Black hole mergers in the universe

SimonF.PortegiesZwart1

and

arXiv:astro-ph/9910061v2 30 Nov 1999StephenL.W.McMillan2

InstituteforAstrophysicalResearchBostonUniversity,725CommonwealthAve.,Boston,MA02215,USAspz@komodo.bu.edu

Dept.ofPhysics,DrexelUniversity,Philadelphia,PA19104,USAsteve@kepler.physics.drexel.edu

Subjectheadings:gravitation—methods:n-bodysimulations—stellardynamics—binaries(includingmultiple):close—stars:evolution—globularclusters:general—

21

–2–ABSTRACT

Mergersofblack-holebinariesareexpectedtoreleaselargeamountsofenergyintheformofgravitationalradiation.However,binaryevolutionmodelspredictmergerratestoolowtobeofobservationalinterest.Inthispaperweexplorethepossibilitythatblackholesbecomemembersofclosebinariesviadynamicalinteractionswithotherstarsindensestellarsystems.Instarclusters,blackholesbecomethemostmassiveobjectswithinafewtensofmillionsofyears;dynamicalrelaxationthencausesthemtosinktotheclustercore,wheretheyformbinaries.Theseblack-holebinariesbecomemoretightlyboundbysuperelasticencounterswithotherclustermembers,andareultimatelyejectedfromthecluster.Themajorityofescapingblack-holebinarieshaveorbitalperiodsshortenoughandeccentricitieshighenoughthattheemissionofgravitationalradiationcausesthemtocoalescewithinafewbillionyears.Wepredictablack-holemergerrateofabout1.6×10−7peryearpercubicmegaparsec,implyinggravitywavedetectionratessubstantiallygreaterthanthecorrespondingratesfromneutronstarmergers.ForthefirstgenerationLaserInterferometerGravitational-WaveObservatory(LIGO-I),weexpectaboutonedetectionduringthefirsttwoyearsofoperation.ForitssuccessorLIGO-II,theraterisestoroughlyonedetectionperday.Theuncertaintiesinthesenumbersarelarge.Eventratesmaydropbyaboutanorderofmagnitudeifthemostmassiveclustersejecttheirblackholebinariesearlyintheirevolution.

1.Introduction

ThesearchforgravitationalwaveswillbegininearnestinJanuary2002,whenLIGO-I(Abramovicietal.1992)becomesfullyoperational(K.Thorne,privatecommunication).Theappearanceofthisnewandwhollyunexploredobservationalwindowchallenges

physicistsandastronomerstopredictdetectionratesandsourcecharacteristics.Mergersofneutron-starbinariesarewidelyregardedasthemostpromisingsourcesofgravitationalradiation,andestimatesofneutronstarmergerrates(perunitvolume)rangefromR∼1.9×10−7h3yr−1Mpc−3(Narayanetal.1991;Phinney1991;PortegiesZwart&Spreeuw1996),whereh=H0/100kms−1Mpc−1,toroughlytentimesthisvalue(Tutukov&Yungelson1993;Lipunovetal.,1997).However,evenwiththemostoptimisticassumptions,wecanexpectaLIGO-Idetectionrateofonlyafewneutronstareventspermillennium.

Inspiralandmergerofblack-holebinariesareconsiderablymoreenergeticeventsthan

–3–

neutronstarmergers,duetothehighermassesoftheblackholes(Tutukov&Yungelson1993;Lipunovetal.1997).Black-holebinariescanresultfromtheevolutionoftwostarswhichareborninaclosebinary,experienceseveralphasesofmasstransfer,andsubsequentlysurvivetwosupernovae(Tutukov&Yungelson1993).Calculationsofeventratesfromsuchfieldbinariesdependsensitivelyonmanyunknownparametersandmuchpoorlyunderstoodphysics,butthemodelsgenerallypredictablackholemergerrate<2×10−9h3yr−1Mpc−3(Tutukov&Yungelson1993;PortegiesZwart&YungelsonR∼

1998;Bethe&Brown1999),substantiallylowerthantherateforneutronstars.Analternativepossibility,whichweexplorehere,isthatblackholesbecomemembersofclosebinariesnotthroughinternalbinaryevolution,butratherviadynamicalinteractionswithotherstarsinadensestellarsystem.

