MNRAS439,566–587(2014) doi:10.1093/mnras/stt2488 AdvanceAccesspublication2014January30 Multimessenger constraints on dark matter annihilation into electron–positron pairs M. Wechakama1,2‹ and Y. Ascasibar3‹ 1Leibniz-Institutfu¨rAstrophysikPotsdam,AnderSternwarte16,D-14482Potsdam,Germany 2DepartmentofPhysics,FacultyofScience,KasetsartUniversity,Chatuchak,Bangkok10900,Thailand 3DepartamentodeF´ısicaTeo´rica,UniversidadAutońomadeMadrid,E-28049Madrid,Spain Accepted2013December20.Received2013December12;inoriginalform2012December11 D o w n lo a d ABSTRACT e d WeinvestigatetheproductionofelectronsandpositronsintheMilkyWaywithinthecontextof fro m darkmatterannihilation.Upperlimitsontherelevantcross-sectionareobtainedbycombining h observationaldataatdifferentwavelengths(fromHaslam,WMAPandFermiall-skyintensity ttp s maps)withrecentmeasurementsoftheelectronandpositronspectrainthesolarneighbourhood ://a c byPAMELA,FermiandHESS.Weconsidersynchrotronemissionintheradioandmicrowave a d e bands,aswellasinverseComptonscatteringandfinal-stateradiationatgamma-rayenergies. m According to our results, the dark matter annihilation cross-section into electron–positron ic.o u pairsshouldnotbehigherthanthecanonicalvalueforathermalrelicifthemassofthedark p .c mattercandidateissmallerthanafewGeV.Inaddition,wealsoderiveastringentupperlimit om on the inner logarithmic slope α of the density profile of the Milky Way dark matter halo /m n (α<1ifm <5GeV,α<1.3ifm <100GeVandα<1.5ifm <2TeV)assumingthat ra dm dm dm s w(cid:2)σitvh(cid:3)ea±th=er3m×al1r0e−li2c6licgmh3tesr−th1.aAn∼lo1gTareiVth,mreigcasrldolpesessotefetpheerdtohmaninαan∼ta1n.5niihsilhaatirodnlychcoanmnpeal.tible /article -a b Keywords: astroparticlephysics–radiationmechanisms:non-thermal–Galaxy:structure– s tra darkmatter. c t/4 3 9 /1 /5 6 6 Wai2004;Hooperetal.2004;Bergstro¨m&Hooper2006;Sańchez- /98 1 INTRODUCTION 4 Condeetal.2007;Strigarietal.2007,2008;Woodetal.2008;Essig, 5 2 Dark matter can be indirectly detected through the signatures of Sehgal&Strigari2009;Martinezetal.2009;Abdoetal.2010;Ac- 7 b standardmodelparticlesproducedbyitsannihilationordecay(see ciarietal.2010;Essigetal.2010;Abramowskietal.(HESSCollab- y g e.g.Bertone,Hooper&Silk2005;Bertone2010).Agreatdealof oration)2011a;Ackermannetal.(TheFermi-LATCollaboration) ue s workhasfocusedontheemissionofgammaraysfromtheGalac- 2011) and galaxy clusters (e.g. Colafrancesco, Profumo & Ullio t o ticCentre(e.g.Berezinsky,Bottino&Mignola1994;Bergstro¨m, 2006;Jeltema,Kehayias&Profumo2009;Ackermannetal.(The n 0 Ullio&Buckley1998;Baltz&Edsjo¨1999;Gondolo&Silk1999; Fermi-LATCollaboration)2010a;Pinzke,Pfrommer&Bergstro¨m 8 A Metoarls.e2l0li0e4t;aPl.ei2r0a0n2i,;MUollhioayeateael.&20d0e2;FSretoiteahsrPeatcahle.c2o00230;04C;ePsarraidnai 2d0et1e1c;tiSoańncinhethz-eCmonicdreowetaavle.2b0a1c1k)g.rPoruonsdpehcatvsefoarlisnodbireeecntdcaornksmidaetrteedr pril 2 0 et al. 2004; Bergstro¨m et al. 2005a,b; Profumo 2005; Aharonian byseveralauthors(e.g.Blasi,Olinto&Tyler2003;Colafrancesco 19 etal.(HESSCollaboration)2006;Zaharijas&Hooper2006;Bo- 2004;Padmanabhan&Finkbeiner2005;Mapelli,Ferrara&Pier- yarskyetal.2008a;Pospelov,Ritz&Voloshin2008;Springeletal. paoli2006;Zhangetal.2006,2007;Cholis,Goodenough&Weiner 2008; Bell & Jacques 2009; Cirelli & Panci 2009; Fornasa et al. 2009b;Gallietal.2009,2011;Slatyer,Padmanabhan&Finkbeiner 2009;Abazajianetal.2010;Bernal&Palomares-Ruiz2010;Cirelli, 2009;Kanzaki,Kawasaki&Nakayama2010;Lavalle2010;Hu¨tsi Panci&Serpico2010;Papucci&Strumia2010;Abramowskietal. etal.2011;McQuinn&Zaldarriaga2011;Delahaye,Bo¨hm&Silk (HESSCollaboration)2011b;Hooper&Goodenough2011;Hooper 2012), as well as X-ray (e.g. Abazajian, Fuller & Tucker 2001; &Linden2011a,b;Abazajian&Harding2012;Ackermannetal. Boyarskyetal.2007;Boyarsky,Ruchayskiy&Markevitch2008b; (The Fermi-LAT Collaboration) 2012b, among many others), the Zavalaetal.2011),radio(e.g.Colafrancesco&Mele2001;Aloisio, Milky Way satellites (e.g. Baltz et al. 2000; Tyler 2002; Baltz & Blasi&Olinto2004;Bergstro¨metal.2009a;Borriello,Cuoco& Miele2009;Ishiwata,Matsumoto&Moroi2009;Fornengoetal. 