Semi-Analytical Models for Lensing by Dark Halos I. Splittin

We use the semi-analytical approach to analyze gravitational lensing of quasars by dark halos in various cold dark matter (CDM) cosmologies, in order to determine the sensitivity of the prediction probabilities of images separations to the input assumption

Semi-AnalyticalModelsforLensingbyDarkHalos:

I.SplittingAngles

Li-XinLiandJeremiahP.Ostriker

PrincetonUniversityObservatory,Princeton,NJ08544–1001,USA

arXiv:astro-ph/0010432v3 7 Jun 2001E-mail:lxl,jpo@astro.princeton.eduABSTRACTWeusethesemi-analyticalapproachtoanalyzegravitationallensingofremotequasarsbyforegrounddarkhalosinvariouscolddarkmatter(CDM)cosmolo-gies,inordertodeterminethesensitivityofthepredictionsforprobabilitiesofimagesseparationstotheinputassumptionsregardingthepropertiesofhalosandcosmologicalmodels.Thepowerspectrumofprimordial uctuationsisnor-malizedbytheclusterabundanceconstraints.ThemassfunctionofdarkhalosisassumedtobegivenbythePress-Schechterfunction.Themassdensitypro leofdarkhalosisalternativelytakentobethesingularisothermalsphere(SIS),theNavarro-Frenk-White(NFW)pro le,orthegeneralizedNFWpro le.Thecosmologiesbeingconsideredinclude:theEinstein-deSittermodel(SCDM),theopenmodel(OCDM),andthe atΛ-model(LCDM).Asexpected,we ndthatthelensingprobabilityisextremelysensitivetothemassdensitypro leoflenses(darkhalos),andsomewhatlesssotothemeanmassdensityintheuniverse,andtheamplitudeofprimordial uctuations.NFWhalosareverymuchlesse ectiveinproducingmultipleimagesthanSIShalos.ForNFWlenses,theSCDMmodel

produceslensingeventsfewerthantheOCDM/LCDMmodelsbytwoordersofmagnitude.ForSISlenses,theSCDMmodelproducesmorelensingeventswithsmallsplittinganglesbutproducesfewerlensingeventswithlargesplittinganglesthantheOCDM/LCDMmodels,whichisduetothefactthatforlargemasshalosthePress-Schechterfunctionisverysensitivetotheamplitudeofpri-mordial uctuations.Inallcasesthedi erencebetweentheOCDMmodelandtheLCDMmodelisnotdramatic.Noneofthesemodelscancompletelyexplainthecurrentobservations:theSISmodelspredicttoomanylargesplittinglenses,whiletheNFWmodelspredicttoofewsmallsplittinglenses.Essentially,theobservedhighratioofsmallsplittingtolargesplittinglensesisnotpredictedcorrectly.Thisindicatesthattheremustbeatleasttwopopulationsofhalosintheuniverse:smallmasshaloswithasteepinnerdensityslopeandlargemass

We use the semi-analytical approach to analyze gravitational lensing of quasars by dark halos in various cold dark matter (CDM) cosmologies, in order to determine the sensitivity of the prediction probabilities of images separations to the input assumption

haloswithashallowinnerdensityslope.AcombinationofSISandNFWhaloscanreasonablyreproducethecurrentobservations,ifwechoosethemassforthetransitionfromSIStoNFWtobe～1013solarmassesasmightplausiblyoccurduetobaryoniccoolingandcontractioninlowermasssystems.Additionally,thereisatendencyforCDMmodelstohavetoomuchpoweronsmallscales,i.e.toomuchmassconcentration.Fromoursensitivitystudiesitappearsthatthecuresproposedforotherapparentdi cultiesofCDMwouldhelphereaswell,anexamplebeingthewarmdarkmatter(WDM)variantwhichisshowntoproducelargesplittinglensesfewerthanthecorrespondingCDMmodelbyoneorderofmagnitude.

Subjectheadings:cosmology:gravitationallensing—galaxies:clusters:general—galaxies:halos

1.Introduction

Gravitationallensingdirectlyprobesthemassdistributionintheuniverse,sotheinves-tigationoflensingeventsofquasarsathighredshiftscanprovideuswithimportantinfor-mationaboutcosmologyandthematterdistributionintheuniverse.Anumberofgroupshaveattemptedtotestcosmologicalmodelsbycomparinglensingprobabilitiespredictedbyvariouscosmologicalmodelswiththoseobtainedfromobservations(Turner,Ostriker,&Gott1984;Narayan&White1988;Cenetal.1994;Wambsganssetal.1995;Maozetal.1997;Wambsganss,Cen,&Ostriker1998;Bartelmannetal.1998;Mortlock&Webster2000;Porciani&Madau2000;Keeton&Madau2001).Inthestatisticalstudyofcosmo-logicalgravitationallensing,twodi erentapproachesareused:oneistostudythelensingprobabilityofsplittinganglesofmultipleimages(Turner,Ostriker,&Gott1984;Narayan&White1988;Cenetal.1994;Kochanek1995;Wambsganssetal.1995;Wambsganss,Cen,&Ostriker1998;Porciani&Madau2000;Keeton&Madau2001),theotheristostudythelensingprobabilityofthelength-to-widthratioofarcsformedbygravitationallensing(Bartelmann&Weiss1994;Bartelmannetal.1998;Cooray1999;Kaufmann&Straumann2000;Meneghettietal.2000).Thetwoapproachesre ecttwodi erentaspectsofgravi-tationallensingandarecomplimentarytoeachother,bothofthemaredirectlyrelatedtoobservationsanddeservedetailedinvestigations.Inthispaperwefocusonthesplittinganglesofmultipleimagescausedbygravitationallensing.Thecaseoflength-to-widthratioofarcswillbeconsideredinanotherpaper.Resultsoftheworkdoneintestingcosmologi-calscenariostodateareinconclusivewithnoclearpreferencefoundforanyofthecurrentmodels,and,overall,onecouldconcludethatnoneofthestudiedmodelsprovidesagood

We use the semi-analytical approach to analyze gravitational lensing of quasars by dark halos in various cold dark matter (CDM) cosmologies, in order to determine the sensitivity of the prediction probabilities of images separations to the input assumption

ttotherapidlyimprovingobservations.Perhapsthereasonforthisindeterminacyisthatlensingalsoprobessmallscalestructure.Whilethisisknown,itsimportancehasperhapsnotbeenappreciatedsu ciently.Gravitationallensingprovidesanexquisitelysensitivetesttothehighfrequencypartofthepowerspectrum,becauseitisthispartwhichestablishesthecentralpro lesofdarkmatterstructures.Sincelensingiscomparablysensitivetoinputassumptionsconcerningthepropertiesoflensesandofcosmologicalmodels,itishazardoustouseobservedlensingstatisticstodrawinferenceswithregardtocosmologybeforedeter-miningthesensitivitytootherfactors.Theprimarypurposeofthispaperistoquantitativelyassessthesensitivityoflensingexpectationstothevariousinputparameterssothatthistoolcanbeusedtogreateste ect.

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