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Dark Matter Spikes and Annihilation Radiation from the
Galactic Center
The annihilation rate of weakly interacting cold dark matter particles at the galactic
center could be greatly enhanced by the growth of a density spike around the central
supermassive black hole (SBH). Here we discuss the effects of hierarchical mergers
o
DarkMatterSpikesandAnnihilationRadiationfromtheGalacticCenter
DavidMerritt,1,?MilosMilosavljevi?c,1,?LiciaVerde,1,2,?andRaulJimenez1,§
1
Dept.ofPhysics&Astronomy,RutgersUniversity,NewBrunswick,NJ
2
Dept.ofAstrophysicalSciences,PrincetonUniversity,Princeton,NJ
Theannihilationrateofweaklyinteractingcolddarkmatterparticlesatthegalacticcenter
couldbegreatlyenhancedbythegrowthofadensityspikearoundthecentralsupermassiveblac
khole(SBH).Herewediscussthee?ectsofhierarchicalmergersonthecentralspike.Mergersb
etweenhaloscontainingSBHsleadtotheformationofSBHbinarieswhichtransferenergytothe
darkmatterparticles,loweringtheirdensity.Thepredicted?uxofannihiliationphotonsfr
omthegalacticcenterisseveralordersofmagnitudesmallerthaninmodelsthatignorethee?e
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ctsofSBHsandmergers.Measurementoftheannihilationradiationcouldinprinciplebeusedt
oconstrainthemergerhistoryofthegalaxy.
2
002Thedarkmatterpuzzleisoneofthecentralchallenges
facingparticlephysicsandcosmology[1].Apopularcan-yadidatefornon-baryoniccolddark
matter(CDM)istheMlightestsupersymmetricparticle,plausiblytheneutralinoχ[2,3].Th
emassoftheneutralinoisconstrainedbyac9celeratorsearchesandtheoreticalconsiderationsofthermalfreeze-outtolieintherange30GeV
10TeV[4,5].Neutralinosaregenericallyfoundtodecoupleatv6atemperaturethatisroughly
Mχ/20,whichmeansthat7theyarenonrelativisticalreadyatdecouplingandhence3behaveli
keCDM.
1Darkmatterparticlesmaybedetecteddirectly,via0laboratoryexperiments[6],orindirec
tly,viatheiran-20nihilationproducts[7].Indirectschemesaretypically/basedonsearch
esforgammaraysfromneutralinoan-hnihilationsinthedarkmatterhalooftheMilkyWayp-(MW
)galaxy[8,9,Sincethephoton?uxdependsoonthesquareddensityofneutralinosintegrateda
longrtthelineofsight,thesignalisgreatlyenhancedindirec-sationswherethedarkmatter
isclumped.Thisincludes:thegalacticcenter,wherethedensityinasmoothhalovwouldbemax
imum,aswellasanylinesofsightinter-iXsectingthecentersofrelichalosthatorbitassubc
lumpsrintheMWhalo[11,12].Thesignalfromthegalacticacenterisfurtherenhancedifthere
isaCDM“spike”as-sociatedwiththecentralsupermassiveblackhole(SBH).Adiabaticgrow
thofaSBHatthecenterofapre-existinghaloproducesapower-lawdistributionofmatterarou
nd
theSBH,a“spike,”withdensityρ?r?γ,2.2
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[13];M???istheSBHmassandσisthe1Dvelocitydispersionofthedarkmatterparticles.Fort
heMWSBH,M??2?5×106M⊙[14,15,16]andrh?1pc.Thesespikesleadtopredictionsofhigher-t
han-observedratesofγ-rayannihilationOneelementmissingfromearlierdiscussionsofda
rkmatterspikesisthedestructivee?ectofhierarchicalmergers.Adarkmatterhaloasmassiv
easthatoftheMW,M?1012M⊙,hasalmostcertainlyexperiencedasigni?cantmergereventsinc
earedshiftofz?2.Fur-thermore,SBHswithmassesof?109M⊙werepresentinatleastsomehalo
salreadyatredshiftsof5?6andSBHsprobablyacquiredmostoftheirmassbyared-shiftof2?3,
theepochofpeakquasarIntheCDMparadigm,bighalosgrowthroughthebuildupofsmalleronesi
fmorethanoneoftheprogenitorhaloscarriedacentralSBH,abinarySBHwillformfol-lowingt
hemergerFormationanddecayoftheSBHbinarytransfersenergytothebackgroundparticlesan
dlowersthedensityofmatterinaregionwithinafew×rharoundthebinaryroughlythescaleof
thespike,beforetheSBHscoalesceduetoemissionofgravitationalradiationInthispaperwe
computethedarkmatterdensitypro?lesofmergingCDMhaloscontainingSBHsandshowthatthen
etresultofincludingtheSBHsistosubstantiallylowerthepredicted?uxofannhilationphotonscomparedwithhaloslackingSBHs.Ourresultsarerelevantalsotomodelsthatrelatetheo
riginofultrahighenergycosmicraystothedecayofunstablesuperheavyrelicparticlesinth
egalactichalo[26].