2012) and multiwavelength signatures (e.g. Regis 2008; Regis & Ullio2008;Bertoneetal.2009;Pato,Pieri&Bertone2009;Crocker (cid:4)E-mail:mwechakama@gmail.com(MW);yago.ascasibar@uam.es(YA) etal.2010;Profumo&Ullio2010). (cid:5)C 2014TheAuthors PublishedbyOxfordUniversityPressonbehalfoftheRoyalAstronomicalSociety Constraintsondarkmatterannihilation 567 Therecentresultsfromindirectdetectionexperimentsintheso- allowed for the positron injection rate or, equivalently, the dark larneighbourhoodhavealsosuggestedthepossibilitythatsucha matterannihilationcross-section. signaturehasbeenseen.Inparticular,thePAMELAexperimenthas Rather than focusing on a particular dark matter candidate, pointedasignificantexcessofelectronsandpositronsabovetheex- we adopt a model-independent approach (see e.g. Wechakama & pectedsmoothastrophysicalbackground(Adrianietal.(PAMELA Ascasibar2011),inwhichalltheinjectedparticlesarecreatedwith Collaboration)2009b).Iftheseresults,confirmedbyFermi(Ack- thesameinitialenergyE ,oftheorderofthemassofthedarkmatter 0 ermann et al. (The Fermi-LAT Collaboration) 2012a) and AMS- particle.Sincethismassisusuallymuchlargerthantherestmass 02 (Aguilar et al. 2013), are interpreted in terms of dark matter of the electron, electrons and positrons will be relativistic at the annihilation, then an abundant population of high-energy e± is momentoftheircreation.However,theycanefficientlylosetheir being created everywhere in the Galactic dark matter halo, with energythroughdifferentprocesses,suchasICS,synchrotronradia- the associated final-state radiation (FSR), as well as synchrotron tion,Coulombcollisions,bremsstrahlungandionization.Through- emission in the Galactic magnetic field and inverse Compton outthispaper,wewilloftenusetheLorentzfactorγ toexpressthe scattering (ICS) of the photons of the interstellar radiation field energyE=γm c2 oftheannihilationproducts,wherem denotes e e D (ISRF). therestmassofelectronandcisthespeedoflight.Wewillfirstdis- o w Althoughthecurrentlymostfavouredexplanationfortheorigin cusstheresultsobtainedfora‘canonical’modeloftheMilkyWay n lo ofGalacticpositrons,tracedbythepositronannihilationemission and then explore the effects of varying the intensity of magnetic a d line at 511 keV (see Prantzos et al. 2011, for a recent review) is field,thediffusioncoefficient,theISRFandtheinnerlogarithmic ed low-mass X-ray binaries (Weidenspointner et al. 2008), and the slopeofdarkmatterdensityprofile. fro m local positron excess at high energies is most likely due to the Theremainderofthispaperisstructuredasfollows:Section2 h contribution of nearby pulsars (see e.g. Profumo 2012), several describestheprocedurefollowedtoestimatetheelectron–positron ttp s workshaveconsideredthepossibilitythatdarkmatterannihilation spectrum,thesurfacebrightnessprofilesandtheparametersofthe ://a makesasizeablecontributiontothepositronbudgetoftheMilky MilkyWaymodel.Ouranalysisoftheobservationaldataisfully c a Way(e.g.Boehmetal.2004;Bœhm&Ascasibar2004;Beacom, describedinSection3(tableswithprecisenumericvaluesarepro- de m Bell&Bertone2005;Picciotto&Pospelov2005;Ascasibaretal. videdasanappendix)andSection4isdevotedtotheconstraintson ic 2006; Beacom & Yu¨ksel 2006; Sizun, Casse´ & Schanne 2006; thedarkmatterannihilationcross-section.Theeffectofthedifferent .o u Finkbeiner & Weiner 2007; Pospelov & Ritz 2007; Barger et al. astrophysicalparametersisdiscussedinSection5,whileSection6 p.c 2009; Bergstro¨m, Edsjo¨ & Zaharijas 2009b; Chen & Takahashi focusesontheconstraintsthatonecanimposeontheslopeofthe om 2009;Cholisetal.2009b;Cirellietal.2009;Donatoetal.2009; darkmatterdensityprofilebyassumingthatdarkmatterparticles /m n Grassoetal.2009;Malyshev,Cholis&Gelfand2009;Mertsch& areproducedasthermalrelicsintheprimordialUniverse.Particu- ra s Sarkar2009;Regis&Ullio2009;Yinetal.2009;Chenetal.2010; larannihilationchannelsarediscussedinSection7,andourmain /a Meade et al. 