Effect of Gravitational Lensing on Measurements of the Sunya

The Sunyaev-Zel'dovich (SZ) effect of a cluster of galaxies is usually measured after background radio sources are removed from the cluster field. Gravitational lensing by the cluster potential leads to a systematic deficit in the residual intensity of unr

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aSubmittedtoApJLetters,October1996E ectofGravitationalLensingonMeasurementsoftheSunyaev-Zel’dovichE ectAbrahamLoeb1,3andAlexandreRefregier2,41.AstronomyDepartment,HarvardUniversity,60GardenSt.,Cambridge,MA021382.ColumbiaAstrophysicsLaboratory,538W.120thStreet,NewYork,NY10027ABSTRACTTheSunyaev–Zel’dovich(SZ)e ectofaclusterofgalaxiesisusuallymeasuredafterbackgroundradiosourcesareremovedfromthecluster eld.Gravitationallensingbytheclusterpotentialleadstoasystematicde citintheresidualintensityofunresolvedsourcesbehindtheclustercorerelativetoacontrol eldfarfromtheclustercenter.Asaresult,themeasureddecrementintheRayleigh–Jeanstemperatureofthecosmicmicrowavebackgroundisoverestimated.WecalculatetheassociatedsystematicbiaswhichisinevitablyintroducedintomeasurementsoftheHubbleconstantusingtheSZe ect.FortheclusterA2218,we ndthatobservationsat15GHzwithabeamradiusof0′.4andasourceremovalthresholdof100µJyunderestimatetheHubbleconstantby6–10%.Ifthepro leofthegaspressuredeclinesmoresteeplywithradiusthanthatofthedarkmatterdensity,thentheratiooflensingtoSZdecrementsincreasestowardstheouterpartofthecluster.Subjectheadings:cosmicmicrowavebackground–di useradiation–galaxies:clusters:general,individual(A2218)–gravitationallensing

1.Introduction

TheSunyaev–Zel’dovich(SZ)e ectdescribesthedistortionintroducedtotheCosmicMicrowaveBackground(CMB)spectrumduetoitsComptonscatteringo freeelectrons,whichareeitherhot(thethermale ect)orpossessabulkpeculiarvelocity(thekinematice ect;seereviewsofbothe ectsinSunyaev&Zel’dovich1980andRephaeli1995).ThethermalSZe ectprovidesanimportantdiagnosticofthehotgasinclustersofgalaxies,andbynowhasbeen

The Sunyaev-Zel'dovich (SZ) effect of a cluster of galaxies is usually measured after background radio sources are removed from the cluster field. Gravitational lensing by the cluster potential leads to a systematic deficit in the residual intensity of unr

measuredinanumberofsystems(seeTable1inRephaeli1995).Thekinematice ecthasanamplitudewhichistypicallyanorder-ofmagnitudesmallerandhasnotyetbeende nitivelydetected(seeRephaeli&Lahav1991andHaehnelt&Tegmark1995,regardingprospectsforafuturedetection).

IthaslongbeenrealizedthatameasurementofthethermalSZe ect,combinedwithX–rayobservations,canbeusedtoestimatethedistancetotheclusterandhencetheHubbleconstant,H0,undertheassumptionthattheclusterisspherical(Cavaliere,Danese,&DeZotti1977,Gunn1978,Silk&White1978,Birkinshaw1979).TheinferredvalueoftheHubbleconstantisinverselyproportionaltothesquareoftheSZtemperaturedecrement.ThisapproachhadledtovaluesoftheHubbleconstantwhicharetypicallyonthelowsideoftherangeinferredfromothermethods(see,e.g.,Table2inRephaeli1995).Anoftencitedsystematice ectthatcouldaccountforthisbiasiselongationoftheselectedclustersalongtheline-of-sight.InthisLetter,weexploreadi erente ectwhichleadstoasystematicbiastowardslow–H0valueseveniftheseclustersareperfectlyspherical.Thee ectresultsfromgravitationallensingbytheclusterpotentials.

MeasurementsofthedecrementintheRayleigh–Jeans(RJ)temperatureofthemicrowavebackgroundduetothethermalSZe ectareroutinelyaccompaniedbytheremovalofbackgroundradiosourcesdowntosome uxthreshold(see,e.g.,Birkinshaw,Hughes,&Arnaud1991).Inthisprocess,itisimplicitlyassumedthatthe uxthresholdfortheremovalofsourcesbehindtheclustercoreisthesameasinacontrol eldfarfromtheclustercenter.However,thisassumptionisnotstrictlytrueduetotheinevitablemagni cationbiaswhichisintroducedbythegravitationallensinge ectoftheclusterpotential.Inreality,theclusteractsasalenswhichmagni esandthusresolvessourcesthatareotherwisebelowthedetectionthreshold.Theresidualintensityofunresolvedsourcesisthereforesystematicallylowerbehindtheclustercore,ascomparedtothatinthecontrol eld.Lensingarti ciallyincreasesthe uxde citbehindtheclustercoreandthusleadstoasystematicunderestimateoftheHubbleconstant.

InthisLetterwecalculatethee ectoflensingonSZmeasurementsoftheHubbleconstant.Ourdiscussiononlensingfollowscloselytheapproachdevelopedinanearlierpaper(Refregier&Loeb1996,hereafterRL)whichfocusedonlensingoftheX–raybackgroundbygalaxyclusters;theinterestedreadershouldconsultthisearlierpaperformoredetails.Here,wedescribeourmodelsforthebackgroundpopulationofradiosourcesandfortheclusterpotentialin§2.Wethenshowin§3howthelensinge ectleadstoasystematicdecrementintheintensityofunresolvedsources.In§4,wepresentnumericalresultsfordi erentvaluesofourmodelparametersandforthespeci cexampleofA2218.Finally,§5summarizesthemainconclusionsofthiswork.

2.Model

WemodelthegravitationalpotentialoftheclusterasaSingularIsothermalSphere(SIS)(e.g.Schneideretal.1992).Thismodelprovidesagood rst–orderapproximationtotheprojected

The Sunyaev-Zel'dovich (SZ) effect of a cluster of galaxies is usually measured after background radio sources are removed from the cluster field. Gravitational lensing by the cluster potential leads to a systematic deficit in the residual intensity of unr

massdistributionofknownclusterlenses(Tyson&Fischer1995,Narayan&Bartelmann1996,Squiresetal.1996a,b).TheSISpotentialcausesbackgroundsourcestoappearbrighterbutdilutedontheskybythemagni cationfactor

αµ(θ)= 1

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