Spherical Gravitational Collapse of Annihilating D

arXiv:astro-ph/0505497v125May2005SphericalGravitationalCollapseofAnnihilatingDarkMatterandtheMinimumMassofCDMBlackHolesR.AliVanderveldandIraWassermanCenterforRadiophysicsandSpaceResearch,CornellUniversity,IthacaNY14853Sphericalgravitationalcollapseofacoldgasofannihilatingparticlesinvolvesacompetitionbetweentwodensitydependentrates:thefree-fallrate∝√ρandthe(s-wave)annihilationrate∝ρ.Thus,thereisacriticaldensityρannabovewhichannihilationproceedsfasterthanfreefall.Gravitationalcollapseofacloudof(initial)massMtoablackholeisonlypossibleifρBH≡3M/4π(2GM)3=3/32πG3M2∼ρann,whichimpliesthatM∼Mann≡(3/32πG3ρann)1/2.Wecomputethecollapseandannihilationofcold,sphericalgascloudswitheitherahomogeneousdensityprofileoraspecificradially-dependentprofilethatwouldleadtoself-similarcollapseiftherewerenoannihilation.Thesecalculationsverifythatthereexistsalowerboundtotherangeofpossibleblackholemassesthatcanformincolddarkmatter(CDM)collapse.Foraparticlemassmandfreeze-outtemperatureTf=m/xf,theminimumblackholemassisMann≈1010M⊙×xf√g⋆/100ΩCDMh2g⋆Sm(Gev)
,whereg⋆Sandg⋆aredegeneracyfactors.Moreover,theformationofablackholeisaccompaniedbytheannihilationofaboutMannoftheoriginalmass.Mostoftheannihilatedmassisreleasedinaburstlastingatime∼GMBH,whereMBHistheblackholemasswhencollapsefirstoccurs.Atotalannihilationluminosity∼1059ergs−1couldeasilybereached,lastingatime∼100seconds.Theabsenceofastronomicalobservationsofsuchspectaculareventssuggestseither:(i)thebranchingratioforCDMannihilationtoe+e−pairsorquarks∼10−10,whilethebranchingratiotoννis∼10−5;or(ii)CDMisnotmadeofannihilatingparticles,butmaybeinsomenon-annihilatingform,suchasaxions;or(iii)CDMblackholesneverform.PACSnumbers:95.35.+d,97.60.LfI.INTRODUCTIONNon-baryonicmatterdominatesthemattercontentoftheUniversetoday,withadensityparameterΩCDMh2≈0.15;therestoftheenergydensityisprimarilyintheformofdarkenergy[1,2].Inthefavoredcolddarkmatter(CDM)scenariobasedonweaklyinteractingmassiveparticles(WIMPS),thedarkmatterisintheformofagasofparticlesthatisverycoldtoday,withanabundancethatwasloweredsubstantiallybyannihilationwhenthetemperatureoftheUniversewas∼m/xf,foraCDMparticlemassmfreezingoutattemperatureTf=m/xf∼0.1m[3,4,5].TheΛCDMpictureprovidesacceptablefitstothetemperatureandE-polarizationfluctuationsofthecosmicmicrowavebackground(CMB)radiation[6,7],aswellastothelargescalestructureoftheUniverse[8].Onsmallerscales,CDMhalosseemtoformcuspycores[9,10];darkmatterannihilationhasbeenproposedasonemeansofsmoothingoutthecusps[11,12].CDMclumpsarefoundtoforminlargescalestructuresimulationsandtoagglomeratetoformhalos[13];annihilationwithintheseclumpswouldleadtoanenhancedfluxofcosmicrayproductsatEarth[14,15].Inthispaper,weinvestigatearelatedquestion:HowdoesannihilationaltersphericallysymmetriccollapseofCDM?Weconsider,forthemostpart,thecollapseofCDMcloudswithzeroorverysmallinternalenergy.Intheabsenceofannihilation,theinevitableresultwouldbecompletegravitationalcollapsetoablackhole.However,toformablackhole,acollapsingsphereofmassMmustreachamassdensityoforderρBH=3M/4π(2GM)3=3/32πG3M2.When(s-wave)annihilationisincluded,therateofCDMannihilationataden-sityρisΓann=ρhσviannm,(1)wherehσviannistheannihilationratecoefficientandmistheCDMparticlemass.Forcomparison,thefreefallrateathighdensitiesisapproximatelyH=r8πGρ3;(2)thenΓann=Hatadensityρann=8πGm23(hσviann)2.(3)2Forρ∼ρann,freefallproceedsfasterthanannihilation,andvice-versa.Therefore,blackholeformationisonlypossibleforamassMprovidedthatρBH∼ρannorM∼3hσviann16πG2m≡Mann.(4)TheannihilationofCDMparticlesintheearlyUniverseleadstoaresidualdensityparameterΩCDMh2providedthathσviann≈8.6×10−9GeV−2xf100ΩCDMh2g⋆S/√g⋆,(5)whereg⋆Sandg⋆aredegeneracyfactorsfortheentropyandmassdensitywhenthetemperatureoftheUniverseis∼Tf,andxf≈lnC−12ln(lnC),(6)whereC=0.038(g/√g⋆)mhσviann/√G,withgbeingthedegeneracyfactorfortheCDMparticlesalone[16].Substi-tutingforhσvianninequation(4),wefindMann≈1010M⊙m(GeV)xf100ΩCDMh2g⋆S/√g⋆≈1010M⊙m(GeV)xf15g⋆S/√g⋆,(7)wherem=m(GeV)GeV;furthermore,inourlastresult,weassumethatΩCDMh2≈0.15,ashasbeendeterminedfromanalysesoftheCMBandoflargescalestructure.OurestimateofManndependsontheCDMparticlemassinseveralways,sinceMann∝xf√g⋆/mg⋆S.Inadditiontotheexplicitinverseproportionalitywithm,thereisalogarithmicdependencefromxfandapiecewiseflatdependenceviag⋆Sandg⋆:forexample,ifm=50GeV,wewouldhaveenoughparticlespeciesforg⋆≈g⋆S≈50,andMann≈4×107M⊙(takingxf≈20forthiscase).Plausibly,thevalueoftheminimumblackholemassforCDMiscomparabletothemassrangededucedforblackholesinthecentersofgalaxies[17].Itisremarkable(althoughperhapsfortuitous)thatthevalueofMannisevenremotelyclosetoanastronomicallysignificantvalue.Sphericalcollapseis,ofcourse,highlyidealized.Morerealistically,wewouldexpectacollapsingcloudtohavesubstantialsubstructureaswellaslargescaleshearflowsinducedbythetidalfieldofnearbyfluctuations.However,wewouldstillexpectthatCDMcloudsthatare,howeverimprobably,nearlysphericalandsmoothwouldhavethebestchancetocollapsetoblackholes.Thus,ourlowerboundontheCDMblackholemassoughttoberobust.Moreover,shearenhancesthe