0047−2808,bysemi–linearinversion
S.DyeandS.J.Warren
arXiv:astro-ph/0411452v2 18 Mar 2005AstrophysicsGroup,BlackettLab.,ImperialCollege,PrinceConsortRoad,London,SW72BW,U.K.
ABSTRACT
WemeasurethemassdensityprofileofthelensgalaxyintheEinsteinringsystem0047−2808usingoursemi–linearinversionmethoddevelopedinanearlierpaper.Byintroducinganadap-tivelygriddedsourceplane,weareabletoeliminatetheneedforregularisationoftheinversion.Thisremovestheproblemofapoorlydefinednumberofdegreesoffreedom,encounteredbyinver-sionmethodsthatemployregularisation,andsoallowsaproperstatisticalcomparisonbetweenmodels.WeconfirmtheresultsofWaythetal.(2004),thatthesourceisdouble,andthatapower–lawmodelgivesasignificantlybetterfitthatthesingularisothermalellipsoidmodel.Wemeasureaslopeα=2.11±0.04.Wefind,further,thatadual–componentconstantM/Lbary-onic+darkhalomodelgivesasignificantlybetterfitthanthepower–lawmodel,atthe99.7%
.69
confidencelevel.Theinnerlogarithmicslopeofthedarkhaloprofileisfoundtobe0.87+0−0.61(95%CL),consistentwiththepredictionsofCDMsimulationsofstructureformation.Wedetermine
.53
anunevolvedB–bandmasstolightratioforthebaryons(only)of3.05+0−0.90h65M⊙/LB⊙(95%CL).ThisisthefirstmeasurementofthebaryonicM/Lofasinglegalaxybypurelygravita-tionallensmethods.Thebaryonsaccountfor65+10−18%(95%CL)ofthetotalprojectedmass,or,
+12
assumingsphericalsymmetry,84−24%(95%CL)ofthetotalthree–dimensionalmasswithinthe
1
meanradiusof1.16”(7.5h−65kpc)tracedbythering.Finally,atthelevelof>3σ,wefindthatthehalomassisrounderthanthebaryonicdistributionandthatthetwocomponentsareoffsetinorientationfromoneanother.
Subjectheadings:astrophysics;gravitationallensing;darkmatter
1.Introduction
TheΛ−CDMmodelhasbeenoutstandinglysuccessfulinexplainingthegrowthofstructureintheUniverse,totheextentthatithasbeenarguedthatweshouldnowtreatthetheoryasestablished(Binney2004).Attemptstofalsifythetheoryhavefocusedmainlyonthepredictedmassprofilesinthecentresofgalaxies.N–bodysimulationshaveestablishedthatasimpleformu-lation,ρ(r)∝(r/rs)−γ(1+r/rs)γ−3,accuratelydescribesthedensityprofilesofdark–matterhalosacrossawiderangeoflengthscales.Atradiimuchsmallerthanthecharacteristicscalers,theden-sityprofileiscuspy,ofpowerlawform,ρ(r)∝r−γ,withvaluesofγintherange1(Navarro,Frenk&White(1996),the‘NFWprofile’)to1.5(Moore
etal.1998,1999)indicated.However,signifi-cantlyshallowerslopesthanpredictedhavebeeninferredfromobservationsoftherotationcurvesoflowsurfacebrightnessgalaxies(LSBs)(deBlok&McGaugh1997;deBloketal.2001).Be-causeofthisSpergel&Steinhardt(2000)arguedthatthecollisionlessCDMpicturerequiresmodi-fication,andthattheparticlesareself–interactingwithalargescatteringcrosssection.
Morerecentworkindicatesthattheseconclu-sionsmaybepremature,andatpresentthesitu-ationisunclear.Evenwithdataofimprovedspa-tialresolution,Swatersetal.(2003)emphasisesourcesofsystematicerrorinthemeasurementofγfromgalaxyrotationcurves.Theyfindthattheirsampleof15dwarfandLSBgalaxyrotationcurvesdoesnotprecludeaslopeγ=1.Atthe1
sametime,themostrecentsimulationsshowthatthecentralmassprofilesdonotreachanasymp-toticvalueoftheslope,butthattheslopecon-tinuestoflattentowardthecentre,downtotheresolutionlimitofthesimulations(Poweretal.2003;Navarroetal.2004).Thismeansthatatthesmallradiioftheobservations,∼1%ofthevirialradius,thefittingformulaeusedfortheinterpretationofthedatamaynotbeappropri-ate.ForthisreasonHayashietal.(2003)ad-vocatecomparingtheobservedrotationcurvesdi-rectlyagainstrotationcurvesmeasuredfromthesimulations.TheyconcludethatthemajorityoftheobservedrotationcurvesareadequatelyfitbyCDMhalos.NeverthelessathirdoftheobservedLSBrotationcurvescannotbesatisfactorilyac-countedfor.Theypostulatethattheinconsistencymightbecausedbytheeffectsofhalotriaxalityonthemotionofbaryonicmaterialandthedifferencebetweencircularvelocityandgasrotationspeed.Thishighlightsthemainproblemencounteredbyalldynamicalanalyses:Theapproachhasmanycomplexitiesthatpreventclearinterpretationoftheobservations.
Theseresultsmotivatethesearchforanalter-nativemethodtomeasuregalaxymassprofiles,freeofsuchambiguities.Gravitationallensingprovidesanattractivesolution,primarilybecausethedeflectionangleofaphotonpassingamassiveobjectisindependentofthedynamicalstateofthedeflectingmass.Thereforelensingisnotsubjecttoanyofthedifficultiesassociatedwithdynami-caltechniques,offeringastraightforwardapproachbasedonwellestablishedphysics.
Stronglensingsystems,whereabackgroundsourceismultiplyimaged,allowparameterisedlensmassprofilestobeconstrained,bysearchingforthebestfittotheobservedimagepositions.Sand,Treu&Ellis(2002)enhancedthistechniquebyincorporatingextraconstraintsfromtheveloc-itydispersionprofileofthelens,andthishassinceseenapplicationtoanumberofsystems(Treu&Koopmans2002;Koopmans&Treu2003;Sandetal.2004).Nevertheless,Dalal&Keeton(2003)havecriticisedtheseresults,arguingthatthetightconstraintsclaimedweredrivenbypriorassump-tionsandthatingeneral,moredetailedmodellingisrequired.