2010; Cline, Frey & Chen 2011; Vincent, Martin & conclusionsaresuccinctlysummarizedinSection8. rtic Cline2012). le-a Thisworkfocusesontheastrophysicalsignaturesofdarkmat- bs terannihilationintoelectron–positronpairs,neglectingotherpro- 2 MODEL PREDICTIONS tra c cesses, such as dark matter decay, or other annihilation products, t/4 2.1 Electron–positronpropagation 3 such as protons and antiprotons (whose contribution is severely 9 /1 constrained by recent observational data; see e.g. Adriani et al. Asinourpreviouswork(Wechakama&Ascasibar2011),theprop- /5 6 (PAMELA Collaboration) 2009a). We try to impose robust, yet agationofelectronsandpositronsthroughtheinterstellarmedium 6 /9 stringent constraints on the relevant cross-section by comparing (ISM)isdeterminedbythediffusion-lossequation 8 the predictions of an analytic model of particle propagation with (cid:2) (cid:3) 45 amultiwavelengthsetofobservationaldataobtainedfromthelit- ∂∂t ddγn(x,γ)=∇ K(x,γ)∇ddγn(x,γ) 27 by erature. More precisely, we compare the expected emission from (cid:2) (cid:3) gu s1y8ncmharoptsroonfrtahdeiastikoyn,aItCdSiffaenrdenFtSfRreqwuietnhciniesth:ethMeilHkyaslWamayrwadiitoh + ∂∂γ b(x,γ)ddγn(x,γ) +Q(x,γ). (1) est on map at 408MHz, the 7yr data from the Wilkinson Microwave 0 Anisotropy Probe (WMAP) in its five bands (23GHz, 33GHz, Weassumeadiffusioncoefficientoftheform 8 A 4L1arGgeHAz,r6e1aGTeHlezsacnodpe94(LGAHTz))bainndnegdaminm1a2-rdaiyffemreanptscfrhoamnntehlesF(ferrommi K(γ)=K0γδ, (2) pril 2 0 0.3 to 300GeV). A straightforward statistical criterion is used in independent of Galactic location. The values of K0 and δ corre- 19 order to mask the most obvious astrophysical signals (i.e. emis- spondingtothethreemodelsdiscussedbyDonatoetal.(2004)and sionfromtheGalactic discandpointsources),andobservational Delahayeetal.(2008)areprovidedinTable1below.Theenergy upperlimitsarederivedfromtheremainingsphericallysymmetric lossrate component. dγ (cid:4) Inadditiontothephotondata,wealsoconsidertherecentmea- b(x,γ)≡−dt (x,γ)= bi(x,γ) (3) surementsofthelocalelectronandpositronspectraperformedby i PAMELA (Adriani et al. (PAMELA Collaboration) 2009b, 2010, isasumovertherelevantphysicalprocesses,andthesourceterm 2011, 2013), Fermi (Ackermann et al. (The Fermi-LAT Collabo- Q(x,γ) represents the instantaneous electron–positron injection ration)2010b,2012a)andHESS(Aharonianetal.(HESSCollab- rate. oration)2008).Aswillbeshownbelow,consideringthepositron Given enough time (of the order of 100 Myr; cf. fig. 2 in spectrum separately (rather than the combined electron+positron Wechakama & Ascasibar 2011), the electron–positron popula- spectrum)yieldsasignificantimprovementonthemaximumvalue tion will approach a steady-state distribution, ∂ dn(x,γ)=0. ∂tdγ MNRAS439,566–587(2014) 568 M.WechakamaandY.Ascasibar Table 1. Three different models of Table 2. Normalization of the grey-body the diffusion coefficient, following the models describing the ISRF, adopted from parametrization K(γ) = K0γδ. The model Cirelli & Panci (2009). In our canonical MED has been proposed by Donato et al. model,weusethevaluesappropriateforthe (2004),andmodelsM1andM2areadopted GalacticCentreinordertocomputetheICS fromDelahayeetal.(2008). andsynchrotronemission.Fortheelectron– positronspectrumatthesolarneighbourhood, Model K0(kpc2s−1) δ weuseISRF(I). MMM1E2D(canonical) 721...497226×××111000−−−111788 000...475605 Model TSL=N3SL481K TIR=N4IR0.6K ISRF(I) 2.7×10−12 7.0×10−5 Assumingthatb(x,γ)variessmoothlyinspace,theparticlespec- CISaRnFon(iIcIa)l 18..79××1100−−1113 71..03××1100−−55 D trumfulfilstherelation o w ∂y(x,γ) + K(γ)∇2y(x,γ)=−Q(x,γ), (4) nloa ∂γ b(γ) synchrotronradiation,Coulombcollisions,bremsstrahlungandion- de where izaTtihoeneonfenrgeyutlroaslshryadterosgdeenpeantodmosn.theenergyoftheparticle.High- d fro m y(x,γ)≡b(γ)ddγn(x,γ). (5) eSnaerargzyine1le9c9t9ro).nWs aencdompopsuittreonthsemtoatianllypolwoseerernaderiagtyedbybyICaSsi(neg.gle. http s Imposing ddγn(x,γ)=0atinfinity,oneobtainstheGreenfunction