Statistical properties of the convergence due to w

arXiv:astro-ph/9911336v212May2000A&Amanuscriptno.(willbeinsertedbyhandlater)Yourthesauruscodesare:Sect.02(12.03.4;12.07.1;12.12.1)ASTRONOMYANDASTROPHYSICSStatisticalpropertiesoftheconvergenceduetoweakgravitationallensingbynon-linearstructuresPatrickValageasServicedePhysiqueTh´eorique,CEASaclay,91191Gif-sur-Yvette,FranceReceived/AcceptedAbstract.Densityfluctuationsinthematterdistribu-tionleadtodistortionsoftheimagesofdistantgalaxiesthroughweakgravitationallensingeffects.Thisprovidesanefficientprobeofthecosmologicalparametersandofthedensityfield.Inthisarticle,weinvestigatethestatisti-calpropertiesoftheconvergenceduetoweakgravitationallensingbynon-linearstructures(i.e.weconsidersmallan-gularwindowsθ∼1′).Previousstudieshaveshownhowtorelatethesecondandthirdordermomentsofthecon-vergencetothoseofthedensitycontrastwhilemodelsbasedonthePress-Schechterprescriptionprovideanesti-mateofthetailofP(κ).Herewepresentamethodtoob-tainanestimateofthefullp.d.f.oftheconvergenceκ.Itisbasedonarealisticdescriptionofthedensityfieldwhichappliestooverdenseaswellasunderdenseregions.WeshowthatourpredictionsagreeverywellwiththeresultsofN-bodysimulationsfortheconvergence.Thiscouldal-lowonetoderivethecosmologicalparameters(Ωm,ΩΛ)aswellasthefullp.d.f.P(δR)ofthedensitycontrastitselfinthenon-linearregimefromobservations.Hencethisgivesaverypowerfultooltoconstrainscenariosofstructureformation.Keywords:cosmology:theory-gravitationallensing-large-scalestructureofUniverse1.IntroductionOneimportantgoalofcosmologyistoobtaintheproper-tiesofthedarkmatterdensityfieldwhicheventuallyledtotheformationofgalaxies,clustersandotherastrophys-icalobjectsweobservetoday.Thetraditionalmethodtoobtainsomeobservationalconstraintsonthedistributionofmatteristobuildlargesurveysofgalaxies,whichmapthedistributionoflight,andtouseatheoreticalmodeltolinkthesegalaxiestothedensityfield.However,thisindirecttechniquepresentsthedisadvantagetointroducesignificantuncertaintiesduetotherelationbetweenmassandlightoneneedstointroduce.Henceitisclearlyim-portanttobuildotherindependentmethodswhichprobeinamoredirectfashionthedistributionofmatter.Suchatoolisprovidedbythegravitationaldistortionoflightrayscomingfromverydistantsources.Indeed,thedensityfluctuationsalong(orcloseto)thelineofsightamplifyandsheartheimagesofdistantgalaxies.Then,onecanmea-sureforinstancetheshearinducedbyweakgravitationallensingfromtheobservedellipticitiesofthesegalaxies,ifweassumethatthesourcegalaxyellipticitiesareuncor-related.Sincesucheffectsarepurelygravitationaltheyprobethetotalmattercontentoftheuniverse,darkmat-teraswellasbaryonicmatter.Inparticular,thestatisti-calpropertiesoftheconvergenceortheshearinducedbygravitationallensingcanbedirectlylinkedtothecharac-teristicsofthedensityfield.Manyauthorshavealreadyconsideredthisproblem,focusingespeciallyonthesecondandthirdordermomentsinthequasi-linearregimerele-vantforlargeangularwindows(e.g.,Blandfordetal.1991;Miralda-Escude1991;Kaiser1992;Bernardeauetal.1997;vanWaerbekeetal.1999).Theadvantageofsuchlargeangularwindows(θ∼10′)isthatonecanuserigorousperturbativeresultstodescribethepropertiesoftheden-sityfield(e.g.,Bernardeau1994).However,smallerangu-larscalesalsopresentastronginterestastheyprobethenon-linearregimewhichisdirectlylinkedtoastrophysicalobjectslikegalaxies,Lyman-αclouds,...Thus,theycouldprovideacheckonthescenariosusedtodescribestructureformationaswellasthebuildingofgalaxies.Moreover,goingtosmallerscalesisalsoanobservationaladvantagesincethesignalismucheasiertomeasurebecauseitbe-comeslargerwhilethesystematicsremainbasicallythesame(e.g.,vanWaerbekeetal.1999).Unfortunately,thenon-linearregimeismuchmoredif-ficulttohandletheoretically.Thus,previoustheoreticalstudieswererestrictedtosecondorthirdordermoments,usingfitstotheresultsofN-bodysimulationsforthenon-linearevolutionofthepower-spectrum(Jain&Seljak1997;Hui1999),ordescribedthedensityfieldasacollec-tionofvirializedhalos,obtainedfromthePress-Schechterprescription(Press&Schechter1974),whichallowsonetomodelthetailofthedistributionoftheconvergence(e.g.,Porciani&Madau1999).However,theseworksdidnotprovidearealisticdescriptionofthefullprobabilitydistri-butionsoftheconvergence.Indeed,thisrequiresaconsis-2P.Valageas:Statisticalpropertiesoftheconvergencetentmodelforthedensityfieldwhichcandescribeunder-denseaswellasoverdenseregions.Ontheotherhand,sev-eralnumericalstudieshavebeenperformedusingN-bodysimulations(e.g.,Jainetal.1999).Inthisarticle,extend-inganearlierworkrelevantforpointsources(Valageas1999a),wepresentananalyticmethodtoobtainthep.d.f.oftheconvergence.Thisisbasedonascalingmodeldevel-opedinBalian&Schaeffer(1989)(seealsoBernardeau&Schaeffer1992;Valageas&Schaeffer1997)whichhasbeenseentoprovideagooddescriptionofthenon-linearden-sityfieldthroughcomparisonswithnumericalsimulations(e.g.,Valageasetal.2000;Bouchetetal.1991;Munshietal.1999).Thisarticleisorganizedasfollows.InSect.2webrieflypresentthemodelweuseforthedensityfieldandwein-troduceafewstatisticaltools.Next,inSect.3wedescribehowweobtainthep.d.f.oftheconvergenceκ,smoothedonsmallangularsc