Ifthebackgroundsourcehasextendedstruc-ture,multiplearcimagesorEinsteinringsare
formed.Withhigh–resolutiondata,theimagewillcomprisealargenumberofresolutionelements.Extendedsourcesthereforehavetheconsiderableadvantagethattheycanprovidemanymorecon-straintsonthelensmassprofilecomparedtoim-agesofpointsources.Acompleteanalysisofim-agesofextendedsourcesrequiresmodellingofthesourcesurfacebrightnessdistribution.Theprop-ertiesofboththesourceandthelensmustbead-justedtogivethebestfittotheobservedring.Additionally,apropersolutiontothisinversionproblemmustalsoaccountfortheconvolutionoftheimagewiththepointspreadfunction(psf).Warren&Dye(2003)(hereafterWD03)provideasummaryofthevariousapproachestothisin-versionproblemthathavebeensuggested.
Thewayinwhichthesourceismodelledcanhavefar–reachingeffects.Becauserealsourceshavecomplicatedstructure,assuminganover–simpleanalyticsourcesurface–brightnessprofilecanbiasthemassmodelsolution,sinceintheminimisationthemassmodelwillattempttocom-pensatefortheshortcomingsofthesourcemodel.Anon–parametricform,forexamplewherethesourcesurfacebrightnessdistributionispixelised,overcomesthisdifficulty.
Wallington,Kochanek&Narayan(1996)de-scribeamethodthatusesapixelisedsourcesurfacebrightnessdistribution.Thesolutionisreachedbysearchingthroughtheparameterspaceofthemassdistributionandthesourcesurfacebrightnessdistributiontofindthebestfit,i.e.thecombinationthatproducesthemodelimagewhich,convolvedwiththepsf,minimisesameritfunction.Themeritfunctionisthesummedχ2ofthefitofthemodelimagetothedata,plusatermproportionaltothenegativeentropyinthesourceplane.Theentropytermisaregularisationtermthatpreventsamplificationofnoiseduetothedeconvolutionandforcesasmooth(andpositive)solutionforthesource.Forthesakeofefficiency,themethodemploystwonestedcycles.Theoutercycleadjuststhelensmassmodel,whiletheinnercycleadjuststhesurfacebrightnessesofthesourcepixels,toproducethebestfitfortheparticularmassmodel.ThemethodwasrecentlyappliedbyWaythetal.(2004)tothesourcemodelledinthispaper,theEinsteinring0047−2808.
InWD03,wepresentedanewmethod,termed‘semi–linear’,forsolvingthisinversionproblem.
2
Algebraicallythemethodisverysimilartothemaximum–entropymethodinthatwesimplyre-placetheentropytermwithalinearregularisationterm.However,themethodisquitedistinctinapplicationbecauseitreplacesthesourceminimi-sationcyclewithasingle,linearmatrixinversion.Thisguaranteesthatthebestsourcefitisobtainedforagivenlensmodelandalsospeedsupthein-version.
Inthispaper,weapplythesemi–linearmethodtoHST–WFPC2observationsoftheEinsteinring0047−2808.OuranalysisbuildsupontheworkofWaythetal.(2004),whousedthesamedata,andinvestigatedarangeofsinglecomponentmassmodels.Oneofthesewasamodelinwhichthemassfollowsthelight,withasinglevariable,themass–to–lightratio(M/L).Thismodelprovidedapoorfit,andthismotivatesananalysiswhichincludesadark–matterhalo,toinvestigatewhatconstraintsthedataprovideontheamountofdarkmatter,andthevalueoftheinnerslopeofthedensityprofile.Accordingly,herewemodelthelenswithtwocomponents,abaryoniccomponent,forwhichthemassfollowsthelight,nestedinadarkhalo.Weshowhowthecontributionfromeachcomponentcanbeseparatedtoallowmea-surementofthebaryonicM/Landtheinnerslopeofthedark–mattermassprofile.Wecomparethismodelagainsttwosingle–componentmodels,thesingularisothermalellipsoid,andthepower–lawellipsoid.
InWD03weincludedadiscussionofthead-vantagesanddisadvantagesofregularisingthein-version.Regularisationallowsasmallsourcepixelsizetobeused,whichisdesirable,toextractfinedetailofthestructureofthesource.Thisispossi-blebecauseregularisationstabilisestheinversion,suppressingtheamplificationofnoiseassociatedwiththedeconvolutionofthepsf.Neverthelesswearguedthatregularisationisnotreliableforquantitativework.Bytheuseofsimulationsweshowedthatinsomecircumstancestheregularisedsolution,whileprovidingasatisfactoryfittotheimage,producedareconstructedsourcelightpro-filethatwasinconsistentwiththeinputmodel.Withrealdataitwouldbeimpossibletoidentifysuchaninconsistency.Afurtherproblemwithreg-ularisationisthat,bysmoothingthesourcelightprofile,iteffectivelyreducesthenumberofparam-etersfittingthesource,byanamountthatcannot
bequantified.Sincethetotalnumberofdegreesoffreedomintheproblemisthenunknown,itisimpossibletocorrectlyassessthegoodnessoffitofthesolution.Thispreventsaproperstatis-ticalcomparisonbetweenmodels(seeadditionalcommentsonthispointbyKochanek,Schneider&Wambsganss(2004)).Forthesereasonswehavesoughttodevelopastableinversionmethodthatavoidsregularisation,butstillmakesmaxi-mumuseoftheinformationintheimage.Aswedemonstrateinthispaper,thekeyistorecognisethatthefixedresolutionoftheimagetranslatestovariableresolutioninthesourceplane,sothatavariablepixelsizeacrossthesourceplaneisre-quired.Afurtheradvantageofunregularisedso-lutionsisthatthecovariancematrixforalltheparametersiseasytocompute(WD03).
Thelayoutofthepaperisasfollows:Inthenextsection,weprovideanoutlineofthesemi–linearmethodanddescribeitsextensiontoincludeadaptivesourceplanepixelisation.InSection3weoutlinetheHSTdatapreparation,providedetailsofthethreelensmodelsfitted,anddescribeourminimisationprocedureandthecomputationoftheuncertainties.Theresultsofthefittingarepresentedin4.Wefindthatthedual–componentmodelprovidesasignificantlybetterfitthantheothertwomodels,andweanalysetheresultsforthismodelinmoredetail.Weprovideabriefdis-cussionandsummaryinSection5
WeadoptacosmologywithΩ=0.3,Λ=0.7,andH◦=65kms−1Mpc−1throughout.2.
Thesemi–linearreconstructionmethod
Forafulldescriptionofthesemi–linearinver-sionmethod,wereferthereadertoWD03.Inthissectionweoutlineonlythemainfeaturesofthemethod.
Theinversionreliesonthefactthatboththesourceplaneandtheimageplanearepixelised.Themannerinwhichthesourceplaneispix-elisedisnotrestricted,allowingtheconcentrationofsmallerpixelsinregionswherestrongercon-straintsexist.Pixelsintheimageplanearela-belledbytheindexj=1,J.Weusedjforthesurfacebrightnessinpixeljandσjforitsuncer-tainty.Forafixedlensmassmodel,onecanformthesetofIpsf-smearedimagesfij,j=1,Jforeachsourcepixelihavingunitsurfacebrightness.
