Multimessenger constraints on dark matter annihilation into electron–positron pairs

22 Pages·2014·5.97 MB·English

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,UniversidadAuto´nomadeMadrid,E-28049Madrid,Spain
Accepted2013December20.Received2013December12;inoriginalform2012December11 D
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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;Sa´nchez- /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;tiSoa´nncinhethz-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:[email protected](MW);[email protected](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(θco<nst1r◦a)initnsoanlmthoestFaSlRl ccaosmese, s://ac
a
spectrum measured by the Fermi (Ackermann et al. (The Fermi- yieldinganullstandarddeviationinFig.3formostvaluesofE0. de
LAT Collaboration) 2010b) and HESS (Aharonian et al. (HESS ForsynchrotronandICSemission,particlediffusionmakesthe mic
Collaboration) 2008) collaborations, as well as the positron-only intensityprofileshallower,especiallyathighinjectionenergies.In .o
u
spectrum determined from Fermi (Ackermann et al. (The Fermi- general,onecansaythatphotonsofagivenfrequencytraceelec- p.c
LATCollaboration)2012a)andPAMELA(Adrianietal.(PAMELA tronsandpositronswithinacertainenergyrange.Ifthatrangeis om
Collaboration)2013)data.Thepositronfractionhasalsobeenre- closetoE ,theseparticleswouldhavejustbeeninjected,andthere- /m
centlymeasuredbytheAMS-02Collaboration(Aguilaretal.2013), fore,thee0ffectsofparticlepropagationshouldbesmall,whereas, nra
anditisforeseenthatelectron,positronandcombinedspectrawill awayfromE0,theseelectronsandpositronswouldhavetravelled s/a
beavailableinthenearfuture. a significant distance from the point of injection, and the surface rtic
brightnessprofilewillbecomeconsiderablyshallower. le-a
ThistrendisindeedevidentinFig.2:surfacebrightnessprofiles bs
4 CONSTRAINTS ON THE DARK MATTER become progressively shallower as one moves from E = 1GeV tra
CROSS-SECTION 0 c
to 10TeV, and the effect is more pronounced for those channels t/4
3
OncetheemissionfromtheGalacticdiscandthemostprominent thattracelow-energyparticles,i.e.Haslam,WMAP,andthelowest- 9
/1
pointsourcesisexcluded,theremainingsphericallyaveragedcom- energy Fermi bands. In the most extreme cases, diffusion keeps /5
6
ponent can be used to place upper limits on the cross-section for theelectron–positronspectrum(andtheensuingintensity)roughly 6
/9
darkmatterannihilationintoelectron–positronpairs. 8
4
First of all, model intensities are computed according to the 52
7
scheme described in Section 2. We consider the injection energy b
y
(i.e.themassofthedarkmatterparticle)asafreeparameterand g
investigatevaluesoftheinitialLorentzfactorγ0between2×103 ues
and2×107,correspondingtoinjectionenergiesE0=γ0mec2from t o
n
1GeVto10TeV.Asanexample,Fig.1displaystheresultsofour 0
8
canonical Milky Way model for the synchrotron, inverse Comp- A
tGoanlaacntdicFCSeRntcreo,natrsisbuumtiionngsatodathrkemphaottteornainnnteihnisliattyioantc1r0o◦ssf-rsoemctitohne pril 2
0
of(cid:2)σv(cid:3)e± =3×10−26cm3s−1. 19
One can readily see that the Haslam radio map will be most
sensitivetosynchrotronemissionbyparticleswithaninitialenergy
between1and10GeV,whereasWMAPdatawillcovertherange
E ∼10–100GeV.Ontheotherhand,thegammaraysobservedby
0
theFermiLATwillconstrainthemaximumICSandFSRemission
allowed.TheFSRissharplypeakedattheinjectionenergy,andit
tracesvaluesofE between1GeVand1TeV.TheinverseCompton
0
spectrumisbroader;itfeaturesthreedistinctemissionpeaks,due
Figure1. Theoreticalphotonspectraofsynchrotronradiation(dashedblack
tothescatteringofCMB,starlightandinfraredphotons,anditis
lines),ICS(dottedbluelines)andFSR(solidmagentalines)forourcanon-
bestsuitedtoprobeinjectionenergiesabove∼10GeV. icalmodelwith(cid:2)σv(cid:3)e± =3×10−26cm3s−1anddifferentinjectionener-
Since the value of the annihilation cross-section only sets the gies,evaluatedat10◦ fromtheGalacticCentre.Greybandsillustratethe
normalization of the spectra, and it does not alter its shape, it is frequencyrangesofHaslam,WMAPandFermi.Theobservationaldataat
relativelyeasytosetanupperlimitbyimposingthatthemodelin- θ=10◦areplottedasredtriangles.