2.Blackholebinariesinstarclusters

Blackholesaretheproductsofstarswithinitialmassesexceeding∼20–25M⊙(Maeder1992;PortegiesZwartetal.1997).AScalo(1986)massdistributionwithalowermasslimitof0.1M⊙andanupperlimitof100M⊙has0.071%ofitsstarsmoremassivethan20M⊙,and0.045%moremassivethan25M⊙.AstarclustercontainingNstarsthusproduces∼6×10−4Nblackholes.KnownGalacticblackholeshavemassesmbhbetween6M⊙and18M⊙(Cowlay1992).Fordefiniteness,weadoptmbh=10M⊙.

2.1.Binaryformationanddynamicalevolution

Ablackholeisformedinasupernovaexplosion.Iftheprogenitorisasinglestar(i.e.notamemberofabinary),theblackholeexperienceslittleornorecoilandremainsamemberoftheparentcluster(White&vanParadijs1996).Iftheprogenitorisamemberofabinary,masslossduringthesupernovamayejectthebinaryfromtheclusterpotentialviatheBlaauwmechanism(Blaauw1962),whereconservationofmomentumcausesrecoilinabinarywhichlosesmassimpulsivelyfromonecomponent.Weestimatethatnomorethan∼10%ofblackholesareejectedfromtheclusterimmediatelyfollowingtheirformation.After∼40Myrthelastsupernovahasoccurred,themeanmassoftheclusterstarsis󰀊m󰀋∼0.56M⊙(Scalo1986),andblackholesarebyfarthemostmassiveobjectsinthesystem.Masssegregationcausestheblackholestosinktotheclustercoreinafraction∼󰀊m󰀋/mbhofthehalf-massrelaxationtime.Foratypicalglobularcluster,therelaxationtimeis∼1Gyr;forayoungpopulouscluster,suchasR136(NGC2070)inthe30Doradus

–4–

regionoftheLargeMagellanicCloud(Massey&Hunter1998),itis∼10Myr.

Bythetimeofthelastsupernova,stellarmasslosshasalsosignificantlydiminishedandtheclustercorestartstocontract,enhancingtheformationofbinariesbythree-bodyinteractions.Singleblackholesformbinariespreferentiallywithotherblackholes(Kulkarnietal.1992),whileblackholesborninbinarieswithalower-massstellarcompanionrapidlyexchangethecompanionforanotherblackhole.Theresultinallcasesisagrowingblack-holebinarypopulationintheclustercore.Onceformed,theblack-holebinariesbecomemoretightlyboundthroughsuperelasticencounterswithotherclustermembers(Heggie1975;Kulkarnietal.1992;Sigurdsson&Hernquist1993).Onaverage,followingeachclosebinary–singleblackholeencounter,thebindingenergyofthebinaryincreasesbyabout20%(Hutetal.1992);roughlyonethirdofthisenergygoesintobinaryrecoil,assumingequalmassstars.Theminimumbindingenergyofanescapingblack-holebinarymaythenbeestimatedas

mbh

Eb,min∼36W0kTisthemeanstellarkineticenergyandW0=󰀊m󰀋|φ0|/kTisthedimensionless

centralpotentialofthecluster(King1966).Bythetimetheblackholesareejected,󰀊m󰀋∼0.4M⊙.TakingW0∼5–10asarepresentativerange,wefindEb,min∼5000–10000kT.

2

WehavetestedandrefinedtheaboveestimatesbyperformingaseriesofN-bodysimulationswithinthe“Starlab”softwareenvironment(PortegiesZwartetal.1999:seehttp::/www.sns.ias.edu/∼starlab),usingthespecial-purposecomputerGRAPE-4tospeedupthecalculations(Makinoetal.1997).Formost(seven)ofourcalculationsweused2048equal-massstarswith1%ofthemtentimesmoremassivethantheaverage;twocalculationswereperformedwith4096stars.Oneofthe4096-particlerunscontained0.5%blackholes;thesmallerblack-holefractiondidnotresultinsignificantlydifferentbehavior.Wealsotestedalternativeinitialconfigurations,startingsomemodelswiththeblackholesinprimordialbinarieswithotherblackholes,orinprimordialbinarieswithlower-massstars.