3
Wemaythenposethequestion:Whatsetofscal-ingssiarerequiredfortheseimagessuchthattheircoadditionyieldsamodelimagewhichprovidesthebestfittotheobservedimage?Thesescalingssiarethenthemostlikelysurfacebrightnessesofthesourceplanepixels,forthegivenmassmodel.Forunregularisedinversion,themeritfunctionis:JG=χ2=
Ii=1sifij−dj
j=1
∂2GL2
Beforedescribingthemethodadopted,itisim-portanttodeterminethesmallestsuitablesourcepixelscaleinaregionoflowmagnification.Be-causetheinversioninvolvesdeconvolution,thesourcepixelsizeshouldbenosmallerthanNyquistsamplingofthepsfinvertedtothesourceplane.1Atthewavelengthofobservation,550nm,theres-olutionofadiffractionlimited2.4mtelescopeis0.06′′FWHM.ThisisamisleadingrepresentationoftheHST-WFPC2imagequalityforthreerea-sons:1)TheHSTpsfincludesbroadlow–levelwingsthatreduceinformationcontentatsmallscales.2)Thecoreofthedeliveredpsfisunder-sampledbythe0.1′′pixelsoftheWFPC2WideFieldCamera.3)Thefinalimageisfurtherde-gradedbythepixelscatteringfunction(Birettaetal.2002).
Tocomputetheappropriatesampling,weusedtheTinyTIMsoftware(Krist1995)tocreateahighlysampledpsfimage.Thisisthepsfinfrontofthedetector.Thisimagewasthenconvolvedwithasquare0.1′′pixel,andthenconvolvedfur-therwiththepixelscatteringfunction.There-alisedimageofapointsourcemaythenbethoughtofasaδ−functionsamplingonthepixelgridofthisconvolvedfunction,withnoiseadded.Toac-countforthelossofinformationduetothebroadwingsofthepsf,ratherthandirectlymeasuretheFWHMofthispixel–convolvedpsf,wemeasuredtheradiuswhichencloses70%oftheenergy,andthencomputedtheFWHMoftheGaussianwhichcontains70%oftheenergywithinthesameradius.TheresultwasaFWHMvalueof0.24′′.Therefore,inanimageoflowS/N,thereislittleinformationatsmallerscalesthanthisvalue,suggestingthatthepixelscaleofthereconstructedsourceshouldbenosmallerthan0.12′′inregionsoflowmag-nification.Itshouldbenotedthatthesub–pixelditheringstrategyusedforourobservations(§3.1)onlyimprovesthesamplingandnottheresolutionofthedata.
Themagnification,µ,givestheratiooftheareaoftheimageofasourceplanepixel(summedoverallcopies)totheoriginalsourceRoughlyspeaking,
√
pixelarea.
1In
fact,inmoredetail,theminimumpixelsizedependsontheS/Nofthedataandonthepsfpowerspectrum,sothatfordataofhigherS/Nitwouldbepossibletouseasmallerpixelsizethanadvocatedhere.
ageplanetothesourceplane.Therefore,onemightexpectthattomaximisetheinformationinthereconstructedsource,thesourcepixelareashouldscaleinverselywithµ.
Toimplementsuchascheme,amagnificationmapforamassmodelclosetothefinalsolutionisneeded.Forthispurposewecomputedthebestfitlensmodelobtainedwitharegularsourcegridofpixelscale0.06′′.Usingthismodel,theadap-tivepixelisationstartswithaninitialgridofsourcepixelscale0.12′′.Dependingonthemagnification,thesepixelsaresplitinto4andsomesub–pixelsfurthersplitinto4,resultinginaminimumsourcepixelscaleof0.03′′.Naively,iftheaveragemag-nificationoverapixelsatisfiesµ>4,thepixelshouldbesplit,andifwithinasub–pixelµ>16,thesub–pixelshouldbesplit.Inrealitythereso-lutionimprovementisdirectiondependent,sincesourcepixelsarenotisotropicallymagnified,andsoamoreconservativecriterionisneeded.Thisisalsodesirablebecausethemagnificationacrossapixelcanvaryrapidly.Thesplittingcriterionreferstotheaveragemagnificationacrossapixel,buttheconditionmaynotbetrueofeachofthesub–pixelsintowhichthepixelissplit.Forthesereasons,insteadofthefactors4and16above,weintroducethe‘splittingfactor’,s,suchthatapixelissplitifµ>sandasub–pixelissplitifµ>4s.Theproblemisthusreducedtoidentifyingtheop-timalvalueofs.Clearlyifthesplittingfactorislarge,onlyafewhighlymagnifiedpixelswillbesplit.Thesourcepixelswillthenbetoolargetomatchallthedetailintheimage,andinformationwillbelost.If,ontheotherhand,thesplittingfac-torissmall,inregionsoflowmagnificationtheywilloversampletheinvertedpsf,resultinginaverynoisyreconstructedsource.
Wedetermineanoptimalvalueofsempirically,bymeasuringtheimprovementinthefitbroughtaboutbythesplittingfordifferentvaluesofs,suc-cessivelyreducingthevalueofstothepointatwhichnosignificantimprovementisobtained.Indetail,startingwithalargevalueofs,thesourceplaneispixelisedasdescribed,andthebestfitmodelisrecomputed(seeSection3.3).Thisgivesasetofminimisedlensmodelparameters,there-constructedsourcesurfacebrightnessdistributionandthevalueofχ2forthefit.Thevalueofsisthenreduced,theprocedurerepeated,andthenewvalueofχ2computed.Ifthereducedvalue5
Fig.1.—Sourceplanepixelisationscorrespondingtosplittingfactorsofs=4,9,and14fromlefttoright.Thegreyscaleshowslog(magnification)calculatedfromthebestfitdual–componentlensmodelinSection4.1.Heavydashedboxshowsthe0.3′′×0.3′′sourceplanesizeusedinthereconstructionthroughoutthispaper.
ofsincreasesthenumberofsourcepixelsby∆I,thenthisisthedecreaseinthenumberofdegreesoffreedom.Thechange∆χ2shouldthereforebedistributedasχ2for∆Idegreesoffreedom,sincewehavesimplyincreasedthenumberoflinearpa-rametersinthefit.If∆χ2issignificantbythistest,thelowersplittingfactorisacceptedandthenextlowervalueofsisthentested.Wesetthesignificancelevelattheconservativevalueof1%,forthereasonsgivenabove.
