneutralinos.
equalmassF . Turning to the viable range of neutralino pair annihilation
s
cross sections, coannihilation processes do not allow us to set
anylowerbound,whileonpurelytheoreticalgroundsageneral
(cid:1) (cid:2)
menWt∆e2MsshionwtheinsoFuirgc.e4funthcetiosncavleirnsgusotfhethsuebahvaeloramgeasesnMhas.nWcee- (uPprpoefrulmimoi2t0o0n5(cid:12))σ.vT(cid:13)h0e<∼o1n0ly−2g2enTMeerVχal−a2rcgmum3/esnhtawshbiecehntireescethnetlyresleict
haveconsideredthethreehalomodelsintroducedintheprevious
abundanceofaWIMPwithitspairannihilationcrosssectionis
section, i.e. the N04, D05 and Burkertprofiles, for the two toy
givenbythenaiverelation
models describing the scaling of concentration parameter with
mass,i.e.theBullocketal.andtheENSschemes,aswellastwo 3×10−27cm3/s
psaamrapmleetvearluinessfuobrhtahleosraatinodFthsabtetiwneheanlotsheoafveeqruaaglemcoanscs.enIntraetaiochn Ωχh2 (cid:5) (cid:12)σv(cid:13)0 (34)
setup,goingtosmallerandsmallervaluesofM ,theaverageen-
s (see Jungmanetal.1995, Eq. (3.4)), which points at a fidu-
hancement∆2Ms increasesandthenflattensoutatthemassscale cial value for (cid:12)σv(cid:13)0 (cid:5) 3 × 10−26cm3/s for our choice of cos-
below which all structures tend to collapse at the same epoch,
mological parameters. The above mentioned relation can be,
andhencehaveequalconcentrationparameter.
however, badly violated in the general MSSM, or even within
In Fig. 5 we show the scaling of the weighted enhance-
minimalsetups,suchastheminimalsupergravityscenario(see
ment∆2 in thesourcefunctiondueto subhalosversustheratio
Profumo2005).
between concentration parameter in subhalos to concentration
SincethirdgenerationleptonsandquarksYukawacouplings
parameterinhalosatequalmassF ;wegiveresultsfortheusual
s are always much larger than those of the first two genera-
setofhaloprofilesconsideredin ourapproach.Analogouslyto
tions, and being the neutralinoa Majorana fermion,the largest
theenhancementforafixedmassshowninthepreviousplot,∆2 BR(χχ → f) for annihilations into a fermion-antifermionpair
isverysensitivetothescalingoftheconcentrationparameterand
areinmostcases1intothethirdgenerationfinalstatesbb,ttand
hence we find a sharp dependence of ∆2 on Fs. The fractional τ+τ−. In the context of supersymmetry, if the supersymmetric
contributionperlogarithmicintervalinsubhalomassM to∆2is
s partners of the above mentioned fermions are not significantly
also showninFig. 5forfoursamplecases. Notethat,although different in mass, the τ+τ− branching ratio will be suppressed,
thefactorizationintheprobabilitydistributionforclumpsinthe
with respect to the bb branching ratio by a color factor equal
radialcoordinateandmass(plustheassumptionthatF doesnot
s to 1/3, plus a possible further Yukawa coupling suppression,
depend on mass) are a crude approximation,what we actually
sincethetwofinalstatessharethesameSU(2)quantumnumber
needinourdiscussionisF andtheradialdistributionforsubha-
s
assignment. Further, the fragmentation functions of third gen-
losatthepeakofthedistributionshowninFig.5:unfortunately
eration quarks are very similar, and give rise to what we will
wecannotreadoutthisfromnumericalsimulations.
dubinthefollowingasa“softspectrum”.Asecondpossibility,
Figure 6 shows the number density of neutralino pairs (we
setheretheneutralinomassto M = 100GeV)asafunctionof when kinematically allowed, is the pair annihilation into mas-
χ sivegaugebosons2,W+W− andZ Z .Again,thefragmentation
the distance from the center of Coma for the three representa- 0 0
functionsforthesetwofinalstatesaremostlyindistinguishable,
tivehaloprofilesintroducedhere,i.e.theN04,D05andBurkert
profileintheirbestfitmodel,andasampleconfigurationforthe
1 Models with non-universal Higgs masses at the GUT scale can
subhalo parameters. For the D05 and N04 profiles, the central
give instances of exceptions to thisgeneric spectral pattern, featuring
enhancementincreases the integratedsourcefunctionby a fac-
lightfirstandsecondgenerationsfermions(seee.g.Baeretal.2005b,
tor ≈6 with respect to the Burkert profile, but this takes place 2005c).
on sucha small angularscale thatfromthe observationalpoint 2 The direct annihilation into photons is loop suppressed in
ofviewitislikeaddingapointsourceatthecenteroftheclus- supersymmetric models (see e.g. Bergstrom&Snellman1988; and
ter. Theenhancementof the annihilationsignalsfromsubhalos Bergstrom&Ullio1997).