Theresultsofoursimulationsmaybesummarizedasfollows.Ofatotalof204blackholes,62(∼30%)wereejectedfromthemodelclustersintheformofblack-holebinaries.Atotalof124(∼61%)blackholeswereejectedsingle,andoneescapingblackholehadalow-massstarasacompanion.Theremaining17(∼8%)blackholeswereretainedbytheirparentclusters.ThebindingenergiesEboftheejectedblack-holebinariesrangedfromabout1000kTto10000kTinadistributionmoreorlessflatinlogEb,consistentwiththeassumptionsmadebyHutetal.(1992).Theeccentricitiesefollowedaroughlythermaldistribution[p(e)∼2e],withhigheccentricitiesslightlyoverrepresented.Abouthalfofthe

–5–

blackholeswereejectedwhiletheparentclusterstillretainedmorethan90%(∼

<2initialrelaxationtimes)ofitsbirthmass,and∼

>90%oftheblackholeswereejectedbeforetheclusterhadlost30%(between4and10relaxationtimes)ofitsinitialmass.Thesefindingsareingoodagreementwithpreviousestimatesthatblack-holebinariesareejectedwithinafewGyr,wellbeforecorecollapseoccurs(Kulkarnietal.1993;Sigurdsson&Hernquist1993).

Weperformedadditionalcalculationsincorporatingarealistic(Scalo)massfunction,theeffectsofstellarevolution,andthegravitationalinfluenceoftheGalaxy.Ourmodelclustersgenerallydissolvedratherquickly(withinafewhundredMyr)intheGalactictidalfield.Clusterswhichdissolvedwithin∼40Myr(beforethelastsupernova)hadnotimetoejecttheirblackholes.However,thosethatsurvivedbeyondthistimeweregenerallyabletoejectatleastonecloseblack-holebinarybeforedissolution.

Basedontheseconsiderations,weconservativelyestimatethenumberofejectedblack-holebinariestobeabout10−4Nperstarcluster,moreorlessindependentoftheclusterlifetime.

2.2.Characteristicsofthebinarypopulation

Theenergyofanejectedbinaryanditsorbitalseparationarecoupledtothedynamicalcharacteristicsofthestarcluster.Foraclusterinvirialequilibrium,wehavekT≡2Ekin/3N=−Epot/3N=GM2/6Nrvir,whereMisthetotalclustermassandrviristhevirialradius.Ablack-holebinarywithsemi-majoraxisahasEb=Gm2bh/2a,so

Eb

2

rvir

M

󰀂–6–

distinctionbetweencore-collapsedglobularclusters(about20%ofthecurrentpopulation)andnon-collapsedglobulars—thepresentdynamicalstateofaclusterhaslittlebearingonhowblack-holebinarieswereformedandejectedduringthefirstfewGyrofthecluster’slife.

3.Productionofgravitationalradiation

AnapproximateformulaforthemergertimeoftwostarsduetotheemissionofgravitationalwavesisgivenbyPeters&Mathews1963):

tmrg≈150Myr

󰀁

M⊙

R⊙

󰀄4

(1−e2)7/2.

(3)

ThesixthcolumnofTable1liststhefractionofblack-holebinarieswhichmergewithinaHubbletimeduetogravitationalradiation,assumingthatthebinarybindingenergiesaredistributedflatinlogEbbetween1000kTand10000kT,thattheeccentricitiesarethermal,independentofEb,andthattheuniverseis15Gyrold(Jhaetal.1999).Thefinalcolumnofthetableliststhecontributiontothetotalblack-holemergerratefromeachclustercategory.

3.1.Mergerrateinthelocaluniverse

Giventheblack-holemergerratecorrespondingtoeachcategoryofstarcluster,wenowestimatethetotalmergerrateRperunitvolume.Table2lists,forvarioustyesofgalaxies,thespacedensitiesandSN,thespecificnumberofglobularclustersperMv=−15magnitude(vandenBergh1995):

SN=NGC100.4(Mv+15)

(4)

(whereNGCisthetotalnumberofglobularclustersinthegalaxyunderconsideration).ThevaluesgivenforSNinTable2arecorrectedforinternalabsorption;theabsorbedcomponentisestimatedfromobservationsinthefarinfrared.Theestimatednumberdensityofglobularclustersintheuniverseis

φGC=8.4h3Mpc−3.

(5)

Aconservativeestimateofthemergerrateofblack-holebinariesformedinglobularclustersisobtainedbyassumingthatglobularclustersinothergalaxieshavecharacteristicssimilar

–7–

tothosefoundinourown.Theresultis

RGC=5.4×10−8h3yr−1Mpc−3.