Figure1illustratesthechangingpixelisationasthesplittingfactorisdecreasedsuccessivelythroughthevalues,14,9,4,fromrighttoleft.Ineachplot,thegreyscaleistheµmap.Wefindthatthemiddlevalues=9correspondstothe1%significancelevelchosen,forallthreelensmodelsdescribedinSection3.2.ComparisonofthegoodnessoffitofeachmodelinSection4.1isthereforecarriedoutwithanadaptivesourcegridconstructedusings=9.
Inordertoensureacompletelyfaircomparisonbetweenmodels,afurthereffectmustbetakenintoconsideration.Thechosenoffsetofthesourceplanecentrewithrespecttothecentreofthelenscausticstructurecaninprinciplebiasthegood-nessoffitofonelensmodelrelativetoanother.Eachofthethreelensmodelstestedhasaslightlydifferentcausticshape.Agivensourceplaneoff-setcanresultinamoreeffectiveadaptivepixelgridforonelensmodelthananotherduetofortu-itousalignmentofpixeledgesrelativetothelenscaustic.Wedealwiththiseffect,essentially,by
includingthesourceplaneoffsetintheminimisa-tion;weperformafulllens+sourceminimisationateverypointonaregulargridofoffsetsandtakethebestoverallfit.ThisisdiscussedfurtherinSection3.3.
OurmethodofadaptivelypixelisingthesourceplaneinthiswaysolvestheproblemnotedbyKochanek,Schneider&Wambsganss(2004)thatplaguescurrentpixelisedsourcebasedmethods.Existingtechniquesusearegularsourcegridandtherebyrelyonsomeformofregularisationtocontrolthebehaviourofthereconstructedsourceinregionsoflowmagnification.Regularisationsmoothsthesourcelightprofile,reducingtheef-fectivetotalnumberofparametersandhencein-creasingthenumberofdegreesoffreedom,byanamountthatcannotbequantified.Thisisespe-ciallyproblematicwhencomparingdifferentlensmodels,asafixedregularisationweightforonemodelgenerallywouldnotgivethesameincreaseinnumberofdegreesoffreedomforanother.Inourscheme,thesplittingfactorhasbeenchosensuchthattheadaptivelysizedpixelsextractmax-imuminformationfromthelensimagewithoutneedofregularisation.Therefore,thenumberofdegreesoffreedomofthefitisawell–definednum-ber.Thisallows,firstly,unambiguousassessmentofwhetheragivenmodelprovidesasatisfactoryfittothedataandsecondly,unbiasedcomparisonofdifferentmodelfits.
6
3.Dataandmethodofanalysis
Inthissectionweprovidedetailsoftheobser-vationaldata(3.1)andthelensmodelsconsid-ered(3.2).Wealsodiscussthepracticalitiesofperformingtheminimisation(3.3)andexplainthecalculationoftheuncertainties(3.4).3.1.
HSTobservations
Thedataanalysedherearethesamedataanal-ysedbyWaythetal.(2004),whogivefulldetailsoftheobservationsanddatareduction.Inordertokeepthecurrentpaperself–contained,thekeyelementsareoutlinedinthissection.
Thefieldof0047−2808wasobservedwithHST’sWFPC2instrumentintheF555Wfilteroverfourorbits.WeusedtheWFC,whichhas0.1′′pixels.ThechosenfilterensuredthatthestrongLyαemissionfromthesourcestar–forminggalaxyatz=3.595(Warrenetal.1996)wasde-tected,therebyenhancingthering:lensfluxratio.Observationswereditheredusinga2×2patternwithahorizontalandverticalstepsizeequaltoN+0.5pixels.Ateachofthefourditherposi-tions,twoexposuresof1200sweretakentoaidcosmicrayremoval.
Aftersubtractingthebackgroundcountsfromeachexposureandeliminatingcosmicrays,pairsofexposuresateachditherpositionwereaveraged.Thesefourcombinedimageswerethenusedtoformaninterlacedimagewithpixelinterval0.05′′.ThisimageisreproducedinFigure2.Inthefig-urethepixelsareshownwithside0.05′′,butthisisonlyforpresentationalpurposes.Intheanaly-sis,weaccountforthetruesizeof0.1′′byfittingsimultaneouslytothefourimagesthatmakeuptheinterlacedimage.
Weconstructedanoiseframeforthisimageforthepurposesofmeasuringχ2,bothforfittingthelightprofileofthelensinggalaxyandforthelens-inganalysis.OurPoissonestimateofthepixelfluxuncertaintiesallowsforphotonnoiseandreadoutnoise,andaccountsfortheremovalofcosmicrays.TheimageofthelensinggalaxywassubtractedbyfittingaS´ersicprofile(S´ersic1968)plusacen-tralpointsource(Waythetal.2004).Thebestfitwasfoundbyminimisingχ2,discountinganannularareacontainingtheimageofthelensedsource.Inthefittingprocedureprofileswerecon-
Fig.2.—Top:InterlacedHSTimageof0047−2808withpixelinterval0.05′′.Bottom:Interlacedimageaftersubtractingtheimageofthelensinggalaxy.Inthisfigure,thepositionangleoftheverticalaxisis27.44◦EofN.
volvedwithamodelWFPC2psfandtheappro-priatepixelscatteringkernel.ThelowerhalfofFigure2showstheresultingEinsteinringafterthelensinggalaxyhasbeensubtracted.Theresidueatthecentreofthesubtractedimageisnotsig-nificant,giventhehighcountsatthecentreofthelensgalaxyimage.Thisistheimageusedinthelensinganalysis.3.2.
Lensmodels
Waythetal.(2004)usedthesamedataanal-ysedinthecurrentpapertotestarangeofcom-monlyusedlensmassmodels.Threemodelsfailed7
toprovideasatisfactoryfit.Thesewere:1)amodelinwhichthemassprofilefollowsthelightprofile,withasinglefreeparameter,M/L,2)
ρs
theNFWmodelρ(r)=
and
ξ(u)2=ux′2+
y′2
[1−(1−q2)u]n+1/2
du(8)
andwhereΣcisthecriticalsurfacemassdensity
expressedhereinunitsofM⊙.ThefittedhalflightluminosityoftheS´ersicprofileisL1/2=(1.99±
−2
0.09)×109h65LB⊙/2′′.Theaxisratiomeasuredfromthelightdistributionisqb=0.69(Waythetal.2004).Sincethe4parametersoftheellipsearethemeasuredvaluesfortheS´ersicfit,thebaryoniccomponentofthemassmodelhasasinglefreeparameterΨ.Notethatthedeflectionanglesforthismodelneedonlybecomputedonce,andthenscaledbyΨasΨisvariedinsearchingforthebest–fitmassmodel.