We?rstcalculatedtheprobabilitythattheMWhalohasexperiencedamajormergereventbygene
ratingmultiplerealizationsofthemergerhistoryofahaloof1012M⊙usingthealgorithmdes
cribedbySomervilleandKolatt[27],asappliedtothestandardΛCDMcosmolog-icalmodel(?0
=0.3,Λ0=0.7).Thisalgorithmaccu-ratelyreproducesthemergerhistoriesofhalosseeninN
-bodysimulationsofstructureformation[28]althoughatavalueofthescalefactorthatmayd
i?erasmuchas20%fromthetruevalueWerecordedallmergereventssuchthatthemassofthesmal
lerofthesubclumpsin-volvedinthemergerwasaboveagivenlimitMlim.From600ofthesereali
zations,wethencomputedtheproba-bilitythatahaloof1012M⊙atredshiftz=0hadinitsmerg
erhistoryatleastonesucheventsincearedshiftofz=2.Fig.1showsthisprobabilityasafunc
tionofMlim.With68%con?dence,ahalothesizeoftheMWhasexperiencedatleastonemergersin
cez=2inwhichthemassofthesmallerofthemerginghaloswasabove2×1011M⊙,implyingamass
ratiobetweensubclumpsof4:1orless.Mergerswithmassratiosof9:1orlessoccurredwithgre
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aterthan90%probability.
Toinvestigatethedetailede?ectsofmergersondarkmatterspikes,weusedanN-bodycodetosi
mulatein-teractionsbetweenCDMhaloscontainingcentralSBHs.
TABLEI:ParametersoftheN-bodyintegrations.M1(M2)isthemassofthelarge(small)halo;ρ
isthecentralhalodensitybeforegrowthoftheSBH;M?isthemassoftheSBHparticle;Nistheco
mbinednumberofDMparticlesinbothhalos;a?nalisthe?nalseparationoftheBHbinary.RunM1
/M2
ρ
1/ρ2M?/MNa?nal(pc)A110.011050.91B110.032×1050.71C310.034×1040.31D31/30.034×1
040.42E510.036×1040.36F
10
1
0.03
1.1×105
0.90
N-bodymodelsweregeneratedfromthesphericalden-sitylaw
ρDM(x)=ρ0x?1.5(1 x1.5)?1,
x≡r/rDM,
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(1)
the“Moorepro?le”
[30],oneofasetofpro?lesfoundtoaccuratelydescribethehalosgeneratedinCDMstruc-ture
formationsimulations;rDMde?nestheradiusoftransitionbetweentheinnercuspandthestee
perouterfallo?.OurchoiceoftheMoorepro?leisconservativeinthesensethatotherpropose
d?ttingfunctions[31,32]haveshallowercentraldensitycuspsimplyinglesspromi-nentspi
kes.TheMoorepro?lehasadivergentmass;wetruncateditspatiallyandgeneratedvelocities
ofthedarkmatterparticlesfromadistributionfunction[33]whichgeneratesanisotropicve
locitydistributionnearthecenterandincreasinglycircularorbitsapproachingthetrunca
tionradius.Thisprocedureguaranteesastateofdetaileddynamicalequilibriuminspiteoft
hemodel’shardedge.Sincethedependenceofthecentraldensi-tiesofCDMhalosonhalomassi
snotwelldetermined,weinvestigatedtwovaluesofthecentraldensityratioρ1/ρ2=1,1/3f
oroneoftheunequal-massmergers,whereρisde?nedastheasymptoticcentraldensityoftheh
alosbeforegrowthoftheSBH.A“blackhole”wasinsertedintoeachN-bodyhalobyslowlyincr
easingthemassofa(Newtonian)centralpointparticlefrom0toM?(seeRef.[34]).Weveri?edt
hattheresultingden-sitypro?lesatis?edρDM?r?2.4,r
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Themassscalingofourmodelswas?xedbythemea-suredmassoftheMWSBH[14,15,16];weassumed
2
FIG.1:12ProbabilitythatthemergertreeofaCDMhaloofmass10M⊙atz=0containsatleastone
mergereventbetweenz=0andz=2inwhichthemassM2ofthesmallersubclumpexceededMlim.
