Numerical wave optics and the lensing of gravitati

arXiv:0710.3140v1[gr-qc]16Oct2007February2,20084:20WSPC-ProceedingsTrimSize:9.75inx6.5inmainNumericalwaveopticsandthelensingofgravitationalwavesbyglobularclustersAndrewJ.Moylan,DavidE.McClelland,SusanM.ScottandAntonyC.SearleCentreforGravitationalPhysics,DepartmentofPhysics,TheAustralianNationalUniversity,CanberraACT0200,[e-mail],[e-mail],[e-mail],[e-mail]AstronomyandAstrophysics,TheAustralianNationalUniversity,CanberraACT2611,Australiageoff@mso.anu.edu.auWeconsiderthepossibleeffectsofgravitationallensingbyglobularclustersongrav-itationalwavesfromasymmetricneutronstarsinourgalaxy.Inthelensingofgravita-tionalwaves,thelongwavelength,comparedwiththeusualcaseofopticallensing,canleadtothegeometricalopticsapproximationbeinginvalid,inwhichcaseawaveopticalsolutionisnecessary.Ingeneral,waveopticalsolutionscanonlybeobtainednumeri-cally.Wedescribeacomputationalmethodthatisparticularlywellsuitedtonumericalwaveoptics.Thismethodenablesustocomparethepropertiesofseverallensmodelsforglobularclusterswithoutevercallinguponthegeometricalopticsapproximation,thoughthatapproximationwouldsometimeshavebeenvalid.Finally,weestimatetheprobabilitythatlensingbyaglobularclusterwillsignificantlyaffectthedetection,byground-basedlaserinterferometerdetectorssuchasLIGO,ofgravitationalwavesfromanasymmetricneutronstarinourgalaxy,findingthattheprobabilityisinsignificantlysmall.1.IntroductionInthisarticleweconsiderwaveopticaleffectsinthegravitationallensingofgravita-tionalwavesbyglobularclusters,anddeterminethepossibleeffectofsuchlensingongravitationalwavesdetectedbyEarth-basedinterferometers.Waveeffectsingravitationallensingarealsorelevantincontextsotherthanthelensingofgrav-itationalwaves,suchasthe‘femtolensing’ofgamma-rayburstsbyhypotheticalcompactobjectsofverylowmass,andhavebeenstudiedbyvariousauthors;seeNakamuraandDeguchi1andreferencestherein.Severalauthorshaveconsideredthepossiblewaveopticaleffectsoflensingongravitationalwaves.2–8BeginningwithNakamura,2manyofthesestudieshaveem-ployedtheKirchhoff-typediffractionintegralfoundinthebookbySchneideretal.9andre-derivedbyNakamuraandDeguchi1usingapathintegralapproach.Taka-hashiandNakamura5investigatedthewaveopticalpropertiesoftwosimplelensprofiles,thesingularisothermalsphereandthepoint-mass,andusedthemtomodeldarkmatterhalosandcompactobjects(suchasblackholes),respectively,obtaininganestimateoftheeffectsoflensingbytheseobjectsongravitationalwavesinthesensitivefrequencyrangeoftheproposedspace-baseddetectorLISA.Inthegeometricalopticslimit,thelensingofgravitationalwavesbyavariety1February2,20084:20WSPC-ProceedingsTrimSize:9.75inx6.5inmain2oflensmodelshasbeenconsideredbyotherauthorsincludingArnaud-Varvellaetal.10andSeto.11Arnaud-Varvellaetal.foundthatthegeometricalopticsapprox-imationisapplicableinthefrequencyrangetowhichEarth-basedinterferometersaresensitive(101-104Hz)forlensmasses&106M⊙.(Foraslightlydifferentcrite-rion,lensmass&108M⊙(f/mHz)−1,seeSuyamaetal.12andreferencestherein.)Themeaningofthiscriterionisunclear,however,forcommonlensmodelsthatdonothaveafinitetotalmass,suchasisothermalspheres,andthegeometricalopticsapproximationremains,nonetheless,invalidwhentheimageslienearcausticsofthelens.Inderivingthegeometricalopticslimitfromthediffractionintegral,NakamuraandDeguchihaveobtainedaconditiononthewavefrequencyandlensingconfigura-tionthatdetermineswhenthegeometricalopticsapproximationisvalidingeneral(Equation(3.3)ofreference1).Thediffractionintegralrelevanttowaveopticallensingmust,ingeneral,becomputednumerically,andisrapidlyoscillatory,makingitdifficulttoevaluate.Thefieldofhighlyoscillatoryintegration,whichiscurrentlyveryactive,13offerspromisingmethodsforthesetypesofintegrals.Afterreviewingwaveopticsingrav-itationallensinginSection2,wedescribeinSection3analgorithm,14basedonamethodoriginallyduetoLevin,15capableofveryefficientlycomputingthediffrac-tionintegralsnecessaryinanalysingthelensmodelsconsideredinlatersections.InSection4wedescribeavarietyofpossiblelensprofilesformodellingglobularclusters,andexploretheirwaveopticalproperties.InSection5weuseourresultstoestimatetheprobabilitythatlensingbyglobularclustersinourgalaxycansig-nificantlyaffectthedetectionbyEarth-basedinterferometersofgravitationalwavesfromasymmetricneutronstarsinourgalaxy.Weusegeometricunitsinwhichc=G=1.Ournotation,whichcloselyfollowsthatofTakahashi,16issummarisedinTable1.Table1.Summaryofnotation.DL(angulardiameter)distancefromobservertolensDLSdistancefromlenstosourceDL+DLS≡DSdistancefromobservertosourceMLlensmassˆxpositioninthelensplaneˆydisplacementofsourcefromopticalaxisˆwangularfrequencyofwaveˆψ(ˆx)2-dimensionallensingpotentialξ0lengthnormalisationconstantˆx/ξ0≡xdimensionlesspositioninlensplane(DL/DSξ0)ˆy≡ydimensionlessdisplacementofsourcefromopticalaxis(DS/DLDLS)ξ20ˆw≡wdimensionlessangularfrequency(DLDLS/DSξ20)ˆψ(ˆx)≡ψ(x)dimensionless2-dimensionallensingpotentialFebruary2,20084:20WSPC-ProceedingsTrimSize:9.75inx6.5inmain32.WaveopticsingravitationallensingInthissect