eKlelecitnro–nNiusshiinngatchreofsos-rsmeacltiisomn.dInestchreibneodn-inreSlaetcivtiiosntic2r.4eg,bimasee,dthoenlothses ://aca (cid:5) (cid:6) d exp −|x−xs|2 functioncanbeapproximatedas em G(x,γ,xs,γs)= (cid:7)2π(cid:7)λ22(cid:7)(cid:8)λ32/2 (cid:9)(γ −γs) (6) bICS(γ)= 34mσTcγ2Urad, (11) ic.oup e .c and either the image charges method or an expansion over the where σ is the Thomson cross-section. The combined radiation om T eigenfunctions of the linear differential operator may be used to energydensityoftheCMB,starlight(SL)andinfrared(IR)light /m n derivetheGreenfunctionforotherboundaryconditions(seee.g. fromthermaldustemission(seee.g.Porter&Strong2005;Porter ra s Baltz&Edsjo¨ 1999;Delahayeetal.2009).Theelectron–positron etal.2008)isrepresentedbythreegreybodies, /a sddpγnec(txr,uγm)i=stbh(uxs1,gγiv)e(cid:9)nγ∞bydγs(cid:9)0∞d3xsex(cid:7)p2π(cid:5)−(cid:7)|λx22−(cid:7)(cid:8)xλ3s2/|22(cid:6)Q(xs,γs), (7) mwUrhaaledirz=eatTi4oinσacSnoBdf(cid:7)eNaTciCh4MrecBpor+mespNeonnSteLtnThtSe4,Lree+fsfpeNecctIiRtviTveIe4Rltye(cid:8),m,apnedraσtuSBreisanthdetShteef(na1on2r–-) rticle-abstract/4 wherethequantity Boltzmann constant. The CMB is modelled as a perfect black- 39 (cid:7)λ2=λ2(γ)−λ2(γs) (8) fbooldloywwCitihretlelim&pePraatnucrie(T2C0M09B)=for2.t7h2e6twKo(oFtihxesrenco2m0p0o9n),enatnsd(sweee /1/566 isrelatedtothecharacteristicdiffusionlengthoftheelectronsand Table 2 below). Expression (11) provides a good approximation /98 positrons,γ denotestheirinitialenergy,andthevariableλisde- for low values of the Lorentz factor γ, but it severely overes- 45 s 2 finedas (cid:9) ∞ 2K(γ) tiimmpaotertsanitt.for γmec2 ≥ TeV, where relativistic effects become 7 by g λ2(γ)= γ b(γ) dγ. (9) atShyingchhreontreorgniersa.diTahtieonexipsreasnsoiothnerfoirmpthoertalnotsslorsastemieschsiamniislmar uest o Consideringthedarkmatterhaloasasphericallysymmetricsource, to that of non-relativistic ICS, substituting the radiation en- n 0 the spatial integral can be reduced to one dimension, and the ergy density in equation (11) by the magnetic energy den- 8 A electron–positronspectrumisfinallygivenbytheexpression sity, U =B2/(8π), where B is the intensity of the magnetic p (cid:5) (cid:6) field: B ril 2 ddγn(r,γ)= b(1γ)(cid:7)e2xπpr2−(cid:7)2λ(cid:7)r22λ(cid:8)21/2 bsyn(γ)= 34mσTcγ2UB. (13) 019 e (cid:10) (cid:9) (cid:9) (cid:11) (cid:12) For lower-energy electrons and positrons, Coulomb interactions ∞ ∞ r2 × dγ dr r exp − s withthethermalplasmamustbetakenintoaccount.Thelossrate γ s 0 s s 2(cid:7)λ2 isapproximately(Rephaeli1979) (cid:13) (cid:5) (cid:6) (cid:5) (cid:6)(cid:14) (cid:15) (cid:2) (cid:3) rr rr ln(γ/n ) × exp (cid:7)λs2 −exp −(cid:7)λs2 Q(rs,γs) . (10) bCoul(γ)≈1.2×10−12ne 1+ 75 e s−1, (14) wheren isthenumberdensityofthermalelectrons. e 2.2 Lossrates Collisionswiththermalionsandelectronsalsoproduceradiation through bremsstrahlung. The loss rate due to bremsstrahlung can Electrons and positrons can lose their energy by several physical beapproximatedas(Blumenthal&Gould1970) processesastheymovethroughtheISM.WeconsiderICSofcos- micmicrowavebackground(CMB),starlightandinfraredphotons, b (γ)≈1.51×10−16n γ[ln(γ)+0.36] s−1. (15) brem e MNRAS439,566–587(2014) Constraintsondarkmatterannihilation 569 Additional energy losses come from the ionization of hydrogen contradiction with observations. Traditionally, it has been argued atoms.ThelossrateisgiveninLongair(1981), that the presence of gas and stars makes the profile even steeper (cid:2) duetotheeffectsofadiabaticcontraction(Blumenthaletal.1986), q4n γ(γ2−1) b (γ)= e (cid:16)H × ln (cid:5) (cid:6) althoughsomerecentclaimshavealsobeenmadeintheopposite ion 8π(cid:11)(cid:10)02m2ec3 (cid:12)1− γ12 2 (cid:11)meIc2 2 (cid:12) (cid:3) dCioreuccthiomnan(e.