Forthedarkmatterhalo,wechooseagen-eralisedNFWmodel(Navarro,Frenk&White1996)whichallowsforavariablecentraldensityprofileslopeγ.Thishasavolumemassdensityprofilegivenby
ρ(r)=
ρs
suchthateverypixelintheringhasbeenreflectedinaplanerunningthroughtheringcentreandperpendiculartothepixel’sradialvector.Bothunderandovermagnifiedsourcereconstructionscanberejectedonthegroundsthattheyproduceextraneousimageswhenlensedtotheimageplane.Topreventconvergencetoanincorrectlocalmin-imum,weinitialisethehalonormalisationρstoavaluewhichisestimatedtolieclosetothecorrectsolution.Thisinitialvalueissetbyafittingfunc-tionderivedfromasimplifiedanalysisinwhichonlyρsisallowedtovaryacrosstheγ−Ψplane.3.4.
Modellinguncertainty
Theuncertaintyonthesetofreconstructedlensmodelparametersisdeterminedfromtheprescrip-tioninWD03.Thisiscalculatedbyinvertingthecurvaturematrixforallparameters,includingthesourcepixels,toobtainthefullcovariancematrix.Theuncertaintyfromthefitforagivenparame-teristhenjusttherelevantdiagonalterminthismatrix.Forthedual–componentmodel,wede-terminetheerroronγandΨdirectlyfromtheirmarginalisedχ2contours.ThetotalerrorthatwequoteforΨincludesanextracontributionfromtheonlysourceofsignificantuncertaintyintheSersicprofile;theparameterL1/2(see3.2.2).
Afinalsourceoferrorinourdual–componentmodelstemsdirectlyfromtheuncertaintyonthescaleradiusrsinthehalocomponent.Dalal&Keeton(2003)arguethatrsshouldbeleftasafreeparameterintheminimisation.optedtoholdrsatthevalueof50h−1
Wehave
pectedfromsimulationsbyBullocket65kpcasex-al.(2001)andsearchforasolutioninthecontextofthismodel.Ourdatacanonlyweaklyconstrainrswhichhastheadvantagethatourresultsdonotdependsensitivelyonitsvalue.Wefindthata10%changeinrsproducesa∼1%changeintheminimisedM/Landanegligiblechangeinγ.Thiserrorisnotincludedinthefinalerrorbudget.4.
Results
Thissectionisdividedintotwohalves.Inthefirst,Section4.1,thethreelensmodelsarecompared.Inthesecond,Section4.2,thedual–componentmodelisconsideredinmoredetailandusingthismodel,wereconstructthesourcesurfacebrightnessdistribution.
4.1.ComparisonofModels
Inthissection,wecomparetheperformanceofthethreemodels,1.SIE,2.power-law,and3.dual–componentmodels,infittingtheobservedringimage.WealsocompareourfindingswiththeanalysisbyKoopmans&Treu(2003)whoanalysed0047−2808usingamethodcombiningdynamicalmeasurementsandlensing.
Allreconstructionsinthissectionareunregu-larised.Asexplainedinsection2.1thisistoallowunbiasedcomparisonofmodels.Theχ2iseval-uatedinanannularmaskedregionshowninthebottomleftpanelofFigure5.Themaskwasde-signedtoensurethatitincludestheimageoftheentiresourceplane,withminimalextraneoussky.Thismeansthatonlysignificantimagepixelsareusedinthefit,makingχ2moresensitivetothemodelparameters.
Table1summarisestheresults,listingthebest–fitvalueofχ2andthenumberofdegreesoffree-dom(NDF)ofthatfit.RecallthattheNDFde-pendsnotonlyonthenumberofparametersofthemassmodel,butalsoontheexactsourcepixelisa-tionused,whichinturnsdependsonthestructureofthecaustics.4.1.1.SIEandpower–lawmodels
FortheSIEmodel,α=2,thebestfitgives
χ2min
=1156.2for1247degreesoffreedom.Thepower–lawmodelgivesχ2min=1157.7with1255degreesoffreedom,andameasuredslopeα=2.11±0.04.TheincreaseintheNDFisbecausefewerpixelsareusedtotessellatethesourceplane.Comparingthepower–lawtotheSIEmodel,theincreaseinχ2of∆χ2=1.5,only,foranincreaseof8degreesoffreedomdiffersfromtheexpecta-tionof∆χ2Thepower–law∼8modelatathereforesignificanceisalevelsignificantof99.3%.im-provementovertheSIEmodel.Thisisreflected
Model
NDF
1156.21157.71161.4
inthemeasuredvalueofα=2.11whichisincon-sistentwithα=2atthe2.7σsignificancelevel.Waythetal.(2004)comparedthesametwomodels,usingamaximum–entropyinversionmethod.Thismethodentailsregularisationoftheinversion.Theyappliedminimalregularisa-tion(suchthattheinversionamountstothenon–negativemin–χ2solution),inordertominimisetheuncertaintyinthechangeintheNDF.Theyfoundthatthepower-lawmodelgivesabetterfitthantheSIE,atanassociatedsignificancelevelof96%.Theyfoundα=2.08±0.03forthepower-lawmodelaswellasorientationsandellipticitiesforbothmodelsconsistentwithourfindings.Theseresultsareingoodagreementwithours,vindicat-ingtheirapproachfordealingwiththeproblemoftheuncertaintyoftheNDF.4.1.2.χ2min
Baryons+darkmatterhalomodel
Withthedual–componentmodel,weobtain
=1161.4for1269degreesoffreedom.Com-paringthisagainstthebestfittingsingularpower–lawmodelgivesanincreaseinχ2of∆χ2=3.7,only,foranincreaseof14degreesoffreedom.Thissmallincreaseinχ2,differsfromtheexpectationof∆χ2∼14atasignificancelevelof99.7%,demon-stratingthatthedual–componentmodelprovidesasignificantlybetterfit.
Figure3showstheχ2contoursintheγ−Ψplane,marginalisedovertheremaining5parame-ters.Thegreyshadedregionsgivethe68%,95%,99%&99.9%one–parameterconfidencelimits.
.69
Weobtainaninnerslopeofγ=0.87+0−0.61to95%confidence(oralimitofγ<1.74to99.9%confi-.53
dence)andaM/LofΨ=3.05+0−0.90h65M⊙/LB⊙to95%confidence,includingthephotometricer-Parameter
3.27×10(0.070′′,0.026′′)
0.82041.7◦
6
1σerrorinfit
θb=41.7±1.5.4.2.
FurtherAnalysis
Intheprevioussectionweestablishedthatthebaryon+darkmatterhalomodelprovidesasignif-icantlybetterfittotheobservedringcomparedtosinglecomponentmodels.Wenowanalysethismodelinmoredetail.4.2.1.
