Constraint On The Cosmological Constant From Gravi

arXiv:astro-ph/9803128v111Mar1998ConstraintOnTheCosmologicalConstantFromGravitationalLensesInAnEvolutionaryModelOfGalaxiesDeepakJain∗,N.Panchapakesan†,S.Mahajan‡andV.B.Bhatia§DepartmentofPhysicsandAstrophysicsUniversityofDelhi,Delhi-110007,IndiaAbstractWestudytheeffectofthecosmologicalconstantonthestatis-ticalpropertiesofgravitationallensesinflatcosmologies(Ω0+λ0=1).Itisshownthatsomeofthelensobservablesarestronglyaffectedbythecosmologicalconstant,especiallyinalow–densityuniverse,anditsexistencemightbeinferredbyastatisticalstudyofthelenses.Inparticular,theopticaldepthofthelensdistributionmaybeusedbestforthispurposewithoutdependingmuchonthelensmodel.Wecalculatetheopticaldepth(probabiltyofabeamencounteringwithalensevent)forasourceinanewpictureofgalaxyevolutionbasedonnumberevolutioninadditiontopureluminosityevolution.Itseemsthatpresentdaygalaxiesresultfromthemergingofalargenumberofbuildingblocks.Wehavetriedtoputlimitonthecosmological∗E–mail:[e-mail]†E–mail:[e-mail]‡E–mail:[e-mail]§E–mail:[e-mail]-tionarymodelofgalaxiespermitslargervalueofcosmologicalconstant.21IntroductionIntheearlyhistoryofmoderncosmologythecosmologicalcon-stantΛwasinvokedtwice.FirstbyEinsteintoobtainstaticmodelsoftheuniverse.NextbyBondi,GoldandHoyletore-solveanagecrisisandtoconstructtheuniversethatsatisfiedthe“PerfectCosmologicalPrinciple”,i.e.,thehypothesisthattheuniverseappearsthesameatalltimesandplaces.Inbothinstancesthemotivatingcrisispassedandthecosmologicalcon-stantwasputaside.Howeverthecosmolgicalconstantremainsafocalpointofcosmologyandofparticletheory.Theformerbecausetodayanunderstandingofawiderangeofobservationsseemtocallforacosmologicalconstant.Thelatterbecauseinthecontextofquantum–fieldtheoryacosmologicalconstantcorrespondstotheenergydensityassociatedwiththevacuumandnoknownprincipledemandsthatitvanish.Variousaspectsofrecentobservationssuggestreconsidera-tionofanonvanishingcosmologicalconstant(Krauss&Turner1995).Theseincludestheageofuniverseonceagain,theforma-tionoflarge–scalestructure(galaxies,clustersofgalaxies,super-clusters)andthemattercontentoftheuniverseasconstrainedbydynamicalestimates,BigBangNucleosynthesisandX–Rayobservationsofclustersofgalaxies.OneofthepossibilitiesofdetectingΛisthegravitationallensingtechnique.Tousethegravitationallensingasatoolforthedeterminationofcosmologicalparameterseitherbyade-tailedstudyofspecificlenssystemsorthroughstatisticalanal-ysisofasampleoflenseshasbeenfrequentlydiscussed(Refsdal31964;Press&Gunn1973).Ithasbeenpointedoutthattheexpectedfrequencyofmultipleimaginglensingeventsisquitesensitivetocosmologicalconstant(Fukugita,Futamse&Kasai1990;Turner1990).ToputalimitonΛ,weneedtofirstcal-culatetheexpectednumberofmultipleimagegravitationallenssystems(producedbytheknowngalaxypopulation)tobeex-pectedinaparticularquasarsamplewithaknowndistributionofredshifts.Thisthenhastobecomparedwiththeobservedfrequencyoflenssystemsfound.Wehaveusedaunifyingmodelofgalaxyformationwhichcananswertwoquestionsraisedbytherecentdataonhigh-redshiftgalaxies.Theseconcerntheagesof,andstarformationhistoryinthedistantradiogalaxies,andthenatureofthelargenumberoffieldgalaxiesrevealedbyfaintgalaxycounts.Thisnewmodelisbasedonthestrongnumberevolutioninadditiontopureluminosityevolutionofthegalaxies(Volmerange&Guiderdoni1990).InSection2,wewritedowntheNewLuminosityFunction(NLF)whichhasstrongnumberevolutioninadditiontopureluminosityevolution.InSection3,wepresentanewcalculationofthetotalmultipleimagelensingcross–sectionsforagalaxyintheSingularIsothermalSphere(SIS)approximation(Turner,Ostriker&Gott1984)(hereafterTOG)basedonthenewgalaxyluminosityfunction,velocity–luminositycorrelationandveloc-itydispersion.InSection4wewritedownthebasicequationsforthestatisticalpropertiesoflensesforeachofthegalaxymod-els.ThequasarluminosityfunctionfortheBSPsample(Boyle,4Shanks&Peterson1988)isdescribedinSection5.Section6containsthenumberoflensedquasarsintheBSPsampleforthePress–SchecterLuminosityFunction(PSLF)andtheNewLuminosityFunction(NLF).InSection7wediscusstheresults.2NewLuminosityFunctionIn1990,VolmerangeandGuiderdoni,proposedaunifyingmodeltoexplainfaintgalaxycountsaswellasobservationalpropertiesofdistantradiogalaxies.Thisnewmodelofgalaxyevolutionisbasedonnumberevolutioninadditiontopureluminosityevolution.Presentdaygalaxiesresultfromthemergingofalargenumberofbuildingblocksandthecomovingnumberofthesebuildingblocksevolvesas(1+z)1.5.ItisarguedthatthepresentluminosityfunctionisthewellknownPress-SchecterLuminosityFunction(PSLF)Φ(L,z=0)=φ∗(L/L∗)αexp(−L/L∗)dL/L∗withL∗beingthecharactersticluminosityatthekneeandφ∗acharactersticdensity.Thesevaluesarefixedinordertofitthecurrentluminositiesanddensitiesofgalaxies.Thenathighz,thecomovingnumberdensityfollowstheNewLuminosityFunction(NLF)Φ(L,z)dL=(1+z)2ηΦ(L(1+z)η,0)dLItisseenthatthevalueη=1.5givesafairfittothedataonhighredshiftgalaxies.Thefunctionalformhasthefollowingproperties:5(i)Self–similarityassuggestedbytheclassicalPress-Schecter(1974)