&g.WEla-Zdsalnety,S2h0l0o6s;mOanh&etHalo.ff2m01an0)2.0G0i1v;eMn athsehcchuernrkenot, − 2 − 1 ln2+ 1 + 1 1− 1 2 , (16) uncertainties,wehavelefttheinnerslopeofthedensityprofileasa γ γ2 γ2 8 γ freeparameterofthemodel. wheren isthenumberdensityofhydrogenatoms,q istheelectron H e charge, (cid:10)0 is the permittivity offree space and Iis the ionization 2.4 Surfacebrightnessprofile energyofthehydrogenatom.Thenumberdensityofthermalelec- Once the electron–positron spectrum is computed, the emission tronsandneutralatomscanbeexpressedintermsofthetotalISM gasdensityρ andtheionizationfractionX as coefficient1forphotonsoffrequencyνisgivenbytheintegral g ion (cid:9) D ne = mρgpXion (17) jν(r,ν)= 41π 1∞ ddγn(r,γ)l(γ,ν)dγ (23) ownloa annHd= mρg(1−Xion), (18) mpoofisntihotresoitneylewlc(itγtrho,nLν–o)proeesmnittizrtotefnadcstapotercfγtrreu.qmTuheneddγnciy(nrt,νeγnbs)iyttyiamfresosinmtghlaeensyepleegccivtirfieoncnldouir-- ded from p h respectively. rtsheoceutreicomeniasinsnidothnbeocuosnkedyffiaircsyiescniomt.npSdliiytnicotehnesw,ieinttawesgsilrulamolneallyaonsdgpehpteehrneidclaionllneytoshyfemsainmggheuttlroaicrf ttps://ac a 2.3 Sourceterm separationθ withrespecttotheGalacticCentre, de (cid:9) m Sincetheelectronsandpositronsinourmodeloriginatefromthe Iν(θ,ν)= ∞jν(r,ν)ds, (24) ic.o annihilationofdarkmatterparticles,theinstantaneousproduction 0 up rateatanygivenpointcanbeexpressedas wheresrepresentsthedistancealongthelineofsight,andtheradial .co m Q(r,γ)=ηndm(r)ndm∗(r)(cid:2)σv(cid:3)e± ddNγe±(γ), (19) disista(cid:17)ncertothecentreoftheMilkyWayatanypointalongtheray /mnra s pwahretircelensdmanadnadntnipdma∗rtidcelenso,teretshpeecntiuvmelbye,r(cid:2)σdevn(cid:3)es±itiiesstohfedthaerkrmmalatatve-r wri=thx=x2s+sinyθ2,,y=scosθ −R(cid:12),andR(cid:12)=8.5kpc(thedista(n2c5e) /article erageoftheannihilationcross-sectiontimesthedarkmatterrela- -a tive velocity, and dNdγe± is the injection spectrum of electrons and ofTthheeScuonnftrriobmutitohneGofalsaycnticchCroetnrotrne)r.adiation, which dominates at bstra pnelodsmsei,t=rnodnnmsdm=in∗ n=thdemmρ∗fiddmmn=aal12nsdmtρaddηmtme.=aFno1dr/η2se=ilnf-1oc.ordnejurgtoataevdoaidrkdmouabttleercpoaurtnitcilnegs;, llosywn(γp,hνot)o=ne√nme3rgqcie3e2Bs,cRa[nχb(eγe)]s,timatedas(seee.g.Sarazin1999(2)6) ct/439/1/5 Weconsiderself-conjugatedarkmatterparticlesthroughoutthis e 6 6 workandassumethateachannihilationeventinjectsoneelectron wherem andq denotetheelectronmassandcharge,respectively, /9 e e 8 andonepositronwithroughlythesameenergyγ0∼mdm/me, B is the intensity of the magnetic field, and the function R(χ) is 45 2 wddNhγeer±e(γδ()γ=−2γδ(γ)d−enγo0t)e,saDiracdeltafunction.Althoughth(is20is) dRe(fiχn)e≡da2sχ(e2.(cid:2)gK.G4/h3i(sχe)llKin1i/,3G(χu)il−ber53tχ&(cid:18)SKv4e2/n3s(sχo)n−19K8128/)3(χ)(cid:19)(cid:3). 7 by gues arathercoarsea0pproximation,ithastheadvantageofbeingmodel (27) t on independent.Forself-conjugatedarkmatterparticles,weobtain Inthisexpression,KreferstothemodifiedBesselfunction,andthe 08 (cid:2) (cid:3) A Q(r,γ)= ρmdm(r) 2(cid:2)σv(cid:3)e± δ(γ −γ0). (21) normaliνzedfrequency pril 2 dm χ ≡ 3γ2νc (28) 019 Weconsiderasphericallysymmetrichalo,describedbyadensity profileoftheform isexpressedintermsofthecyclotronfrequency ρ q B ρdm(r)= (cid:5)r(cid:6)α(cid:5)1s+ r(cid:6)3−α, (22) νc ≡ 2πemec. (29) r r s s At high photon energies (i.e. gamma rays), we consider the con- where rs and ρs denote a characteristic density and radius of the tributionsofICSandFSR.ForICS(seee.g.Blumenthal&Gould halo,respectively,andαistheinnerlogarithmicslopeofthedensity 1970) (cid:9) profile.Localinhomogeneitiesthatwouldboosttheexpectedsignal, 3σ chν ∞ n(ν ) such as small-scale clumpiness or the presence of subhaloes, are l (γ,ν)= T 0 F((cid:16),q)dν , (30) ICS 4γ2 ν 0 nottakenintoaccount.Theshapeofthedarkmatterdensityprofile 0 0 in the inner regions is far from being a settled question. N-body simulations suggest that, at least in the absence of baryons, the 1Energyradiatedperunitvolumeperunitfrequencyperunittimeperunit profileshouldbequitesteepnearthecentre(α ∼1),inapparent solidangle. MNRAS439,566–587(2014) 570 M.WechakamaandY.Ascasibar wheren(ν )isthephotonnumberdensityoftheISRFbeingscat- Table 3. Characteristic density and radius of 0 tered,whichwerepresentedasthesumofthreegreybodies thedarkmatterdensityprofile(equation22)as (cid:2) a function of its asymptotic logarithmic inner n(ν )= 8πν02 1 slopeα. 