MNRAS439,566–587(2014)
572 M.WechakamaandY.Ascasibar
D
o
w
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lo
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d
e
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h
ttp
s
://a
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ic
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tra
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t/4
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/1
/5
6
6
/9
8
4
5
2
7
b
y
Figure2. Surfacebrightnessprofilesofsynchrotron,ICSandFSRasafunctionoftheangularseparationθfromtheGalacticCentre.Redtrianglescorrespond g
u
tothemeanobservationalintensityafterdiscardingthecontributionoftheGalacticdiscandprominentpointsourcesasdiscussedinSection3.Theoretical e
s
profilesarenormalizedtothemaximumvalueoftheannihilationcross-section(seeFig.4)allowedbythesedata.Theangularseparationthatprovidesthe t o
n
tightestconstraint–i.e.thetangentpointbetweenmodelsandobservations–isdepictedinFig.3.ForsynchrotronandICSemission,theintensitiesobtained 0
forthecanonicalMilkyWaymodelareexpressedingreytoblacklines,whereadarkercolourrepresentsahighervalueoftheinjectionenergyE0.The 8 A
normalizedintensityofFSR,shownasagreensolidline,doesnotdependonE0. pril 2
0
constantwithintheinnermost10◦–20◦.Forsynchrotronemission, derived from the combined electron+positron data. For the sake 1
9
thetightestconstraintsontheannihilationcross-sectioncomefrom of comparison, we also show these for PAMELA (Adriani et al.
θ ∼5◦–13◦,whereasforICStheoptimalangleincreasesfrom1◦ (PAMELA Collaboration) 2011), Fermi (Ackermann et al. (The
to12◦(seeFig.3). Fermi-LAT Collaboration) 2010b) and HESS (Aharonian et al.
InadditiontothephotonsarrivingfromthecentreoftheMilky (HESSCollaboration)2008).Notethat,inthelattercase,themea-
Way, the dark matter annihilation cross-section (cid:2)σv(cid:3)e± is also surementsareabletoprobehigher(∼TeV)energies,butitisnot
stronglyconstrainedbytheobservedabundanceofrelativisticelec- possible to discriminate between the electron and positron signa-
trons and positrons in the solar neighbourhood. In particular, we tures.
considertherecentmeasurementsofthepositronspectrumbythe Ourconstraintsarederivedbyimposingthatthepredictedamount
FermiCollaboration(Ackermannetal.(TheFermi-LATCollabora- ofelectronsand/orpositronsdoesnotexceedtheobservedvalues
tion)2012a)andthePAMELAexperiment(Adrianietal.(PAMELA foranyLorentzfactorγ.Giventheenergydependenceoftheob-
Collaboration)2013).Sincethepositronfractionisoftheorderof servedspectrum,[dn] ∼E−3,andtheenergylosses,b(E)∼E2,
dE obs
10percentorlessattheenergiesbelow∼10GeV,theconstraints the most restrictive constraint comes from the spectrum near the
from the positron-only spectrum will be much tighter than those injection energy, where propagation can be safely neglected and
MNRAS439,566–587(2014)
Constraintsondarkmatterannihilation 573
Fermi.Ascanbereadilyseeninthefigure,thetightestconstraints
are provided by FSR and ICS for injection energies above 20–
30GeV,whereasthepositronspectruminthesolarneighbourhood
andsynchrotronemissionlimittheproductioncross-sectionatlower
energies.