Baryonicmass&light
AsFigure3shows,wehavebeenabletocon-strainthebaryonicM/Lwithouttheneedfordy-namicalmeasurements.Thisisthefirsttimeapurelensinganalysisofasinglesystemhasmea-suredthebaryonicM/Ldirectly.Beingalenses-timatedquantity,thisisfreeoftheuncertaintiesassociatedwithdynamicalmethods(seeSection1).
Koopmans&Treu(2003)showthatthelensgalaxyin0047−2808isoffsetfromthelocalfun-damentalplanebyafactor0.37dex.Thisvalueisincloseagreementwiththeexpectedpassiveevo-lutionofthisgalaxy,estimatedfrompopulationsynthesismodelsmatchedtothemeasuredV−Icolour(Waythetal.2004).Correctingbythisfactor,ourderivedM/LΨ=7.1isremarkablysimilartothelocalaveragevalueforthecentresofellipticalsof7.3±2.1h65M⊙/LB⊙(Gerhardetal.2001;Treu&Koopmans2002),determineddynamically.Eitherthisisacoincidence,oritisanindicationthatthevariouselementsgoingintothiscomparison,i.e.ourlensinganalysis,thelo-caldynamicalanalysis,thepopulationsynthesismodels,andtheassumptionofpassiveevolution,areallquiteaccurate.4.2.2.
Haloandbaryonicfractions
Wecalculatethefractionalcontributionofthebaryonstothetotalprojectedmassbyintegratingtheprojectedmassdistributionofeachcomponentinsideacircularapertureofradius1.16′′placedatthelenscentre(theoffsetbetweencomponentsissmallenoughtodisregard).+10Withinthisaperture,thebaryonsaccountfor65−18%(95%CL)ofthetotalprojectedmass.
Thefractionofdarkmatterinsideasphereofthesameradiusisobtainedbydeprojectingbothsurfacemassdensityprofiles.Becauselensing
measuresonlyprojectedmass,wehavenoinfor-mationregardingitsdistributionalongthelineofsightandsothismustbeassumed.Ourapproachistofirstcalculateacircularlyaveragedsurfacedensityprofileforeachcomponentandthende-projectassumingsphericalsymmetry.Deprojec-tioniscarriedoutusingtheapproachgivenbyBinney&Tremaine(1987).
Wefindthatthebaryonsaccountfor84+12−24%(95%CL)ofthetotalmasswithinasphereofdius1.16′′(7.5h−1
ra-65kpc).Figure4plotsboththecir-cularlyaveragedsurfacemassdensityprofileandthecumulativedeprojectedmassprofileforthehaloandbaryons.Inthecaseofthebaryons,thepointmassisincluded.
Fig.4.—Circularlyaveragedprojectedsurfacemassdensityprofileofhalo(solid)&baryons(dashed),andcumulative3–dimensionalmasspro-fileofhalo(dot–dash)andbaryons(dotted).Ver-ticaldashedlineismeanradius1.16′′oftheringtracedbythelensedimages.4.2.3.
Sourcereconstruction
Thereconstructedsourceandimageforthebest-fitdualcomponentmodelareshowninFigure5.Thetoprowofthisfigureshowstheunregu-larisedresult.Notethecorrespondencebetweenthepixelisationinthesethreepanels,andthatinthemiddlepanelofFigure1.Thetop–leftpanel(side0.3′′)showsthereconstructedsource,andthebottommiddlepanel(side4.2′′)istheimageofthissource,convolvedwiththeWFCpsf.This12
isthebest–fittotheactualimageshowninthebottomleft–handpanel.Theuncertaintiesonthesourcepixelsurfacebrightnessesareshowninthetopmiddlepanel,andthecorrespondingsignifi-cancemap(ie.surfacebrightnessdividedbystan-darderrors)isshowninthetopright–handpanel.Notethat,incontrastwithsimilarmapsshowninWD03,wheretheuncertaintieswerenoticeablysmallerinsidethecaustic,heretheyaremoreuni-formacrossthesourceplane,asaconsequenceofthepixelsizevaryingwiththemagnification.ThelimitedsourceplaneresolutionallowedforthesedatabyouradaptivepixelisationschemeisaconsequenceoftherelativelylargeWFCpixels,andtherelativelylowS/Noftheimage.Neverthe-lessthereareclearlytwoareasinthesourceplane,ofhighsignificance,wherethesourcefluxiscon-centrated,oneoneithersideofthecaustic.ThisisthesameconclusionreachedbyWaythetal.(2004).AbettersampledimageofhighS/Nwouldallowformationofaclearerpictureofthenatureofthesourcegalaxy.Obviously,thesourcelightpro-filedoesnotfollowasimpleparametricform.Thismeansthatattemptingtomodel0047−2808byformingmodelimagesusingasimplesinglesourcewouldbiasthefittedmodelparameters.
Purelyforthepurposesofvisualisation,wealsoperformedaregularisedinversion.TheresultsareprovidedinthemiddlerowofFigure5.Forthis,theregularisationweight,λ,waschosentomakeχ2andtheregularisationterminequation(4)contributeequallytothefigureofmeritG.Settingtheminimumallowedsourcepixelsizeto0.01′′×0.01′′,wefollowedthesameprocedure,setoutinSection2.1,toselectthesplittingfactor.Theclarityofthesourceissomewhatimproved.Theoutersourceappearstobeextended,andtostraddlethecaustic.Thesignificanceplot,right–handpanel,middlerow,identifiesbothsourcecomponentsasbeinghighlysignificant.There-constructedimage,bottomright–handpanel,issomewhatsmoother,asexpected.5.
