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arXiv:astro-ph/0307524v226Mar2005InternationalJournalofModernPhysicsATheinfluenceofdarkmatterhaloontoevolutionofsupermassiveblackholeM.I.Zelnikov∗E.A.Vasiliev†I.E.TammTheoreticalDepartmentLebedevPhysicalInstituteLeninskypr.53,Moscow,RussiaTheinfluenceofdarkmatter(DM)onthegrowthofsupermassiveblackholes(SMBHs)isstudied.ItisshownthatgravitationalscatteringofDMparticlesonbulgestarsleadstodiffusionofDMinphasespace{m,mz,I}(mdenotestheangularmo-mentumandIistheradialaction).Appropriatediffusioncoefficientsarecalculatedfordifferentbulgemodels,anditisarguedthatthediffusionalongmaxisisthemostimpor-tanteffect.ItisshownthatthisprocessleadstonoticeableflowofDMintotheblackhole(BH),resultinginitspower-lawgrowth:Mbh∝t9/16.Comparisonwithobservationaldatashowsthat,inprinciple,thiseffectmayexplainobservedmassesofSMBHs.SpecialattentionispaidtothecorrectionsrelatedtotheinnermostregionofBHgravitationalinfluenceandthediffusionalongIaxis.TheirinfluenceontheBHgrowthlawisshowntobenegligible.1.IntroductionTheinteractionofasupermassiveblackholeinagalaxycenterwithdarkmatterhaloisalreadymuchinvestigated1,2,3.Thereanadiabaticinvariantapproachisappliedfordifferenthalostructure.Theinitialhaloprofilesaretakentobeself-similar(power-law)ρ∼r−α(0α2),isothermal(ρ∼r−2)andNFWprofiles4(ρ∼δcr/rs(1+r/rs)2).InRef.1themethodofadiabaticinvariantisusedtocalculatesmallchangesinorbitalparametersofparticlescausedbyslowvariationofgravitationalpotentialduetotheblackholegrowth.Thisapproachhastwodrawbacks:absorptionbyBHisnottakenintoaccount,andappropriatechangeofDMdistributionfunctionisneglected.Thelattermeansthatthelossconeisalwaysfull,thusleadingtooverestimationofdarkmatterflow.MorecorrectapproachisusedinRefs.5,6.TheretheevolutionofDMdistri-butionfunctionduetoabsorptionbyBHandchangesinBHmassandlossconeparametersareconsideredconsistently.Theauthorscometotheconclusionthatforcurrentvaluesofblackholemassesthefractionofdarkmatterinsidethemisrathersmall.∗zelnikov@lpi.ru†eugvas@lpi.ru1InternationalJournalofModernPhysicsAThisapproachwasdevelopedinRef.7,wherethechangeofdarkmatterdistri-butionfunctionoutsidethelossconeduetodiffusioninphasespacewastakenintoaccount.Itwasdemonstratedthatthisdiffusioncausedbygravitationalscatter-ingofDMparticlesonstarseffectivelyrefillsthelossconeanddeterminestheBHgrowthlaw.Thiseffectwasshowntogivereasonableestimateforobservedblackholesmasses.Thepresentworkfurtherdevelopsthementionedapproachbyaccountingforthreeotherfactors:changeofgravitationalpotentialinthevicinityofblackhole,modificationofstardistributioninthisregion,andtheconditionsunderwhichthediffusionbecomeseffectivelytwo-dimensional.Thepaperisorganizedasfollows.Inthefirstsectionweintroduceourmodelofdarkmatterhaloandthegalacticbulge.Inthesecondsectionwewritedownthekineticequationwhichdescribestheevolutionofdarkmatterhaloinphase-space,andcalculateseveralquantitiesrelatedtostardistributionanddarkmattermotion.Calculationofthediffusioncoefficientsisperformedinthethirdsection.Alsoestimatedarethediffusiontimescalesandcorrectionsrelatedtothecentralregion.Inthefourthsectionthesolutionofthediffusionequationinone-dimensionalformispresented,theblackholegrowthlawisestablished,andthecorrectionstotheDMflowduetotwo-dimensionaldiffusionandthecentralregionareinvestigated.Itisshownthatthesecorrectionsinmostcasesplaynosignificantroleinthediffusionprocess.Finally,inthesection5wepresentsomenumericalresultsanddiscussthem.1.1.DarkmatteringalaxiesWhilethemostpartofmatterintheUniverseisnowprovedtobedark,thenatureofthedarkmatterstillremainsunclear.Inmostmodels,however,thesignificantordominantpartofdarkmatteristakentoconsistofcolddarkmatter(CDM),i.e.non-relativisticparticlesinteractingonlygravitationally.ThisleadstotheabsenceofthermodynamicalequilibriuminDMgasandthenecessityofkineticapproach.FollowingthisapproachitwasshownthattheinitialinhomogenitiesinDMdistribu-tiongrowandformspherically-symmetricstructures–non-dissipativegravitationalsingularities(NGSs)ofdifferentscales,havingthesameinternalstructure.GeneralanalytictheoryofNGSformation8(consistentwithnumericalsimulations9,10andotheranalytictheories11)predictsthatallthesestructureshavesimilardensityprofilesdescribedbythefollowingformula:ρ(r)=Ar−ξ,ξ=12/7.(1)ThedensityprofilesofDMhaloesappearinginnumericalsimulations(seeRefs.4,10)havedifferentforms,typicallyshallowerinthecentre.Whilebeingmorerealisticintheouterpartsthantheprofile(1)duetohierarchicalformationmechanism,theycannotdescribewelltheveryinnerregionbecauseoflimitedspatialresolution,therebyallowingdifferentvaluesoftheinnerslope12,14rangingfrom−1to−1.5,2InternationalJournalofModernPhysicsAallofwhicharewithintheerrorsinthesimulations.Furthermore,observationsdonotexcludecuspydensityprofiles13,butstillareunabletodeterminetheinnerslopeprecisely.Therefore,theadoptedprofiledoesnotcontradictobservationsandisconvenientforanumberofreasons(simplicityofdistributionfunctionandtightrelationbetweenangularmomentumandenergy).Asinitialinhomogenitiesarenotsphericallysymmetric,theparticlesofNGSpossessangularmomenta(thoughtotalangularmomentumofNGSiszero).Highdegreeofc