Detection of Anisotropies in the Gravitational-Wav

arXiv:gr-qc/9607068v223Dec1996DetectionofAnisotropiesintheGravitational-WaveStochasticBackgroundBruceAllenDepartmentofPhysicsUniversityofWisconsin-MilwaukeePOBox413Milwaukee,WI53211,USAemail:[e-mail]ematicalPhysics,UniversityCollegeDublin,Belfield,Dublin4,Irelandemail:[e-mail](February7,2008)Bycorrelatingthesignalsfromapairofgravitational-wavedetectors,onecanundertakesensi-tivesearchesforastochasticbackgroundofgravitationalradiation.Ifthestochasticbackgroundisanisotropic,thenthiscorrelatedsignalvariesharmonicallywiththeearth’srotation.Wecalculatehowtheharmonicsofthisvaryingsignalarerelatedtothemultipolemomentswhichcharacterizetheanisotropy,andgiveaformulaforthesignal-to-noiseratioofagivenharmonic.ThespecificcaseofthetwoLIGO(LaserInterferometricGravitationalObservatory)detectors,whichwillbe-ginoperationaroundtheyear2000,isanalyzedindetail.Weconsidertwopossibleexamplesofanisotropy.Ifthegravitational-wavestochasticbackgroundcontainsadipoleintensityanisotropywhoseorigin(likethatoftheCosmicBackgroundRadiation)ismotionofourlocalsystem,thenthatanisotropywillbeobservablebytheadvancedLIGOdetector(with90%confidenceinoneyearofobservation)ifΩgw5.3×10−8h−2100.Wealsostudythesignalproducedbystochasticsourcesdistributedinthesamewayastheluminousmatterinthegalacticdisk,andinthesamewayasthegalactichalo.TheanisotropyduetosourcesdistributedasthegalacticdiskorasthegalactichalowillbeobservablebytheadvancedLIGOdetector(with90%confidenceinoneyearofobservation)ifΩgw1.8×10−10h−2100orΩgw6.7×10−8h−2100,respectively.I.INTRODUCTIONThedesignandconstructionofanumberofnewandmoresensitivedetectorsofgravitationalradiationiscurrentlyunderway.TheseincludetheLIGOdetectorbeingbuiltintheUnitedStatesbyajointCaltech/MITcollaboration[1],theVIRGOdetectorbeingbuiltnearPisabyanItalian/Frenchcollaboration[2],theGEO-600detectorbeingbuiltinHannoverbyanAnglo/Germancollaboration[3],andtheTAMA-300detectorbeingbuiltnearTokyo[4].Therearealsoseveralresonantbardetectorscurrentlyinoperation,andseveralmorerefinedbarandinterferometricdetectorspresentlyintheplanningandproposalstages.Whentwoormoreofthesedetectorsareoperating,itwillbecomepossibletocorrelatetheirsignals,andinthisway,tosearchforastochasticbackgroundofgravitationalradiation.ThetechniqueforsuchasearchwasoriginallydescribedinworkbyMichelson[5],Cristensen[6]andFlanagan[7].Areviewofthesetechniquesmaybefoundin[8].Suchradiationmightbetheresultofprocessesthattookplaceduringtheveryearlyuniverse.Itmightalsoresultfromtheincoherentsuperpositionofmanyfaintunresolvablepresent-daysourcessuchascoalescingbinarysystems.Thestochasticgravitational-wavebackgroundmightbeisotropiconthesky,oritmightbeanisotropic.Forexample,ifthebackgroundresultsfromearly-universeprocesses,thenitmightbeisotropictoaboutthesamedegreeasthe2.7◦Kelectromagneticbackgroundradiation.Ontheotherhand,ifthebackgroundisduetowhite-dwarfbinariesinourowngalaxy,thentheymightbedistributedinapancakeorbarwhichmimicstheshapeoftheobservedluminousmatterinourgalaxy.Inthispaper,weshowhowthecorrelatedsignalfromapairofgravitationalwavedetectorsisrelatedtomultipolemomentswhichcharacterizetheanisotropy.Thisshouldpermitasignaltobeanalyzedtosearchfor(orplaceupperlimitson)themultipolemomentswhichcharacterizetheanisotropy.Inthispaper,wewillassumethatthereaderisalreadyfamiliarwiththeworkpreviouslycited(references[5–8])onstochasticbackgrounddetection.Thesepaperisorganizedasfollows.InSectionIIweshowhowabackgroundofstochasticgravitationalradiationmaybedecomposedinaplane-waveexpansion,withthecoefficientsoftheexpansiontreatedasstochasticrandom1variables.InSectionIIIthepropertiesoftheserandomvariablesarerelatedtothe(frequency)spectrumandspatialdistributionoftheradiation,andasetofmultipolemomentsareintroducedwhichcharacterizetheanisotropiesofthestochasticbackground.Theseanisotropiesmaybesearchedforbystudyingthevariationsofthedetectoroutputsastheearthrotatesrelativetothefixedcosmicframe.InSectionIVweshowhowthecorrelationbetweenapairofdetectorsfixedontheearthvarieswithtimeastheearthrotates,anddetailhowthatcorrelationisrelatedtotheanisotropiesofthestochasticgravitationalbackground.Thevariationofthecorrelationwiththeearth’srotationmaybedecomposedintoharmonicsoftheearth’speriod.InsectionSectionVweintroduceasetoffunctionsγℓm(f)whicharegeneralizationsofthewell-knownoverlapreductionfunctionγ(f)ofreferences[5–8].Thesefunctionscharacterizetheeffectoftheℓ,manisotropymultipoleonthem’thharmonicofthedetectorcorrelation.TheprincipalresultofthispaperistocomputethesefunctionsfortheLIGOpairofdetectors.Thisisdonebyintroducingtwospecialframesofreference,fixedwithrespecttotheearth,inSectionVI,andthenperformingasetofintegrationsinSectionVII.Thefinalintegrationsareperformedandexplicitformulafortheγℓm(f)areobtainedinSectionVIII.InSectionIXweanalyzethesignal-to-noiseratiosassociatedwiththem’thharmonic,andgiveaformulawhichmaybeusedtodetermineifagivenanisotropyisdetectableornot.Followingthis,weconsidertwospecificexamplesofanisotropy.InSectionXweconsideradi