S.Colafrancescoetal.:DMannihilationsinComa 29
0.12
2 2
∆ 108 σ8 = 0.89 ∆o N04 profile Fs = 1
n t 0.1 Fs = 2
o
107 ti Bullock et al. ENS
u
b
ri 0.08
t
n
106 o
c
B. et al. al 0.06
ti
r
105 pa
0.04
ENS N04
104 D05
0.02
Burkert
103 0
-7 -5 -3 -1 3 5 7 9 11 13 15
1 10 10 10 10 10 10 10 10 10 10 10 10 10
Fs ≡ 〈 cs 〉 / 〈 cvir 〉 Ms [ M ❍⋅ ]
Fig.5.Left:scalingoftheweightedenhancementinsourcefunctionsduetosubhalosversustheratiobetweenconcentrationparameterinsubhalos
toconcentrationparameterinhalosatequalmassF .Resultsareshownthethreehaloprofilesintroduced,i.e.theN04,D05andBurkertprofile,
s
thetwotoymodelsforthescalingofconcentration parameterwithmass,i.e.theBullocketal.andtheENS.Right:fractionalcontributionper
logarithmicintervalinsubhalomassM to∆2infoursamplecases.Anormalizationofthefluctuationspectrumσ =0.89isadopted.
s 8
-6 [ cm ]airs11001---321 N0D405 BMu vlilro =c k9. (cid:127)e t1 a0l1.,4 fhs -1= M50 ❍%⋅ ,, cF vs ir= = 2 10 -1-3-1V cm s]1100--2265 W+ W− σNv p a=ir s1 =0 -126 c cmm-63 s-1
Np10 Ge -27
10--54 E) [ 10 τ+ τ−
10-6 Burkert Q (e10-28
10 R
radio
10-7 10-29
-8 R vir
10 -30 –
-9 total 10 Mχ = 100 GeV b b
10 smooth
-10 subhalos -31
10 10
1 10 102 103 10-2 10-1 1
r [ kpc ] E / Mχ
Fig.6. Thenumber densityof neutralinopairs(neutralinomassset to Fig.7.Thespectralshapeoftheelectronsourcefunctionincaseofthree
M = 100 GeV) as a function distance for the center of Coma for samplefinalstates(seetextfordetails).
χ
the threehaloprofilesintroduced, i.e.the N04, D05and Burkert pro-
fileintheirbestfitmodel,andasampleconfigurationforthesubhalo
parameters. give a neutralino thermal relic abundance exactly match-
ing the central cosmologically observed value. To this ex-
tent, we refer to the so-called minimal supergravity model
(Goldberg1983;Ellisetal.1983,1984),perhapsoneofthebet-
and will be indicated as giving a “hard spectrum”. The occur-
rence of a non-negligible branching fraction into τ+τ− or into terstudiedparadigmsoflow-energysupersymmetry,whichen-
ables, moreover, a cross-comparison with numerous dedicated
lightquarkswillgenerallygiveraisetointermediatespectrabe-
studies,rangingfromcolliders(Baeretal.2003)toDMsearches
tweenthe“hard”and“soft”case.
(Edsjoetal.2004;Baeretal.2004).
Figure 7 shows the spectral shape of the electron source
The assumptions of universality in the gaugino and in the
function in the case of the three sample final states bb, τ+τ−
scalar (masses and trilinear couplings) sectors remarkably re-
and W+W− for Mχ = 100 GeV, and clarifies the previous dis- duce, in this model, the number of free parameters of the gen-
cussion. In what followswe will thereforeemploysample DM
eral soft SUSY breaking Lagrangian(Chungetal.2003) down
configurations making use of either soft (bb) or hard (W+W−) to four continuous parameters (m , M ,A ,tanβ) plus one
0 1/2 0
spectra,keepinginmindthatotherpossibilitieswouldlikelyfall sign (sign(µ)). The mSUGRA parameter space producing a
inbetweenthesetwoextrema. sufficiently low thermal neutralino relic abundance Ω h2 has
χ
In order to make a more stringent contact with supersym- been shown to be constrainedto a handfulof “regions”featur-
metry phenomenology, we will however also resort to real- ing effective Ω h2 suppression mechanisms (Ellisetal.2003).