(6)

Irregulargalaxies,starburstgalaxies,earlytypespiralsandblueellipticalgalaxiesallcontributetotheformationofyoungpopulousclusters.Intheabsenceoffirm

measurementsofthenumbersofyoungpopulousclustersinothergalaxies,wesimplyusethesamevaluesofSNasforglobularclusters.ThespacedensityofsuchclustersisthenφYPC=3.5h3Mpc−3,andtheblackholemergerrateis

RYPC=2.1×10−8h3yr−1Mpc−3.

Wefindthatgalacticnucleicontributenegligiblytothetotalblackholemergerrate.Basedontheassumptionsoutlinedabove,ourestimatedtotalmergerrateperunitvolumeofblack-holebinariesis

R=7.5×10−8h3yr−1Mpc−3.

(8)(7)

However,thismaybeaconsiderableunderestimateofthetruerate.First,asalreadymentioned,ourassumednumber(∼10−4N)ofejectedblack-holebinariesisquite

conservative.Second,theobservedpopulationofglobularclustersnaturallyrepresentsonlythoseclustersthathavesurviveduntilthepresentday.ThestudybyTakahashi&PortegiesZwart(2000)indicatesthat∼50%ofglobularclustersdissolveinthetidalfieldoftheparentgalaxywithinafewbillionyearsofformation.Wehavethereforeunderestimatedthetotalnumberofglobularclusters,andhencetheblack-holemergerrate,byaboutafactoroftwo.Third,averysubstantialunderestimatestemsfromtheassumptionthatthemassesandradiiofpresent-dayglobularclustersarerepresentativeoftheinitialpopulation.Whenestimatedinitialparameters(Table1,bottomrow)areused,thetotalmergerrateincreasesbyafurtherfactorofsix.Takingalltheseeffectsintoaccount,weobtainanetblack-holemergerrateof

R∼3×10−7h3yr−1Mpc−3.(9)Wenotethatthisfigureissignificantlylargerthanthecurrentbestestimatesofthe

neutron-starmergerrate.

3.2.LIGOobservations

ThemaximumdistancewithinwhichLIGO-Icandetectaninspiraleventisestimatedtobe󰀃

Mchirp

Reff=18Mpc

–8–

(K.Thorne,privatecommunication).Here,the“chirp”massforabinarywithcomponentmassesm1andm2isMchirp=(m1m2)3/5/(m1+m2)1/5.Forneutronstarinspiral,

m1=m2=1.4M⊙,soMchirp=1.22M⊙,Reff=21Mpc,andweobtainthedetectionratementionedintheintroduction.Forblack-holebinarieswithm1=m2=mbh=10M⊙,wefindMchirp=8.71M⊙,Reff=109Mpc,andaLIGO-Idetectionrateofabout1.7h3peryear.Forh∼0.65(Jha1999),thisresultsinaboutonedetectioneventeverytwoyears.LIGO-IIshouldbecomeoperationalby2007,andisexpectedtohaveReffabouttentimesgreaterthanLIGO-I,resultinginadetectionrate1000timeshigher,oraboutoneeventperday.

4.Discussion

Black-holebinariesejectedfromgalacticnuclei,themostmassiveglobularclusters

>106M),andglobularclusterswhichexperiencecorecollapsesoonafter(masses∼⊙

formation,tendtobeverytightlybound,havehigheccentricitiesandmergewithinafewmillionyearsofejection.ThesemergersthereforetracetheformationofdensestellarsystemswithadelayofafewGyr(thetypicaltimerequiredtoformandejectbinaries),makingthesesystemsunlikelycandidatesforLIGOdetections,asthemajoritymergedlongago.Thiseffectmayreducethecurrentmergerratebyanorderofmagnitude,althoughmoresensitivefuturegravitationalwavedetectorsmayseesomeoftheseearlyuniverseevents.Weestimatethatthemostmassiveglobularclusterscontributeabout90%ofthetotalblackholemergerrate.However,whiletheirblack-holebinariesmergepromptlyuponejection,thelongerrelaxationtimesoftheseclustersmeanthatbinariestendtobeejectedmuchlaterthaninlowermasssystems.Consequently,wehaveretainedthesebinariesinourfinalmergerrateestimate(Eq.9).Butwenotethatrepresentsasignificantsourceifuncertainty.

Bythetimetheblackholebinaryisejectedithasexperienced∼40–50hardencounterswithotherblackholes,aswellasasimilarnumberofencounterswithotherstarsorbinaries.Duringeachoftheselatterencounters,thereisasmallprobabilitythatalow-massstarmaycollidewithoneoftheblackholes.Suchcollisionstendtosoftentheblackholebinarysomewhat(seePortegiesZwartetal.1999),buttheyareunlikelytodelayejectionsignificantly.Acollisionbetweenamain-sequencestarandablackholemay,however,leadtobriefbutintenseX-rayphase.