0 c3 exp(hν /kT )−1 0 CMB (cid:3) + NSL + NIR , (31) α ρsc2(GeVcm−3) rs(kpc) exp(hν /kT )−1 exp(hν /kT )−1 0 SL 0 IR 0.00 2.346 8.64 and 0.20 1.737 9.56 (cid:2) (cid:3) ((cid:16)q)2(1−q) 0.50 1.042 11.41 F((cid:16),q)≡ 2qlnq+(1+2q)(1−q)+ 0.70 0.702 13.08 (cid:11) (cid:12) 2(1+(cid:16)q) 1.00 0.349 16.67 1 1.20 0.197 20.33 ×(cid:9) q− (cid:9)(1−q), (32) 4γ2 1.25 0.169 21.49 1.50 0.066 29.81 D with(cid:16)≡ 4γhν0,(cid:16)q ≡ hν ,σ theThomsoncross-section,h 1.70 0.025 42.57 ow thePlanckmceoc2nstant,ktγhmeecB2−ohltνzmaTnnconstantandtheproductof 1.90 0.006 70.30 nlo a Heavisidefunctionsensuresthatonlykinematicallyallowedcolli- Einasto 0.17 0.060 20.00 de sioFnosr4Fγ1S2R≤,tqhe≤e1maisrseiotankecnoeifnfitociaecnctofuonrtp.hotonsoffrequencyν is Cirelli & Panci 2009) where the photon intensity is represented d from givenby (cid:2) (cid:3) by three grey-body components (see equations 12 and 31). The http jν(r,ν)= 4hπν ρmdmd(mr) 2 d(cid:2)σdvν(cid:3)FSR, (33) nthoerMmGoaaslltiazicmattiipcoonsrtstaaransntaldyn,dewfdfeeucastltisvaoerienteqvmeusopttieegrdaattieunrteThseaboelffefet2hc.etolfigthhetienmnietrtesdlobpye s://acad e witheachannihilationeventyieldingaphotonspectrumgivenby of dark matter density profile on the production rate of electron– m d(cid:2)σdvν(cid:3)FSR=(cid:2)σv(cid:3)e± πα κ2−2νκ+2ln(cid:20)(cid:11)2mmdm(cid:12)2(1−κ)(cid:21), (34) p1o.5s.itWrohnepnavirasr.yWingeαvawryetahlesoinmnoerdilfoygtahreithcmhaircacsltoerpiestiαcfdreonmsit0y.5antod ic.oup.c e radiusinexpression(22)sothatthedarkmatterdensityatthesolar o m whereαisthefine-structureconstantandκ =hν/m c2 (seee.g. radiusisequalto0.3GeVcm−3andthevirialmassoftheGalaxyis /m Peskin&Schroeder1995). dm 1012M(cid:12).TheappropriatevaluesofρsandrsarequotedinTable3 nra forseveralvaluesoftheinnerlogarithmicslopeα.Inaddition,we s/a alsoconsidertheso-calledEinastoprofile rtic 2.5 Astrophysicalparameters (cid:10)−2(cid:2)(cid:11)r (cid:12)α (cid:3)(cid:15) le-a TheemissioncoefficientassociatedwithFSRisfullyspecifiedby ρ(r)=ρs exp α r −1 , (35) bs theinitialenergyandinjectionrateoftheelectron–positronpairs, s tra related to the nature of the dark matter particle (mass and cross- whereα=0.17. ct/4 3 section) and the parameters describing the density profile of the 9 /1 Galactichalo.Incontrast,thephotonintensityfromthesynchrotron /5 3 OBSERVATIONAL DATA 6 andICSemissionalsodependsontheastrophysicalparametersthat 6 /9 determinethepropagationandenergylossesoftherelativisticpar- In order to constrain the production of relativistic electrons and 8 4 ticles.Wewillfirstdefineacanonicalmodelbasedonobservations positronsintheMilkyWay,weconsiderobservationsofthewhole 52 7 oftheMilkyWayandtheninvestigatetheeffectofeachindividual skyatverydifferentwavelengths.Moreprecisely,theHaslammap b y componentbyvaryingthevaluesoftheadoptedparameters.Inall in the radio band, the five WMAP channels at microwave wave- g u cases,wecalculatetheelectron–positronspectrumasdescribedin lengths and 12energy bins oftheFermiLAT observations inthe e s expression(10),andthenestimatethephotonintensityaccording gamma-rayregime.TheHaslamandWMAPmapsaredominated t o n toexpression(24). bysynchrotronemission,whereasFermitracesICSandFSR. 0 8 Our canonical model assumes a dark matter density profile TheHaslam408MHzradiocontinuumall-skymap(Haslametal. A ρwsicth2 α==0.315(GNeaVvarcrmo,−3F,recnokns&isteWnthiwteith199d7y)n,amrsic=al1m7okdpeclsanodf 1w9e8re1,o1b9ta8i2n)edcofmrobmintehsedaartcahifvroesmoffotuhredNifafteiroennatlsCuervneteyrs.foTrhSeudpaetra- pril 20 1 the Milky Way (e.g. Dehnen & Binney 1998; Klypin, Zhao & computingApplicationsAstronomyDigitalImageLibrary(NCSA 9 Somerville2002).ThevirialmassoftheGalaxyisthus1012M(cid:12), ADIL)inequatorial1950coordinates,andtheyweresubsequently andthelocaldarkmatterdensityisρ (r(cid:12))c2 =0.3GeVcm−3. processedfurtherintheFourierdomaintomitigatebaselinestrip- dm TheISMismainlycomposedofneutralhydrogenatoms(X =0) ing and strong point sources. For the WMAP data, we take the ion with number density ρ /m ∼ 1 cm−3 (Dehnen & Binney 1998; full-resolutionco-addedtemperaturemapsforeachofthefivefre- g p Ferrie`re2001;Robinetal.2003),anditispermeatedbyatangled quency bands (23, 33, 41, 61 and 94GHz) corresponding to the magneticfieldwhoseintensityisB ∼6μGthroughouttheGalaxy 7yrobservations(Jarosiketal.2011).TheFermigamma-raymaps (Beck2001;Ferrie`re2001;Ascasibar&D´ıaz2010). werecomputedbyDobleretal.