χ
istic benchmark SUSY models: by this we mean thoroughly The latter are coannihilations of the neutralino with the next-
defined SUSY setups which are fully consistent with ac- to-lightest SUSY particle (“Coannihilation” region), rapid an-
celerator and other phenomenological constraints, and which nihilations through s channel Higgs exchanges (“Funnel”
30 S.Colafrancescoetal.:DMannihilationsinComa
Table1.TheinputparametersofthefourmSUGRAbenchmarkmodels energiesE >∼ 1GeVandthisemissionisdirectlyradiatedsince
weconsiderhere.TheunitsforthemassparametersareGeV,andthe the π0 → γγ e.m. decay is very fast. This gamma-ray emis-
universal trilinearcoupling A0 issetto0forallmodels(seeBattaglia sion is dominant at high energies, >∼0.3−0.5 of the neutralino
etal.2004,fordetails). mass, but needs to be complemented by other two emission
mechanisms which produce gamma-rays at similar or slightly
Model M1/2 m0 tanβ sign(µ) mt lowerenergies:thesearetheICSandthebremsstrahlungemis-
B(cid:4)(Bulk) 250 57 10 >0 175 sionbysecondaryelectrons.Wewilldiscussthefullgamma-ray
D(cid:4)(Coann.) 525 101 10 >0 175
emission of Coma induced by DM annihilation in Sect. 4 be-
E(cid:4)(FocusP.) 300 1653 10 >0 171
K(cid:4)(Funnel) 1300 1070 46 <0 175 low. Secondaryelectronsare producedthroughvariousprompt
generationmechanismsandbythedecayofchargedpions(see,
e.g.,Colafrancesco&Mele2001).Infact,chargedpionsdecay
through π± → µ±ν (ν¯ ), with µ± → e± + ν¯ (ν ) + ν (ν¯ ) and
region),theoccurrenceoflightenoughneutralinoandsfermions µ µ µ µ e e
produce e±, muons and neutrinos. Electrons and positrons are
masses (“Bulk” region) and the presence of a non-negligible
producedabundantlybyneutralinoannihilation(seeFig.8,left)
bino-higgsinomixing(“FocusPoint”region).
andaresubjecttospatialdiffusionandenergylosses.Bothspa-
With the idea of allowing a direct comparison with the ex-
tialdiffusionandenergylossescontributetodeterminetheevo-
isting research work in a wealth of complementary fields, we
lution of the source spectrum into the equilibrium spectrum of
restrict ourselves to the “updated post-WMAP benchmarks for
these particles, i.e. the quantity which will be used to deter-
supersymmetry”proposedandstudiedbyBattagliaetal.(2004).
minetheoverallmulti-wavelengthemissioninducedbyDMan-
Allofthosesetupsaretunedsoastofeatureaneutralinothermal
nihilation.Thesecondaryelectronseventuallyproduceradiation
relicdensitygivingexactlythecentralWMAP-estimatedCDM
bysynchrotroninthemagnetizedatmosphereofComa,Inverse
density3. As a preliminary step, we computed the electrons,
Compton Scattering of CMB (and other background) photons
neutrinos, gamma-rays and protons source spectra for all the
13A(cid:4)-M(cid:4)models.Remarkablyenough,althoughtheSUSYpar- and bremsstrahlung with protons and ions in the atmosphere
oftheComacluster(see,e.g.,Colafrancesco&Mele2001;and
ticlespectrumisratherhomogeneousthroughoutthemSUGRA
Colafrancesco2003, 2006,for a review). These secondarypar-
parameterspace,theresultingspectraexhibitatleastthreequal-
ticles also produceheatingofthe intra-clustergasby Coulomb
itatively different shapes, according to the dominant final state
collisionswiththeintra-clustergasparticlesandSZeffect(see,
in neutralino pair annihilation processes. In particular, in the
e.g. Colafrancesco2003, 2006). Other fundamental particles
Bulk and Funnel regions the dominant final state is into bb¯,
and,withasub-dominantvariablecontribution,τ+τ−.Thelatter which mighthave astrophysicalrelevance are also producedin
DMannihilation.Protonsareproducedinasmallerquantitywith
channel is instead dominant, for kinematic reasons, in the stau respectto e± (see Fig. 9, right),butdo notloose energyappre-
Coannihilation region.Finally, a third, and last, possibility is a
ciablyduringtheirlifetimewhiletheycandiffuseandbestored
dominantgauge bosons final state, which is the case along the
intheclusteratmosphere.Theseparticlescan,inprinciple,pro-
FocusPointregion.Inthisrespect,intheefforttoreproduceall
duce heating of the intra-cluster gas and pp collisions provid-
ofthementionedspectralmodes,andtoreflecteverycosmolog- ing,again,asourceofsecondaryparticles(pions,neutrinos,e±,
ically viable mSUGRA region,we focused on the four models
muons,...)incompleteanalogywiththesecondaryparticlepro-
indicated in Table 1, a subset of the benchmarks of Battaglia
ductionbyneutralinoannihilation.Neutrinosarealsoproduced
et al. (2004) (to which we refer the reader for further details).
