Measuring gravitational waves from binary black ho

arXiv:gr-qc/9701039v24Nov1997Measuringgravitationalwavesfrombinaryblackholecoalescences:I.Signaltonoiseforinspiral,merger,andringdown.´Eanna´E.FlanaganNewmanLaboratory,CornellUniversity,Ithaca,NY14853-5001.ScottA.HughesTheoreticalAstrophysics,CaliforniaInstituteofTechnology,Pasadena,California91125.Weestimatethesignal-to-noiseratios(SNRs)thatonewouldexpecttomeasurefromcoalescingbi-naryblackhole(BBH)systemsforthefollowingthreebroadbandgravitational-waveobservatories:initialandadvancedground-basedinterferometers(LIGO/VIRGO)andspace-basedinterferome-ters(LISA).Wefocusparticularlyonthehighlyrelativisticandnonlinearmergerportionofthegravitational-wavesignal,whichcomesaftertheadiabaticinspiralportionandbeforetheringdownportionduetothequasinormalringingofthefinalKerrblackhole.Ground-basedinterferometerscandomoderateSNR(afewtens),moderateaccuracystudiesofthedynamicsofmergingblackholesinthemassrange(afew)M⊙to∼2000M⊙.LISA,bycontrast,candohighSNR(afew×104),high-accuracystudiesofBBHsystemsinthemassrange105M⊙∼(1+z)M∼108M⊙,wherezisthebinaries’cosmologicalredshift.OurestimatedSNRssuggestthatcoalescingblackholesmightwellbethefirstsourcesdetectedbytheLIGO/VIRGOnetworkofground-basedinterferometers.Becauseoftheirlargermasses,theycanbeseenouttomuchgreaterdistances(upto∼250MpcforM∼50M⊙fortheinitialLIGOinterferometers)thancoalescingneutronstarbinaries(heretoforeregardedasthe“breadandbutter”workhorsesourceforLIGO/VIRGO,visibleto∼30MpcbytheinitialLIGOinterferometers).Low-massBBHs(M∼30M⊙forthefirstLIGOinterferometers;M∼80M⊙fortheadvanced;(1+z)M∼3×106M⊙forLISA)arebestsearchedforviatheirwell-understoodinspiralwaves;moremassiveBBHsmustbesearchedforviatheirmorepoorlyunderstoodmergerwavesand/ortheirwell-understoodringdownwaves.Asearchforlow-massBBHsbasedontheinspiralwaves,atasensitivitylevelroughlyhalfwaybetweenthefirstLIGOinterferometersandtheadvancedLIGOinterferometers,shouldbecapableoffindingBBHsoutto∼1Gpc.AsearchformassiveBBHsbasedontheringdownwavescanbeperformedusingthemethodofmatchedfilters.Ifonewantsthereductionintheeventrateduetothediscretenessofthetemplatefamilytobenomorethan10%,thenthenumberofindependenttemplatesneededinthesearchwillonlybeabout6000orless.SuchasearchwiththefirstLIGOinterferometersshouldbecapableoffindingBBHsinthemassrangefromabout100M⊙toabout700M⊙outto∼200Mpc;withadvancedLIGOinterferometers,fromabout200M⊙toabout3000M⊙outtoz∼1;andwithLISA,BBHswith106M⊙∼(1+z)M∼3×108M⊙shouldbevisibleouttoz∼100.Theeffectivenessofasearchbasedonthemergerwaveswilldependonhowmuchonehaslearnedaboutthemergerwaveformsfromnumericalrelativitysimulations.Withonlyaknowledgeofthemergerwaves’rangeoffrequencybandsandrangeoftemporaldurations,asearchbasedonmergerwavescanbeperformedusinganonlinearfilteringsearchalgorithm.SuchasearchshouldincreasethenumberofdiscoveredBBHsbyafactorofroughly10overthosefoundfromtheinspiralandringdownwaves.Ontheotherhand,afullsetofmergertemplatesbasedonnumericalrelativitysimulationscouldfurtherincreasethenumberofdiscoveredBBHsbyaadditionalfactorofuptoabout4.PACSnumbers:04.80.Nn,04.25.Dm,04.30.Db,95.55.Ym1I.INTRODUCTIONANDSUMMARYA.CoalescencesofblackholebinariesIthaslongbeenrecognizedthatcoalescencesofbi-narysystemsoftwoblackholescouldbeanimportantsourceofgravitationalwaves[1,2],bothforthegroundbasedinterferometricdetectorsLIGO[3]andVIRGO[4]currentlyunderconstruction,andalsoforthepossiblefuturespace-basedinterferometerLISA[5,6,7].Theor-bitsofbinaryblackholes(BBHs)graduallydecayfromenergyandangularmomentumlosstogravitationalradi-ation.Eventually,theholescoalescetoformafinalblackhole.ForgravitationalradiationreactiontosuccessfullydrivethebinarytomergeinlessthanaHubbletime,theinitialorbitalperiodmustbe∼0.3days(M/M⊙)5/8,whereMisthetotalmassofthebinary;thusthecriticalorbitalperiodisoforderdaysforsolarmassblackholes,andoforderyearstohundredsofyearsforsupermassiveblackholes(106M⊙∼M∼109M⊙).Theprocessofcoalescencecanbedividedupintothreemoreorlessdistinctphases:•Anadiabaticinspiral,duringwhichthegravita-tionalradiationreactiontimescaleismuchlongerthantheorbitalperiod.Theinspiralendswhenthebinaryorbitbecomesrelativisticallydynami-callyunstableatanorbitalseparationofr∼6M(inunitswhereG=c=1)[8,9].Thegravitationalwavesfromtheinspiralcarryencodedwithinthemthemassesandspinsofthetwoblackholes,someoftheorbitalelementsofthebinaryandthedistancetothebinary[1,10].•Towardstheendoftheinspiral,theblackholesencounterthedynamicalinstabilityandmakeagradualtransitionfromaradiation-reactiondriveninspiraltoafreely-fallingplunge[8,11,12],afterwhich,eveniftheradiationreactioncouldbeturnedoff,theblackholeswouldstillmerge.Wewillcallthesubsequentplungeandviolentcollisionthemergerphase.Gravitationalwavesfromthemergercouldberichwithinformationaboutthedynamicsofrelativisticgravityinahighlynonlin-ear,highlydynamicregimeaboutwhichwehaveapoortheoreticalunderstandingtoday.•AsthefinalblackholesettlesdowntoastationaryKerrstate,thenonlineardynamicsofthemergergraduallybecomemoreandmoredescribableasoscillationsofthisfinalblackhole’squasinormalmodes[13,14].Thecorrespondin