Similar (or stronger) constraints can also be obtained from the
analysis of the CMB (e.g. Galli et al. 2009, 2011; Slatyer et al.
2009) and the gamma-ray emission from the Galactic Centre at
∼TeV energies. In particular, a stringent upper limit in this mass
range has been derived by comparing HESS measurements from
suitably defined ‘source’ and ‘background’ regions (Abramowski
etal.(HESSCollaboration)2011b;Abazajian&Harding2012).
AccordingtoFig.4,thetypicalvalueforthermalrelics,(cid:2)σv(cid:3)e± =
3×10−26cms−1,isruledoutforparticlemasseslighterthanafew Do
w
GeV.Bothparticlephysicsprocessesandastrophysicalboostfactors n
lo
havepreviouslybeenadvocatedtoincreasethecurrentannihilation a
d
rateintheMilkyWaybymorethanafactorof10withrespectto ed
Figure 3. Angular separation θ that provides the upper limits for syn- theearlyUniverse.Suchmodels wouldbeexcluded foranydark fro
cpr(cid:2)eθheg(cid:3)rni2odoteanroncsrntslo,yhsosIfCwodrSitfehfaaeencraehdvneotFrfcaShtgRhaeen(cid:2)enoθmeb(cid:3)liss=se,srri1veo/esnNdp.eλwTc(cid:22)tahivveieeθllo(eyλpn.tigi)mtahnasdl.Ssvtoaallniuddealriinsdedcseoavmniadptuisothenad(cid:2)dθoin2w(cid:3)de−ed- misneatrovttaeetlrievcceatrtnordenai–dtpmaoteesnibttreoolfnotwphaetiahrsse.trS∼oipnhTcyeesVoicuarrelagsniimgalnyeaslai,snmnineivhreoilllayvteiensxgcalvpuerdirimyngacrotihnlye- m https://a
emissionfromthediscandprominentpointsources,itisexpected c
a
thatadeeperunderstandingoftheastrophysicalsourcesofelectrons de
m
andpositronswouldmakepossibletoprobetheinterestingregion ic
oftheparameterspacebelow(cid:2)σv(cid:3)e± =3×10−26cms−1. .ou
p
.c
o
m
5 EFFECT OF THE ASTROPHYSICAL
/m
PARAMETERS n
ra
s
AlltheconstraintsrepresentedinFig.4arebasedonthe‘canoni- /a
cal’MilkyWaymodeldiscussedinSection2.5.TheFSRfromthe rtic
GalacticCentreandthelocalpositronspectrumdirectlytracethe le-a
instantaneousinjectionrate,andtherefore,theydonotdependon bs
the propagation parameters. However, the surface brightness pro- tra
c
filesofsynchrotronandICSemissionaresensitivetotheprecise t/4
3
valuesadoptedfortheintensityofthemagneticfield,thediffusion 9
/1
coefficientandtheISRF.Theinnerlogarithmicslopeofdarkmatter /5
6
densityprofilehasaverystrongimpactontheinjectionrateclose 6
/9
tothecentre,andthus,itaffectsallthetracersconsideredinthis 8
4
Figure4. Upperlimitsonthedarkmatterannihilationcross-sectionderived workexceptthepositronspectruminthesolarneighbourhood. 52
bycomparingthepredictedsynchrotron(reddashedline),ICS(bluedotted Here, we investigate the effect of the various astrophysical pa- 7 b
line)andFSR(blacksolidline)emissionwithmultiwavelengthobserva- rametersofourpropagationmodelontheupperlimitsobtainedfor y g