in the process of neutralino annihilation (see Fig. 9, left) and
We collect in Table 2 the branching ratios for the final states
propagatewithalmostnointeractionwiththematteroftheclus-
of neutralinopairannihilations.In the lastcolumnofthistable
ter.However,theresultingfluxfromComaisfoundtobeunob-
we also provide the thermally-averagedpair annihilation cross
servablebycurrentexperiments.
sectiontimestherelativevelocity,atT =0,(cid:12)σv(cid:13) .Table2isan
0 To summarize, the secondary products of neutralino anni-
accurateguidelinetointerprettheresultingsourcespectraforthe
hilationwhichhavethemostrelevantastrophysicalimpactonto
fourbenchmarksunderconsiderationhere,whichare shownin
themulti-frequencyspectralenergydistributionofDMhalosare
Figs.8and9.Figure8showsinparticularthedifferentialelec-
neutralpionsandsecondaryelectrons.
tron (left) and photon (right) yields per neutralino annihilation
multiplied by(cid:12)σv(cid:13) , i.e. the sourcefunction Q(r,E) dividedby
0
thenumberdensityofneutralinopairsNpairs(r)asafunctionof 4. Neutralino-inducedsignals
theparticles’kineticenergy.Asmentionedabove,theBulkand
Funnelcasesareverysimilarbetweeneachother,thoughinthe AcompletedescriptionoftheemissionfeaturesinducedbyDM
lattercaseonehasaheavierspectrumandalargervalueof(cid:12)σv(cid:13) . must take, consistently, into account the diffusion and energy-
0
Figure9showsthesamequantityforneutrinosandprotons. loss propertiesofthese secondaryparticles.Thesemechanisms
The productsof the neutralinoannihilationwhichare more are taken into account in the following diffusion equation (i.e.
relevanttoourdiscussionaresecondaryelectronsandpions.The neglectingconvectionandre-accelerationeffects):
(cid:12) (cid:14) (cid:12) (cid:14)
secondaryparticlesproducedbyneutralinoannihilationaresub-
∂ dn dn ∂ dn
ject to various physical mechanisms: i) decay (which is espe- e = ∇ D(E,x)∇ e + b(E,x) e
cially fast for pions and muons); ii) energy losses which can ∂t dE dE ∂E dE
be suffered by stable particles, like electrons and positrons; +Q (E,x), (35)
e
iii) spatial diffusion of these relativistic particles in the atmo-
wheredn /dE istheequilibriumspectrum,D(E,x)isthediffu-
sphere of the cluster. Gamma-rays produced by neutral pion e
decay, π0 → γγ, generate most of the continuum spectrum at sion coefficient, b(E,x) is the energyloss term and Qe(E,x) is
thesourcefunction.Theanalyticalsolutionofthisequationfor
3 Weadjustedherethevaluesofm giveninBattagliaetal.(2004) thecaseoftheDMsourcefunctionisderivedintheAppendixA.
0
inordertofulfillthisrequirementmakinguseofthelatestIsajetv.7.72 Inthelimitinwhichelectronsandpositronsloseenergyon
releaseandoftheDarkSUSYpackage(Edsjoetal.2003,seeTable1). atimescalemuchshorterthanthetimescaleforspatialdiffusion,