Simulations of weak gravitational lensing

arXiv:astro-ph/0312133v213May2004SimulationsofWeakGravitationalLensingMartinWhitea,b,1,ChrisValeb,2aDepartmentofAstronomy,UniversityofCalifornia,Berkeley,CA,94720bDepartmentofPhysics,UniversityofCalifornia,Berkeley,CA,94720AbstractWedescribethesimulationdataproducedbyapilotprogrammetocomputemockweakgravitationallensingmapsforarangeofcurrentlypopularcosmologicalmod-elsbyraytracingthroughhigh-resolutionN-bodysimulations.Theprogrammerequiredonlyamodestinvestmentincomputertimetoproducemapsaccuratetoarcminutescalescoveringhundredsofsquaredegreesofskyfor4cosmologicalmodels.Keywords:Cosmology,Lensing,Large-ScalestructuresPACS:98.65.Dx,98.80.Es,98.70.Vc1IntroductionWeakgravitationallensingbylarge-scalestructurehasbecomeanindispens-abletoolformoderncosmology,usedalreadytosetconstraintsonthemassdensity(Ωmat,inunitsofthecriticaldensity)andthefluctuationamplitude(σ8)(seee.g.vanWaerbeke&Mellier,2003;Hoekstra,Yee,&Gladders,2002,forthecurrentstatus)andtoutedforitspotentialtoconstrainclusterscalingrelations(Huterer&White,2002)anddarkenergy(Benabed&Bernardeau,2001;Huterer,2002;Hu,2002;Heavens,2003;Abazajian&Dodelson,2003;Refregier,2003;Jain&Taylor,2003;Bernstein&Jain,2004;Takada&Jain,2004;Takada&White,2004).Liketheanisotropiesinthecosmicmicrowavebackground(CMB),thetheoryofweakgravitationallensingiswellunderstood(Mellier,1999;Bartelmann&Schneider,2001).UnliketheCMBhoweverthecalculationinvolvesmodelingthenon-linearevolutionofthemassintheuni-verse.Thismakesthepredictionsofweaklensingveryrich,butalsomeansthatanaccuratetreatmentrequiresnumericalsimulations.1E-mail:[e-mail]2E-mail:[e-mail]sevierScience2February2008Inthispaperwegivesomedetailsofapilotprogramdesignedtoprovideweaklensingray-tracingsimulationsofasmallgridofcosmologicalmodelsinthecurrentlyfavoredregionofparameterspace.Thesemodelscanbeusedbyobserverswishingtotesttheiralgorithmsorfittotheirdataandbytheoristswishingtotestorcalibratefastandflexible,butapproximate,methodsofcalculation.Theinitialgridissmall,buttheprocessofitscreationisalmostentirelyautomaticallowingittobeexpandedsimplyastheneedarises.Thispaperfocusesonthecreationofthemodelgrid.Wedescribethechoiceofcosmologicalparametersin§2.1,theN-bodysimulationsin§2.2andtheraytracingandnumericalissuesin§2.3and§2.4.Wegivesomerepresentativeresultsin§3beforeconcludingin§4.2Cosmologicalmodels2.1ChoosingparametersWechooseasmallnumberofcosmologicalmodelswhichprovidegoodfitstothecurrentCMBandlarge-scalestructuredata.Forsimplicityallofthemodelsinthepilotprogramarevariantsofthecolddarkmattermodelwithadarkenergycomponent,andtheirparametersareshowninTable1.TheparametervariationsaroundabasemodelaredesignedtokeeptheCMBfluctuationsalmostinvariant,inanticipationofincreasinglyprecisedatafromWMAPandPlanck,andtheentireprocessisautomatedusingPerlscriptsandCcode.Webeginwitharelativelysmallnumberofmodelstodemonstratethecostandfeasibilityofmakingsuchgrids.Aswegainexperienceinusingthesedataproductsandunderstandthedriversbetterwecanextendthemodelgridand/orincreasethefidelityofthesimulations.Areourprioritiestohavelargermaps?higherresolution?moreredshiftrange?more‘sky’?etc.Wefirstpickthephysicalmatterdensityωmat≡Ωmath2,the(comoving)distancetolastscattering,dls,anda(constant)equationofstateofthedarkenergy,w.Weapproximatetheredshiftoflastscatteringasz=1080,ignoringtheslightmatterandbaryondensitydependence.ThisthenallowsustosolvefortheHubbleconstant,h,andthusΩmatandΩdeassumingspatialflatness.Becausetheyarereasonablywellknown,comparedtosomeotherparameters,wefixωmat=0.145anddls=13.7Gpcforallofthemodelsinourgrid.ThesenumbersareclosetothebestfitforarecentanalysisofWMAPandSDSSdata(Tegmark,2003).2ModelΩmatwhnσ8τχ21&20.296-1.00.701.000.930.159773&40.357-0.80.641.000.880.159755&60.296-1.00.700.950.850.109797&80.357-0.80.640.950.810.10976Table1Parametersforthemodelsrun.Foreachcosmologicalmodeltwoindependentsetsofinitialconditionsaregenerated.Allmodelsarespatiallyflat,soΩde=1−Ωmat.Forallmodelsthematterdensityisωmat=0.145andthebaryondensityωb≡Ωbh2wasfixedat0.023.Ourresultsareveryinsensitivetothislatterchoice.Allmodelshavepower-lawspectra(norunning)withindexnandthedarkenergyhasaconstantequationofstatew.Thenormalization,σ8,isfromafittotheWMAPTTpowerspectrumdataandtheχ2isforthisfitwith893degreesoffreedom.Thenextstepistospecifytheothermodelparameters,forexampletheopticaldepthtoThomsonscattering,τ,thespectralindex,nandthebaryondensityωb.(TheopticaldepthentersprimarilythroughitseffectonthenormalizationofthepowerspectrumwhenwefittoWMAP.)Wecurrentlyholdthenumberoflightneutrinosfixed,andincludenomassiveneutrinos.Wedealwithpurepower-lawspectrawithnorunningspectralindex.Ourprimaryvariationsareintheopticaldepth,thespectralindexandtheequationofstateofthedarkenergy(seeTable1).Thefirsttwoaffecttheamplitudeandshapeofthedensityorpotentialfluctuationsatlatetimeswhilewisaparameterofgreatinteresttothecosmologycommunity.Weuseafixedequationofstateinthisinitialsurvey,thoughitiseasytoincludeanyknownfunctionalforminthefuture.Foraspecifiedmodelwecomputethe