tionaldata.Theareasshadedinblueshowtheconstraintsobtainedfrom thedarkmatterannihilationcross-section.Aswedidforthecanon- ue
ntheeigmhbeaosuurrheomodenbtsyoPfAthMeEcLoAm,bFineermdielaencdtroHnE+SpSo.sTithroenuspppeecrtrluimmitastothbetasionleadr icalmodel,weconsiderdifferentinitialenergiesE0from1GeVto st on
fromthepositronspectrumareshownbytheredareas.Thehorizontaldotted 10TeVandcomparethepredictedemissionwiththefullobserva- 08
l[idndEne]imndodicelat≈estQhb0ev∝aluEe−(cid:2)σ2.vT(cid:3)eh±e=m3ax×im10u−m26pcrmodsu−c1.tionrateallowedby tiinoLtnuaerltnduiasntasotsraedrtet,rbwtuoitthnasotswheeswsinethtveeanirrsyiitneyaflcouhfenothcfeethomenaagtshtnreeotrpiechsyufisleitcsld.alBp.arTahmisetpears- April 20
thedatacanthenbeexpressedas rameterplaysanimportantroleintheenergylosses,anditsetsthe 19
(cid:2) (cid:3)
dn totalamountofenergythatisradiatedawayassynchrotronemis-
Q (r(cid:12))<b(γ ) (γ ), (37) sion. The top panel on Fig. 5 shows the upper limits derived by
0 0 dE 0
obs combiningtheconstraintsobtainedfromsynchrotronandICS.The
andonearrivestothecondition resultsobtainedforB=1,6(ourcanonicalmodel),30and100μG
(cid:2) (cid:3) (cid:2) (cid:3)
m 2 dn areplottedasdotted,solid,dash–dottedanddashedlines,respec-
(cid:2)σv(cid:3)e±(γ0)< ρ (drm(cid:12)) b(γ0) dE (γ0) (38) tively.Alltheotherconstraints(FSRandlocalpositronspectrum)
dm obs areindependentofB,andareshownbytheshadedarea.
inordernottooverproducetheobservedsignal. Synchrotronconstraintsaremostimportantatthelowestinjection
The results are plotted in Fig. 4, together with the upper lim- energies(E ∼1–30GeV),whiletheupperlimitsathigherinitial
0
its on the dark matter annihilation cross-section derived from the energies(from∼30GeVto10TeV)areduetoICSinthegamma-
comparisonofthepredictedsynchrotron,ICSandFSRemission, rayregime.Theintensityofthemagneticfieldaffectsbothprocesses
assuming our canonical Milky Way model for particle propaga- inanoppositeway:forlowvaluesofthemagneticfield,allenergy
tion, with multiwavelength observations by Haslam, WMAP and is lost by ICS, and synchrotron emission is almost irrelevant; as
MNRAS439,566–587(2014)
574 M.WechakamaandY.Ascasibar
InthemiddlepanelofFig.5,weinvestigatetheupperlimitsof
synchrotronandICSfordifferentmodelsoftheISRF.Asmentioned
inSection2.5,weadoptedtheparametrizationproposedbyCirelli
&Panci(2009)intermsofthreeblackbodycomponents.Thetem-
peratures and normalizations of each component are summarized
inTable2.TheeffectoftheISRFissimilartothatofthemagnetic
field,butintheoppositedirection:ahigherphotondensityresultsin
alargeramountofenergybeinglostbyICSratherthansynchrotron
emission.Nevertheless,forreasonablevaluesofthemodelparam-
eters,theupperlimitson(cid:2)σv(cid:3)donotvarybymorethanafactor
of3.