Discussionandsummary
Oneofthemaingoalsofthispaperwastoinvestigatetheextenttowhichapuregravita-tionallensanalysisoftheimageofanextendedsource,usingalltheinformationintheimage,couldconstraintheinnerslopeofthedarkmat-
terdensitydistributioninthelensgalaxy.Apply-ingthemethodtothelens0047−2808,wehavesucceededinshrinkingtheuncertaintiesconsider-ably,comparedtothelens+dynamicalanalysisofKoopmans&Treu(2003),whichusedonlythepositionsofthefourbrightestpeaksintheimageaslensconstraints.Ourmeasurementofthein-nerslopeofthedarkmatterhaloofγ=0.87+0−0..69
61(95%CL)isconsistentwiththecuspypredictionoftheCDMmodel.Nevertheless,wefindthatthedarkmattermakesonlyaminorcontribu-tiontothetotalmasswithinasphericalradiusequaltotheEinsteinradiusofthelens.Thereismountingevidencefrombothdynamicalandlens-ingmethodsthatthisisfairlytypicalofbrightandintermediate–luminosityearly–typegalaxies.Forexample,Romanowskyetal.(2003)comparedthemeasuredmotionsofplanetarynebulaearoundthreenearbyellipticals,outtoseveraleffectiveradii,withdynamicalmodelswithoutdarkmat-terandfoundsatisfactoryagreement.Asimilar,morequantitative,conclusionwasreachedbyastatisticalanalysisof22multiply–imagedquasars,byRusin,Kochanek&Keeton(2003).Althoughasinglemultiply–imagedquasarisnotusefulforconstrainingdual–componentmodels,byassum-ingafixedratioofdarkmattertobaryons(withintwoeffectiveradii),thesamepower–lawslopeγforthedarkmatterinallsystems,andbyinvokingarelationbetweenΨandgalaxyluminosity,Rusin,Kochanek&Keeton(2003)wereabletoderiveusefulconstraintsondual–componentmodels.ByfixingtheinnerslopetotheNFWvalueγ=1,theyfindthatthebaryonsaccountfor78%±10oftheprojectedtotalmasswithintwoeffectiveradii,
average7h−1
65kpc.Thisisconsistentwithoursurementof65%+10−1mea-−18(95%CL),within7.5hagainin2D.Notethatweareabletoachieve65kpc,sim-ilarconstraintsfromasinglesystem,withfewerassumptions.Thishighlightstheusefulnessofim-agesofextendedsources.
AsFigure4shows,thebaryonsaremorecon-centratedthanthedarkmatterinthislensgalaxy,anddominatethemassdistributionwithinthere-gionoftheimage.Thebaryonswillaltertheshapeofthedarkmatterhalo,predictedbythepuredarkmattersimulations,inanon–trivialway,de-pendentonthehistoryofassemblyofthevariouscomponents,thesequenceofstarformation,andtheextenttowhichgasisblownoutofthegalaxy
13
SurfaceBrightnessStandardErrorSignificanceSurfaceBrightnessStandardErrorSignificanceSurfaceBrightnessSurfaceBrightnessSurfaceBrightnessFig.5.—Reconstructedsourcefrombestfitlensmodel(causticshownbydashedheavyline).Topleft:UnregularisedreconstructedsourcesurfacebrightnessdistributionobtainedinSection4.2.3.Topmiddle:Standarderrorsonunregularisedsourcepixels.Topright:Significanceofunregularisedsource.Middleleft:Regularisedsourcefrombestfitlensmodel.Middle:Standarderrorsonregularisedsource.Middleright:Significanceofregularisedsource.Bottomleft:Maskedobservedringimage.Bottommiddle:Lensedimageofunregularisedsource.Bottomright:Lensedimageofregularisedsourcebywinds.Asimpletreatmentforestimatingtheinfluenceofbaryonsisthesocalled‘adiabaticap-proximation’ofBlumenthaletal.(1986).Inthisapproximation,theexpectedprofileofacollapsedhalocanbeestimatedfromitsinitialprofileandtheinitialandcollapsedbaryonicprofiles,assum-ingthehaloadiabaticallycontracts.Byapplyingthisinreverse,theinitialhaloprofilecanbees-timatedfromthecollapsedprofileasdeterminedfromatwocomponentmodelsuchasours.This
initialprofilecanthenbecompareddirectlywiththepuredarkmatterCDMsimulations.Treu&Koopmans(2002)usedthisreversemethodontheirtwocomponentmodelofthelenssystemMG2016+112.Theyfoundthattheinnerslopeoftheinitialhalocanbesubstantiallyshallowerthanthemeasuredcollapsedslope.Ifthisisavalidapproximation,thenasimilareffectwouldbeex-pectedfor0047−2808.Thisresultisofinterestforthecasewherethemeasuredslopeγisalreadysig-
14
nificantlysmallerthanthepredictedvalue,γ∼1,sinceitactstowidenthediscrepancy.Butgiventhesimplificationsinvolved,andthereforetheun-certaintyinthemagnitudeoftheeffect,itisoflim-itedinterestinthepresentcasewhereourbest–fitvalueisconsistentwiththeCDMvalue.
Forourdualcomponentmodelwefoundthatthedarkhaloisoffsetinorientationfromthebaryonsby6.7◦icantdifference±in1.axis7,andratioisrounder,of∆q=with0.13a±signif-0.02.Ourmodelisasimpleone,giventherelativelylowS/Noftheimage,anddoesnotincludeexter-nalshear.Nevertheless,thegalaxydoesnotlieinadenseenvironment,asevidencedbytheimageprovidedinWarrenetal.(1999).Therewouldbesomedegeneracyinasolutionincludingshear,betweentheamountofshearandtheorientationoffset(seeKeetonetal.(1997)foradiscussion).Adeeperimage,withsmallerpixels,wouldjus-tifyamoresophisticatedanalysis.Togaugethepotentialimprovement,wehaveundertakenex-tensivesimulationsofobservationsof0047−2808withtheHSTAdvancedCameraforSurveys.Us-ingtheHighResolutionChannelover10orbits,withapplicationofthesemi–linearmethodwean-ticipateareductionintheerroronγbyafactorof∼5,sufficienttostronglytesttheCDMex-pectation.TheimprovementisaconsequenceofthehighthroughputofACS,andespeciallythesmallerpixelsize.
Weconcludewithasummaryofthemainpointsofthepaper:
1.Wehaveextendedthesemi–linearmethodofWD03forinvertinggravitationallensimagesofextendedsources,toincludeadaptivepix-elisationofthesourceplane.Wehaveiden-tifiedamethodfortessellatingthesourceplanethatappliesanobjectivecriterionforsub–dividingpixels,whichmaximisesthein-formationaboutthesourceextractedfromtheimage,andisalsostable.Becauseofthistheinversiondoesnotrequireregular-isation.Thiseliminatestheproblemthatwithregularisedinversionthenumberofde-greesoffreedomisilldefined.Properstatis-ticalcomparisonofdifferentmassmodelsistherebyenabled.2.Wehaveappliedoursemi–linearmethodtoHST-WFPC2observationsoftheEinstein
ringsystem0047−2808todeterminethelensgalaxymassprofile.WeconfirmtheresultofWaythetal.(2004)thatapower–lawmodel,α=2.11±0.04producesasignifi-cantlybetterfitthanthesingleisothermalellipsoidmodel.Furthermoreouranalysisshowsthatadual–componentmodel,com-prisingabaryonicS´ersicprofile+pointmassnestedinadarkmattergeneralisedNFWhalo,givesasignificantlybetterfit(3.0σ)tothedatathanthebestpower–lawmodel.Wedemonstratedthatusing100%oftheinfor-mationcontainedintheEinsteinringimage,withanadaptivesourceplanepixelscale,providessignificantlybetterconstraintsthanthemodellingofKoopmans&Treu(2003),whousedthemeasuredradialvariationofthestellarvelocitydispersion,plusthelensconstraintsprovidedbythepositionsofthefourbrightestpeaksinthering.
