Gravitational Lensing of Supernovae Type Ia by Pse

arXiv:astro-ph/0608153v17Aug2006GravitationalLensingofSupernovaeTypeIabyPseudoEllipticNFWHaloesHamedBagherpour1HomerL.DodgeDepartmentofPhysicsandAstronomy,UniversityofOklahoma,Norman,OK73019,USAABSTRACTWepresenttheeffectsofellipticityofmatterdistributioninmassivehalosontheobservationofsupernovae.ApseudoellipticalNavarro-Frenk-White(NFW)massmodelisusedtocalculatetheintroducedgainfactorsandobservationratesoftypeIasupernovaeduetothestronglensing.Weinvestigatehowandtowhatextenttheellipticityinmassdistributionofthedeflectinghaloscanaffectsurveyslookingforcosmologicallydistantsupernovae.Weusehalomassesof1.0×1012h−1M⊙and1.0×1014h−1M⊙atredshiftszd=0.2,zd=0.5,andzd=1.0,withellipticitiesofuptoǫ=0.2.Subjectheadings:gravitationallensing—supernovae:general1.IntroductionSupernovaehaveemergedasthemostpromisingstandardcandles.Duetotheirsignifi-cantintrinsicbrightnessandrelativeubiquitytheycanbeobservedinthelocalanddistantuniverse.Observationaleffortstodetecthigh-redshiftsupernovaehaveprovedtheirvalueascosmologicalprobes.Thesystematicstudyandobservationofthesefaintsupernovae(mainlytypeIa)hasbeenutilizedtoconstrainthecosmicexpansionhistory(Goobar&Perlmutter1995;Perlmutteretal.1999;Schmidtetal.1998).Lightemittedfromanyce-lestialobjectissubjecttolensingbyinterveningobjectswhiletraversingthelargedistancesinvolved(Kantowski,Vaughan,&Branch1995)andthefartherthelightsource,thehigheritschanceofbeingsignificantlylensed.Apartfromthefactthatgravitationallensingcanlimittheaccuracyofluminositydistancemeasurements(Perlmutter&Schmidt2003),[e-mail]–2–Studyingsupernovaeandtheirratesathighredshiftsprovideuswithmuchneededinformationforconstrainingthemeasurementsoftheellusivedarkenergy,aswellasunder-standingthecosmicstarformationrateandmetalenrichmentathighredshifts.Inordertoobserveand,hence,studythefainthigh-redshiftsupernovae,onecanraisethechanceofobservationbylookingthroughclustersofgalaxiesorevenmassivegalaxies(seeSmail,etal.(2002)andthereferencestherein).These‘gravitationaltelescopes’amplifythehigh-redshiftsupernovaeandtherebyincreasethechanceoftheirdetection.However,thisboostinob-servationisoffsetbythecompetingeffectofdepletion(Fig.1),duetothefieldbeingspreadbythedeflector(amplificationbias).Foranassumedlensmodelandagivenfieldofviewitisnotobviouswhicheffectdominatestheobservationofsupernovaethroughthehalo.Thenetresultdependsonthedeflectorandsourceparametrsaswellastheobservationalsetup(Gunnarsson&Goobar2003).Someresearchhasbeenconductedonthefeasibilityofobservingsupernovaethroughclusterofgalaxies(see,forinstance,Saini,Raychaudhary,&Shchekinov2002;Gal-Yam,Maoz,&Sharon2002;Gunnarsson&Goobar2003).Thesestudieshavenottakenintoac-counthowthemorphology(mainlytheellipticity)oftheseclustersasgravitationaltelescopescouldchangetheexpectedsupernovarate.Inthispaper,weinvestigatewhetherintroduc-ingellipticityintothemassdistributionofthedeflectinghaloscanaffecttheobservationofsupernovae.Forthispurpose,weuseapseudoellipticalNavarro-Frenk-White(NFW)halomodelwithdifferentvaluesofellipticity.Throughoutthepaperweassumetheso-calledconcordancecosmologywhereΩm=0.3,ΩΛ=0.7,andh100=0.67,withh100=H0/100kms−1Mpc−1.In§2webrieflygoovertheNFWmodelandshowhowananalyticalformalismforapseudoellipticalNFWmassprofilecanbeintroduced.Stronglensingbythindeflectorsaswellasthewayellipticitycanafffecttheamplificationisexplainedin§3.Wepresentanddiscusstheresultsofourcalculationsin§4.2.TheNFWHaloModelProfile2.1.NFWHaloesHighresolutionN-bodynumericalsimulations(Navarro,Frenk,&White1995,1996,1997)haveindicatedtheexistenceofauniversaldensityprofilefordarkmatterhalosresultingfromthegenericdissipationlesscollapseofdensityfluctuations.Thisdensityprofiledoesnot(strongly)dependonthemassofhalo,onthepowerspectrumofinitialfluctuations,oronthecosmologicalparameters.Thesehalomodelswhichareformedthroughhierarchicalclusteringdivergewithρ∝r−1nearthehalocenterandbehaveasρ∝r−3initsouterregions.Insidethevirialradius,thisso-calledNFWhaloprofileappearstobeaverygood–3–descriptionofthemassdistributionofobjectsspanning9ordersofmagnitudeinmass:rangingfromglobularclusterstomassivegalaxyclusters(seeWright&Brainerd(2000)andreferencestherein).TheNFWhalomodelissimilartoHernquistprofile(Hernquist1990)thatgivesagooddescriptionofellipticalgalaxyphotometry.However,thetwomodelsdiffersignificantlyatlargeradii,possiblyduetothefactthatellipticalgalaxies,countrarytothedarkhalos,arerelativelyisolatedsystems.ThesphericallysymmetricNFWdensityprofiletakestheformofρ(r)=δcρcrrs(1+rrs)2(1)whereρc=[3H2(z)]/(8πG)isthecriticaldensityforclosureoftheuniverseattheredshiftzofthehalo,H(z)istheHubbleparameteratthesameredshift,andGistheuniversalgravityconstant.Thescaleradiusrs≡r200/cisthecharactristicradiusofthehalowherecisadimensionlessnumberreferedtoastheconcentrationparameter,andδc=2003c3ln(1+c)−c1+c(2)isacharactristicoverdensityforthehalo.Thevirialradiusr200isdefinedastheradiusinsidewhichthemassdensityofthehaloisequalto200ρc.ItistheneasytoseethatM(r200)≡M200=8003ρcr3200.(3)Therefore,NFWhalosaredefinedbytwoparamete