AsshownonthebottompanelofFig.5,theeffectofthediffusion
coefficientisevensmaller.Theupperlimitsareslightlymorestrin-
D
gentwhentheelectronsandpositronsareallowedtotravelashorter o
w
distancefromtheplacewheretheywereinjected,butthedifference n
lo
between the three propagation models isbarely noticeable. Thus, a
d
we conclude that our results are not severely affected by the as- ed
trophysicaluncertaintiesassociatedwithparticlepropagation.An fro
m
additionalsourceofuncertaintywouldberelatedtoourchoiceof h
sphericalboundaryconditions.Althoughwehavenotinvestigated ttp
s
thisissueindetail,comparisonwithotherstudiesbasedoncylin- ://a
dricalboundaryconditions(e.g.Ackermannetal.(TheFermi-LAT c
a
Collaboration)2012b;Fornengoetal.2012;Mambrinietal.2012) de
m
suggestthattheeffectofthischoiceontheannihilationcross-section ic
isrelativelyminor(seeAppendixB). .o
u
Incontrast,theexactvalueoftheinnerslopeαofthedarkmatter p.c
density profile plays a very important role in setting the actual om
constraints on (cid:2)σv(cid:3). We have investigated several values in the /m
interval0<α<2(theappropriatevaluesofρs andrs arequoted nra
s
in Table 3) as well as the Einasto profile given by equation (35). /a
WereportinFig.6theupperlimitsobtainedfromthecomparison rtic
ofthepredictedFSR,synchrotronandICSemissionforα=0.50, le-a
1.00, 1.25 and 1.50 with our multiwavelength observational data bs
set.ResultsforFSRandthecombinationofsynchrotronandICS tra
c
emissionareplottedseparately.Constraintsfromthelocalpositron t/4
spectrumareindependentofαandareshownasasolidarea. 39
/1
The top panel of the figure shows the upper limits obtained /5
6
bythesameprocedureappliedtothecanonicalmodel,i.e.choos- 6
ing the angular separation θ that provides the tightest constraint. /98
4
Notsurprisingly,largervaluesofαresultinlowervaluesofθ.The 52
constraints from FSR and synchrotron+ICS emission come from 7 b
innermost1◦forα>0.5andα>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,dfowriaththtehremmalasrseliocfwdaitrhkmmdamtt<er 9/1/5
6
whereasalltheconstraintsonthebottompanelareobtainedfromtheob- 100GeV,thedarkmatterdensityprofileoftheMilkyWaymustbe 6
servedemissionatθ =10◦fromtheGalacticCentre.Inbothcases,black shallowerthanα∼1.3inordernottooverproducetheobservedsig- /98
andredlinesrepresentthelimitsassociatedwithFSRandsynchrotron+ICS nal.Itisworthnotingthat,sinceFSRonlydependsontheinjection 452
e(imndisespieonnd,ernetspoencαtiv)ealrye.sThhoewncobnysttrhaeinsthsadfroowmedthaerelaosc.al positron spectrum rate,thisconstraintontheinnerlogarithmicslopeαisindependent 7 by
ontheotherastrophysicalparameters.SynchrotronandICSyield g
u
strongerlimitsthanFSRatlowandhighinjectionenergies,respec- e
s
on the dark matter annihilation cross-section, we would like to tively,althoughofcoursetheseresultsdependmuchmoreonthe t o
stressthatanypriorknowledgeofthedarkmatterdensityprofile detailsoftheadoptedpropagationmodel(mostnotably,theinten- n 0
mayleadtomuchstrongerconstraintsiftheinnerslopewassteeper sity of the magnetic field). For our canonical set-up, synchrotron 8 A
thaFninαal=ly,1,leatsuevsidneontecetdhaint tthheeulpopcaelrpdaanrkel.matter density is sub- roafddiaatrikonmiamttpeorsmesasesxetrse,mareolyuntdighatfceownGsteraVin(tsobfsoerrvaatliiomniatelddaratangaet pril 2
0
ject to relatively large uncertainties (cf. Dehnen & Binney 1998; lower frequencies would probably make possible to extend these 19
Klypin et al. 2002; Salucci et al. 2010; Iocco et al. 2011), which constraintstowardslowermasses).Inparticular,thestandardcase
translatetriviallytotheupperlimitsonthecross-section.Inaddi- α =1wouldbeexcludedforE <5GeV.Athighenergies,ICS
0
tion,departuresfromsphericalsymmetry(includingthepresence emission rules out slopes steeper than α = 1.5 for dark matter
ofsubstructures)willalsohaveasignificanteffectonthederived massesbelow∼2TeV.Theregimeα>1.8seemstobeexcludedin
constraints(seee.g.Diemand,Moore&Stadel2005;Lavalleetal. anycase.
2008). Thefactthatweareconsideringthetotalradioandgamma-ray
emission, without taking into account the contribution of astro-
physical 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)