(2010)fromall‘Class3’(diffuse) Apartfromthecanonicalmodel,weconsidertheeffectthatthe photon events in the first-year data release. We use the 12 loga- magneticfield,thediffusioncoefficientandtheISRFhaveonthe rithmically spaced frequency bands, from 0.3 to 300GeV, of the synchrotronandICSemission.Theintensityofthemagneticfield smoothedmapswithoutpointsourcesubtraction. B is varied from 1 to 100 μG. For the diffusion coefficient (see Sinceweareinterestedinasphericallysymmetriccomponent, equation 2), we consider the three models discussed by Donato wemayfollowasimple,conservativeprocedureinordertomaskthe et al. (2004) and Delahaye et al. (2008), summarized in Table 1. emissionfromtheGalacticdiscandindividualpointsourceswithout WewillalsousethreedifferentmodelsoftheISRF(adoptedfrom relying on any particular foreground model. For each frequency, MNRAS439,566–587(2014) Constraintsondarkmatterannihilation 571 wecomputetheaverageintensityI(θ)in180binsasafunctionof tensitiesdonotexceedtheobservedvalues(redtrianglesinFig.1) theangularseparationθfromtheGalacticCentre.Wealsoestimate atanyangularseparationθ.Notsurprisingly,thetightestconstraint thestandarddeviationσ(θ)withineachbin,aswellastheaverage will always be provided by a small value of θ, i.e. close to the standarddeviation GalacticCentre.Thedarkmatterdensityandthustheinjectionrate (cid:22) σave= ni=1nσ(θi), (36) aserervheidghinetretnhseirtyeathlsaonraenaycwhehsearemealsxeiminutmheaGtθal=ax0y,.aHnodwpeavrteirc,ltehsemoaby- diffusefromtheirinjectionpoint,effectivelysmoothingthedensity where n = 180 is the total number of the bins. We then start an cusp.Thepredictedsurfacebrightnessprofilesofsynchrotron,ICS iterative procedure, where all pixels more than 3σave away from andFSRemission,normalizedaccordingtosuchprescription,are I(θ)arediscardeduntilconvergenceisachieved. plottedinFig.2togetherwiththeobservationaldata,andtheangle This method seems to correctly identify and remove the most thatsetsthemaximumnormalizationthatwouldbecompatiblewith obviousstructuresinallbutthetwohighest-energyFermibands, theobservations(i.e.theupperlimitof(cid:2)σv(cid:3)e±)isplottedinFig.3. wherethephotonstatisticsissopoorthatitisextremelydifficult FSRisproducedattheverymomentofpaircreation,andthus todistinguishdiffuseemissionfromindividualpointsources.For it directly traces the positron injection profile, which is, in turn, Do thesetwobands,weoptedtousetheoriginalaverageintensityI0(θ) proportional to the square of the dark matter density. Therefore, wn withoutapplyinganymask.Rawintensitymaps,maskedresidual theintensityoftheFSRemissiondoesnotdependontheinjection loa maps,i.e.I−I(θ),andtheaverageintensityI(θ)foreachwavelength energyoftheparticlesoranyastrophysicalparameterotherthanthe ded areshowninFigsA1–A3ofAppendixA.NumericvaluesofI0(θ), innerlogarithmicslopeαofthedarkmatterdensityprofile.Forthis fro I(θB)easniddeσs(θth)easreeoqbusoetrevdatiinoTnaalbldeastAa,1w–Ae4a.lso consider the energy rineaFsiogn.,2tdhoennootrmdeapliezneddosnuErf0a.cIenboruigrhctanneosnsicparolfimloedseolf(wFShReredeαp=ict1e)d, m http sbpoeucrthroaoodf;cinospmaircticrauylare,lewcetrounsseatnhde pcoomsitbrionnesdinelethcetrosno+lapronseitirgohn- farnodmevtehnemveorryescoeniftrαe>of1t,hteheGtaiglahxtyest(θco0.5andα>1.25,respectively. y g u Since the particle production rate near the centre of the Milky e s Wayincreasesdramaticallywiththevalueoftheinnerslopeofthe t o n density profile, this is, by far, the most