M?=3×106M⊙.ThelengthscalewasdeterminedbyassumingthatthedarkmatterintheMWproduc
esacircularrotationvelocityof90kms?1atthesolarcircle,R⊙=8kpc[37]andthatitsdensi
typro?leisρDM∝r?1.5inwardofR⊙.
Simulatedmergersofthehalomodelswerecarriedoutusinganew,general-purposeN-bodyalgo
rithm[38]thatcombinestheelementsofahashedtreecodewithaquadrupolarexpansionofforc
emomentsforthebulkdynamics,andthehighlyaccurateHermitepredictor-correctorschemef
ornear-neighborandmassive-particleinteractions.Thecodeimplementsindividualblockt
imesteps,individualsteppingandsofteningcriteria,andfullfunctionalparallelization
.InteractionsbetweenSBHanddark-matterparticleswereunsoftened.Calculationswerecar
riedoutusing16processorsontheRutgersSunHPC-10000computer.Integrationsweretermina
tedwhentheseparationbetweentheSBHparticleswaslessthan1pc(seeTable1).
Fig.2showsthee?ectofmergersonthedark-matterdensitypro?les.Thesteep,ρ?r?2.4densi
tyspikesaredestroyedineachcasebythetransferofkineticenergyfromthebinarySBHtothed
arkmatterparticles.(Merg-erswithoutcentralSBHstendtopreservespikeslopes[24,39].)
Theenergytransfertakesplaceintwostages[24]:beforethetwoSBHsformabinarysystem,dyn
am-icalfrictionactingontheindividualSBHscausestheirorbitstodecay;andafterformati
onofaboundpair,darkmatterparticleswhichpassthebinarywithinafewtimesthebinary’ss
emi-majoraxisareejectedbythegravita-
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FIG.2:Densitypro?lesofthemergeddarkmatterhalos.Theoriginisde?nedasthecenterofmas
softhebinarySBH;sphericalsymmetryisassumed.ThecurveslabelledMandSarethedensitypr
o?lesofthelargehalobeforeandaftergrowthoftheSBH.
tionalslingshot.Theresultisalowereddensityouttoa“core”radiusrcof?10?100pcandad
ensitypro?lethatrisesinwardofrcasaweakpower-law,ρ?r?0.5.Theamountofdamagedoneto
thepre-existingspikein-creaseswiththemassofthesecondarySBH,consistentwiththeexpe
ctationthatthemassejectedbythebinaryisoforderafewtimesM2[40].Finalmeandark-matte
rdensitieswithin100pcare5?10M⊙pc?3foralloftheruns,risinginwardtoρ?102M⊙pc?3fo
rthe1:1mergersand?103M⊙pc?3forthe10:1merger.
Thedi?erentialphoton?uxalongadirectionthatmakesanangleψwithrespecttothegalactic
centeris
dΦγ
4πMχ2??
ρ2dl,(2)ψwhere?isthesolidangle,ρistheneutralinodensity,and
σ
vistheannihilationcrosssection(independentofvfornonrelativisticparticles);thesum
isoverallannihilationchannels.Nγ=2forχχ→γγandNγ=1forχχ→Zγ[10].Wearepr
incipallyconcernedwiththe?nalline-of-sightintegral,J(ψ)=theinformationaboutthe?
?haloψρ2
dl,whichcontainsallofdensitypro?le.Followingearlierauthors[7,10],wewriteJindimen
sionlessformas
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J(ψ)=1
0.3GeV/cm3
??2??
ρ2dl(3)ψwherethenormalizingfactorsforlengthanddensityare
roughlytheradiusofthesolarcircleandthelocaldensityofdarkmatterrespectively.Final
lyweaveragethe
?ux
3
FIG.3:Dimensionlessintegrated?uxofthemergermodelsasafunctionoftheangularacceptan
ceofthedetector??(Eq.4).ThecurvelabelledMisthe?uxpredictedbyaMoore-pro?lehalowit
houtaSBH-inducedspike(Fig.2).
overthe?eldofviewassumingacircularapertureofsize??centeredatψ=0:
??J??≡
1
4
emissionofgravitationalradiation[25].Theradiationsoemittedcarrieslinearmomentuml
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eadingtoarecoiloftheSBHatavelocityvrecoil?102?103km/s[43,44,45],andpossiblyevenh
igheriftheSBHswererapidlyspin-ningpriortocoalescence[46].Recoilvelocitiesofthiso
r-derwouldejectthecoalescedSBHfromthenucleusanditssubsequentinfallwoulddisplaced
arkmatterparticles(e.g.[47,48]).Quantitativeevaluationofthise?ectwillrequiremore
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4.WethanktheCenterforAdvancedInformationProcessingatRutgersUniver-sityfortheirge
nerousallocationofCPUtime.
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