Finally,wehaveassumedthatthemassofastellarblackholeis10M⊙.Increasingthismassto18M⊙decreasestheexpectedmergerratebyabout50%—highermassblackholestendtohavewiderorbits.However,thelargerchirpmassincreasesthesignaltonoise,and

–9–

thedistancetowhichsuchamergercanbeobservedincreasesbyabout60%andtheoveralldetectionrateonEarthincreasesbyaboutafactorofthree.For6M⊙blackholes,thedetectionratedecreasesbyasimilarfactor.Forblack-holebinarieswithcomponentmasses>

∼12M⊙,thefirstgenerationofdetectorswillbemoresensitivetothemergeritselfthantotheinspiralphasethatprecedesit(Flanagan&Hughes1998).Sincethestrongestsignalisexpectedfromblack-holebinarieswithhigh-masscomponents,itiscriticallyimportanttoimproveourunderstandingofthemergerwaveform.Evenforlower-massblackholes(with

>10M),theinspiralsignalcomesfromanepochwhentheholesaresoclosetogethermbh∼⊙

thatthepost-Newtonianexpansionsusedtocalculatethewaveformsareunreliable(Bradyetal.1998).

AcknowledgmentsWethankPietHut,JunMakinoandKipThorneforinsightfulcommentsonthemanuscript.ThisworkwassupportedbyNASAthroughHubbleFellowshipgrantHF-01112.01-98Aawarded(toSPZ)bytheSpaceTelescopeScienceInstitute,whichisoperatedbytheAssociationofUniversitiesforResearchinAstronomy,Inc.,forNASAundercontractNAS5-26555,andbyATPgrantNAG5-69(toSLWM).SPZisgratefultoDrexelUniversity,TokyoUniversityandtheUniversityofAmsterdam(underSpinozagrant0-08toEdwardP.J.vandenHeuvel)fortheirhospitality.CalculationsareperformedontheGRAPE-4computersatTokyoUniversityandDrexelUniversity,andontheSGI/CrayOrigin2000supercomputeratBostonUniversity.

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–11–

–12–

Table1:Overviewofselectedparametersforyoungpopulousclusters(Massey&Hunter1998),globularclusters(Djorgovski&Meylan,1994)andgalacticnuclei(McGinnetal.19).Thefirstthreecolumnslisttheclustertype,thetotalmass(insolarunits)andthevirialradius(inpc).Thetotalmassandvirialradiusaregivenasdistributionswithameanandthestandarddeviationaroundthemean.Theorbitalseparation(insolarunits)fora1000kTbinaryconsistingoftwo10M⊙blackholesisgiveninthefourthcolumn.Thefifthandsixthcolumnslisttheexpectednumberofblack-holebinariesthatareformedbytheclusterandthefractionofthesebinarieswhichmergewithin12Gyr(allowing∼3Gyrfortheformationandejectionofthebinariesandassuminga15Gyrolduniverse,Jha1999).Thecontributionstothetotalblackholemergerrateperstarclustersperyear(MR)aregiveninthefinalcolumn.Thebottomrowcontainsestimatedparametersforthezero-agepopulationofglobularclustersintheGalaxy,indicatedwith⋆.clustertype

PopulousGlobularNucleusGlobular⋆

Mrvir1000kT[log][log][R⊙]4.5-0.44205.5±0.50.5±0.3315

<<∼7∼0∼3.3

6.0±0.50±0.333

Nb

fmerge

MR

[Myr−1]0.00610.000.210.038

7.97.7%15051%2500100%50092%

Table2:Galaxymorphologyclass,spacedensities,averageabsolutemagnitude(Heyletal.1997),andthespecificfrequencyofglobularclustersSN(fromvandenBergh,1995andMcLaughlin1999).Thefinalcolumngivesthecontributiontothetotalnumberdensityofglobularclusters.GalaxyTypeE–S0SabSbcScdBlueESdm/StarB

φGN

[10−3hMpc−3]

3.492.192.803.011.870.50

Mv-20.7-20.0-19.4-19.2-19.6-19.0

SNh210710.2140.5

GCspacedensity

[h3Mpc−3]

6.651.530.160.031.810.01