relevant astrophysical pa- 0 rameter.Forα>1.25,across-sectionlargerthan3×10−26cm3s−1 8 A Oisnruthleedcoonuttrfaoryr,ainfythdeadrkarkmmatatettrecradnednisdiatyteplriogfihlteerotfhtahneM∼1il0k0yGWeaVy. pril 2 0 was shallow, with a logarithmic slope significantly below α = 1, 19 Figure5. Upperlimitsonthedarkmatterannihilationcross-sectionfrom the positron spectrum in the solar neighbourhood would provide synchrotronandICS,fordifferentvaluesofthemagneticfield(top),ISRF the most stringent limits on dark matter annihilation, and there- (middle)anddiffusioncoefficient(bottom).ConstraintsfromFSRandthe fore,theconstraintswouldnotdependatallontheactualvalueof localpositronspectrumareindicatedbytheshadedarea. thelogarithmicslope. One may remove the dependence of the results on the precise shape of the dark matter density profile by fixing θ = 10◦ when oneincreasesthevalueofB,synchrotronconstraintsbecomemore comparingmodelpredictionswithobservationaldata.Asshownin importantattheexpenseofICSemission.Inthemostextremecase the bottom panel of Fig. 6, we find, in agreement with previous (B =100μG),gamma-rayconstraintsarenegligible,andtheupper work(e.g.Serpico&Zaharijas2008;Ackermannetal.(TheFermi- limits derived from synchrotron radiation are well approximated LATCollaboration)2012b),thattheuncertaintyassociatedwiththe by a pure power law. For large values of the magnetic field, the precisevalueofαreducestoaboutafactorof2whenthecomparison synchrotron constraints are more stringent than the upper limits isrestrictedtothephotonintensityatθ=10◦.Whilethisistherefore derivedfromthepositronspectrum. agoodchoicewhenthegoalistoprovideaconservativeupperlimit MNRAS439,566–587(2014) Constraintsondarkmatterannihilation 575 D o w n lo a d e d Figure7. Upperlimitsontheinnerlogarithmicslopeofdarkmatterdensity fro m profileα,obtainedbyimposingthatFSR,ICSandsynchrotronemissiondo h notoverproducetheobservedsignal(accordingtotheobservationaldata) ttp forathermaldarkmatterrelic(i.e.(cid:2)σv(cid:3)e± =3×10−26cm3s−1). s://a c a d e m Byassumingagivenvalueofthecross-section,onecanconstrain ic thevalueofα fromthetotalintensityandthemorphologyofthe .o u observedsurfacebrightness. p.c In this work, we will focus only on the total intensity in order om toderivearobustupperlimit.Moredetailedconstraintscouldbe /m n obtainedfromtheshapeofthesurfacebrightnessprofilesatdiffer- ra s entwavelengthsoncetheastrophysicalcontributionisadequately /a subtracted. We set the dark matter annihilation cross-section into rtic electron–positronpairstothevalueexpectedforathermallypro- le-a duced relic, (cid:2)σv(cid:3)e± =3×10−26 cm3 s−1, and compute the value bs ofαforwhichthepredictedemissionrisesabovetheobservedlevel. tra Figure6. Upperlimitsonthedarkmatterannihilationcross-sectionfor c differentvaluesoftheinnerlogarithmicslopeαofthedarkmatterdensity ThecorrespondingupperlimitsareplottedinFig.7asafunction t/43 pprroovfiildee.sOtnhethmeotsotpstprianngeeln,tthliemcito(nsseteraFinitgs.3arfeodrethrievecdanforonmicatlhceasaengαle=th1a)t, coafntdhiedaitnei.tiOaluernreersguyltsE0shaoswsotchiaatte,dfowriaththtehremmalasrseliocfwdaitrhkmmdamtt1.8seemstobeexcludedin constraints(seee.g.Diemand,Moore&Stadel2005;Lavalleetal. anycase. 2008). Thefactthatweareconsideringthetotalradioandgamma-ray emission, without taking into account the contribution of astrophysical origin, implies that these are conservative upper limits, 6 CONSTRAINTS ON THE INNER SLOPE OF and therefore, we can conclude that, if dark matter particles an- THE DENSITY PROFILE nihilateprimarilyintoelectronsandpositrons(or,moregenerally, As pointed out in Ascasibar et al. (2006), the photons from the anyotherparticle;seebelow),anyscenariowheretheMilkyWay centralregionoftheGalaxycontaininformationonboththedark features a steep density profile (due to e.g. adiabatic contraction) matterannihilationcross-sectionandtheshapeofthedensityprofile. maybefirmlyruledout. MNRAS439,566–587(2014)

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