Can Strong Gravitational Lensing Constrain Dark En

arXiv:0707.1730v1[astro-ph]12Jul2007CanStrongGravitationalLensingConstrainDarkEnergy?SeokcheonLee1andKin-WangNg1,21InstituteofPhysics,AcademiaSinica,Taipei,Taiwan11529,R.O.C.2InstituteofAstronomyandAstrophysics,AcademiaSinica,Taipei,Taiwan11529,R.O.C.AbstractWediscusstheratiooftheangulardiameterdistancesfromthesourcetothelens,Dds,andtotheobserveratpresent,Ds,forvariousdarkenergymodels.ItiswellknownthatthedifferenceofDssbetweenthemodelsisapparentandthisquantityisusedfortheanalysisofTypeIasupernovae.Howeverweinvestigatethedifferencebetweentheratiooftheangulardiameterdistancesforacosmologicalconstant,(Dds/Ds)Λandthatforotherdarkenergymodels,(Dds/Ds)otherinthispaper.Ithasbeenknownthatthereislensmodeldegeneracyinusingstronggravitationallensing.Thus,weinvestigatethemodelindependentobservablequantity,Einsteinradius(θE),whichisproportionaltobothDds/Dsandvelocitydispersionsquared,σ2v.Dds/Dsvaluesdependontheparametersofeachdarkenergymodelindividually.However,(Dds/Ds)Λ−(Dds/Ds)otherforthevariousdarkenergymodels,iswellwithintheerrorofσvformostoftheparameterspacesofthedarkenergymodels.Thus,asinglestronggravitationallensingbyuseoftheEinsteinradiusmaynotbeapropermethodtoinvestigatethepropertyofdarkenergy.However,betterunderstandingtothemassprofileofclustersinthefutureorothermethodsrelatedtoarcstatisticsratherthanthedistancesmaybeusedforconstraintsondarkenergy.1IntroductionRecentobservationsofhighredshiftTypeIasupernovae(SNeIa)suggestedthattheexpansionoftheUniverseiscurrentlyaccelerating[1].Thecosmicmicrowavebackground(CMB)anisotropydata,indicatingaspatiallyflatuniverse[2]containingalowvalueforthecolddarkmatter(CDM)densityparameter[3],hasconfirmedthattheUniverseisdominantlymadeupofacomponentwithnegativepressure(darkenergy)tomakeupthecriticaldensitytoday.Thecosmologicalconstantand/oraquintessencefieldarethemostcommonlyacceptedcan-didatesfordarkenergy.Althoughthecosmologicalconstantissimpleandfavoredbycurrentcosmologicalobservations,thereis50to120ordersofmagnitudediscrepancybetweentheoryandthemeasuredvalue[4].Thequintessence,whichmightalleviatethisproblemisadynamicalscalarfieldleadingtoatimedependentequationofstateparameter[5].Variousscalarfieldpotentialsforthequintessencehavebeeninvestigated[6].ItisimportanttousevariouswaysofcheckingfortheexistenceofdarkenergyinadditiontoSNeIaandCMBanisotropyconstraintsondarkenergy.Anumberofothertestshavebeenconsideredincluding“geometric”testsusingstandardcosmologicalmethods(thegalaxyclustergasmassfraction[7],thelocationofCMBpeaks[8,9],theredshift-angularsize[10],thestronggravitationallensing[11]-[14],fluctuationsoftheluminositydistance[15],etc.).Thestatisticsofgravitationallensingofquasars(QSOs)byinterveninggalaxiescanconstrainonthecosmologicalconstant[12].Lensedimagesofdistantgalaxiesincluster,arcsorrings,mayprovideaboundontheequationofstateparameterofdarkenergy[13].WhileSNeIaisusedtodeterminetheluminositydistanceitself,agravitationallensingsystemcanbeusedmeasuretheratioofangulardiameterdistances.Thus,thegravitationallensingsystemisregardedasanindependenttoolthatcomplementsSNeIaasaprobeofdarkenergy[14].However,thelensingobservationsprimarilydependontheparametersoflensmodelswithminordependenceoncosmologicalparameters[16].Thereisthelensmodeldegeneracyinboththeprojectedmassdensityprofileandthecircularvelocityprofile.ItisshownthatweneedtomeasuretheEinsteinradiusandthevelocitydispersionwithinO(1)%accuracyinordertoputaconstraintonωDE.Inthegravitationallensing,oneoftheobservablequantitieswithouthavinganymodeldepen-denceistheEinsteinradius(θE),whichisproportionaltothevelocitydispersionsquared(σ2v)andtheratiooftheangulardistancesDds/Ds,whereDdsisthedistancefromthelenstothesourceandDsisthatfromthesourcetotheobserver.Withdifferentvaluesofcosmologicalparameters,wecanhavedifferentvaluesofDds/Ds,i.e.differentvaluesofθE.Thus,itmightbeusedforprobingthepropertyofdarkenergy,ωDE.However,thereisanambiguityinmeasuringσv.IftheerrorofσvmeasurementisnotwithinthedifferencesofDds/Dsbetweendifferentcosmologicalmodels,thenwecannotdistinguishthedifferencesbetweenmodelsbymeasuringθE.Thispaperisorganizedasfollows.Inthenextsectionwereviewthegravitationallensing1NSSsI′ISdS0ˆαβθDdsDdDsLOFigure1:Agenerallensingsystem.Listhecenterofthelens,andthelinethroughLandtheobserverOistheopticalaxis.βisanunperturbedangularpositionofthesourcerelativetothat.ˆαisthedeflectionangleofalightray,thusanimageofthesourceisobservedatpositionθ.Howeverallanglesareverysmall,wecanreplacethereallightray(SI′O)byitsapproximationSIO.systemwiththebasicequationsusedinlensingobservations.Wealsobrieflymentionthemostpopularlensmodels.WereviewthevariousaspectsoferrorsinmodelinglensinSec.3.InSec.4,wecheckboththedifferencesofDsandthedifferencesofDds/Dsbetweenthecosmologicalconstantandotherdarkenergymodels.Ourconclusionisinthelastsection.2GravitationalLensingandIsothermalGalaxyModelsFigure1showsasimplelensingsystem[17].ConsiderthesourcesphereSs,i.e.aspherewithradiusDs,centeredattheobserverOandthedeflectorsphereSdwithradiusDd,i.e.thedistancetothecenterofthelensL.Inaddition,considerthe