3.Forthedual–componentmodel,wefindthatthebaryoniccomponenthasanrest–frameB–bandM/Lof3.05+0−0.unevolved
.53
90h65M⊙/LB⊙.ThisM/Lwasobtainedwithoutanydy-namicalmeasurementsandisthereforenotsubjecttotheusualuncertaintiesassociatedwiththisapproach.Theerrorsquotedhereincludethephotometricuncertainty.Evolv-ingthisvaluetozeroredshiftgivesaresultconsistentwiththedynamically–measuredM/Linthecentresofnearbyellipticals.4.Themeasuredinnerslopeofthedark–
matterhaloisγ=0.87+0−0..69
61(95%CL),consistentwiththepredictionsofCDMsim-ulations.5.Wefindthatthebaryonsaccountfor65+10−18%(95%CL)ofthetotalprojectedmassor,assumingsphericalsymmetry,84+12−24%(95%CL)ofthetotaldeprojected−masswithina
radiusof1.16”(7.5h1
65kpc)tracedbythering.6.Wefindthatthedark–matterhaloissignifi-cantlymisalignedwiththestellarlight,andalsoissignificantlyrounder.7.Thereconstructedsourcesurfacebright-nessdistributionshowstwodistinctsourceobjectsinagreementwiththefindingsof
15
Waythetal.(2004).Thishighlightstheneedfornon–parametricsourcestoobtainunbiasedlensmassprofiles.
Acknowledgements
WewouldliketothankPaulHewett,GeraintLewis,LeonLucy,andRandallWaythforhelpfuldiscussions.SDissupportedbyPPARC.REFERENCES
Binney,J.,2004,in’DarkMatterinGalaxies’,IAUSymposium220,ed.StuartRyder,D.J.Pisano,MarkWalker&KenFreeman,Publ.Astron.Soc.PacBinney,J.&Tremaine,S.,1987,’GalacticDy-namics’,PrincetonUniversityPressBiretta,J.,Lubin,L.,etal.,2002,HSTWFPC2InstrumentHandbookVersion7.0,Baltimore,STScIBlumenthal,G.R.,Faber,S.M,Flores,R.,Pri-mack,J.R.,1986,ApJ,301,27Bullock,J.S.,Kollat,T.S.,Sigad,Y.,Somerville,R.S.,Kravtsov,A.V.,Klypin,A.A.,Primack,J.R.,Deckel,A.,2001,MNRAS,321,598Dalal,N.&Keeton,C.R.,2003,astro-ph/0312072deBlok,W.J.G.,McGaugh,S.S.,Bosma,A.,Ru-bin,V.C.,2001,ApJ,552,23deBlok,W.J.G.&McGaugh,S.S.,1997,MNRAS,290,533Gerhard,O.,Kronawitter,A.,Saglia,R.P.,&Bender,R.,2001,AJ,121,1936Hayashi,E.,Navarro,J.F.,Power,C.,Jenkins,A.,Frenk,C.S.,White,S.D.M.,Springel,V.,Stadel,J.,Quinn,T.,2003,submittedtoMN-RAS,astro-ph/0310576Keeton,C.R.,2001,astro-ph/0102341
Keeton,C.R.,Kochanek,C.S.,Seljak,U.,1997,ApJ,482,604Kochanek,C.S.,Schneider,P.,Wambsganss,J.,2004,Part2ofGravitationalLensing:Strong,Weak&Micro,Proceedingsofthe33rdSaas-FeeAdvancedCourse,G.Meylan,P.Jetzer&P.North,eds.(Springer-Verlag:Berlin)
Koopmans,L.V.E.&Treu,T.,2003,ApJ,583,606Krist,J.E.,1995,”WFPC2Ghosts,Scatter,andPSFFieldDependence,”inCalibratingHubbleSpaceTelescope:PostServicingMission,eds.A.KoratkarandC.Leitherer,p.311Moore,B.,Governato,F.,Quinn,T.,Stadel,J.,Lake,G.,1998,ApJ,499,L5Moore,B.,Quinn,T.,Governato,F.,Stadel,J.,Lake,G.,1999,MNRAS,310,1147Navarro,J.F.,Hayashi,E.,Power,C.,Jenkins,A.,Frenk,C.S.,White,S.D.M.,Springel,V.,Stadel,J.,Quinn,T.R.,2004,MNRAS,349,1039Navarro,J.F.,Frenk,C.S.&WhiteS.D.M.,1996,ApJ,462,563Power,C.,Navarro,J.F.,Jenkins,A.,Frenk,C.S.,White,S.D.M.,Springel,V.,Stadel,J.,Quinn,T.,2003,MNRAS,338,14Press,W.H.,Teukolsky,S.A.,Vetterling,W.T.,Flannery,B.P.,2001,’NumericalRecipesinFortran77,2ndEdition’,CambridgeUniversityPressRomanowsky,A.J.,Douglas,N.G.,Arnaboldi,M.,Kuijken,K.,Merrifield,M.R.,Napolitano,N.R.,Capaccioli,M.,Freeman,K.C.,2004,Sci-ence,301,1696Rusin,D.,Kochanek,C.S.,&KeetonC.R.,2003,ApJ,595,29Sand,D.J.,Treu,T.,&Ellis,R.S.,2002,ApJ,574,L129Sand,D.J.,Treu,T.,Smith,G.P.,Ellis,R.S.,2004,ApJ,604,88Schramm,T.,1990,A&A,231,19
S´ersic,J.L.,1968,Cordoba,Argentina:Observa-torioAstronomico(1968)Spergel,D.N.&SteinhardtP.J.,
2000,
Phys.Rev.Lett.,84,3760
Swaters,R.A.,Madore,B.F.,vandenBosch,F.C.,Balcells,M.,2003,ApJ,583,732
16
Treu,T.&Koopmans,L.V.E.,2002,ApJ,575,87Wallington,S.,Kochanek,C.S.&Narayan,R.,1996,ApJ,465,64Warren,S.J.,Hewett,P.C.,Lewis,G.F.,Møller,P.,Iovino,A.,Shaver,P.A.,1996,MNRAS,278,139Warren,S.J.,Lewis,G.F.,Hewett,P.C.,Møller,P.,Shaver,P.A.,Iovino,A.,1999,A&A,343,L35Warren,S.J.&Dye,S.,2003,ApJ,590,673(WD03)Wayth,R.B.,Warren,S.J.,Lewis,G.F.,Hewett,P.C.,2004,MNRAS,submitted,astro-ph/0410253
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