Pixelated Lenses and H_0 from Time-delay QSOs

arXiv:astro-ph/9911231v1 12 Nov 1999PixelatedLensesandH0fromTime-delayQSOs

LiliyaL.R.Williams

DepartmentofPhysicsandAstronomy

UniversityofVictoria

Victoria,BC,V8P1A1,Canada

and

PrasenjitSaha1

DepartmentofPhysics

OxfordUniversity

KebleRoad,

Oxford,OX13RH,UKReceived–2–

ABSTRACT

ObservedtimedelaysbetweenimagesofalensedQSOleadtothe

determinationoftheHubbleconstantbyRefsdal’smethod,providedthemass

distributioninthelensinggalaxyisreasonablywellknown.Sincethetwoorfour

QSOimagesusuallyobservedarewoefullyinadequatebythemselvestoprovide

auniquereconstructionofthegalaxymass,mostpreviousreconstructions

havebeenlimitedtosimpleparameterizedmodels,whichmayleadtolarge

systematicerrorsinthederivedH0byfailingtoconsiderenoughpossibilities

forthemassdistributionofthelens.Weusenon-parametricmodelingofgalaxy

lensestobetterexplorephysicallyplausiblebutnotoverlyconstrainedgalaxy

massmaps,allofwhichreproducethelensingobservablesexactly,andderive

thecorrespondingdistributionofH0’s.Blindtests—whereoneofussimulated

galaxylenses,lensingobservables,andavalueforH0,andtheotherappliedour

modelingtechniquetoestimateH0indicatethatourprocedureisreliable.For

foursimulatedlensedQSOsthedistributionofinferredH0haveanuncertainty

of≃10%at90%confidence.Applicationtopublishedobservationsofthe

twobestconstrainedtime-delaylenses,PG1115+080andB1608+656,yields

H0=61±11kmsec−1Mpc−1at68%confidenceand61±18kmsec−1Mpc−1at

90%confidence.–3–

1.Introduction

MostwaysofmeasuringtheHubbleconstantinvolveaformofdistanceladder,which

utilizesanumberofastrophysicalstandardcandleandstandardrulerrelations,andis

calibratedlocallybyageometricaltechniquesuchasparallax(e.g.,Madoreetal.1999,

Madoreetal.1998,Kennicutt1995).Arecentexcitingdevelopmentinthisfieldistoextend

thereachofthegeometricalrungofthedistanceladderbyusingmasersinorbitaround

galaxycenterstogetdistancestonearbygalaxiesthusbypassingCepheids(Herrnstein

etal.1999).Afewmethodsinvolvenodistanceladder:goodexamplesare(i)inferring

thedistanceofTypeIIsupernovaefromtheirlightcurvesandspectrabymodelingtheir

expandingphotospheres(Schmidtetal.1992),and(ii)comparingtheH0-independent

angularextentofgalaxyclusterstotheirH0-dependentdepthasdeducedbytheX-ray

emission,andtheSunyaev-Zeldovichmicrowavebackgrounddecrementduetothecluster

(Hughes&Birkinshaw1998).

Butthemost‘one-step’methodofallwasproposedbyS.Refsdalin1964,thoughit

hasonlyrecentlybecomefeasible.TheprincipleofRefsdal’smethodissimple.Inasystem

whereahigh-redshiftQSOissplitintomultipleimagesbyaninterveninggalaxylens,the

differenceinlighttraveltimebetweendifferentimages(observableastimedelaysifthe

QSOisvariable)isproportionaltothescalefactoroftheuniverse.Thetimedelayisgiven

bytheschematicformula

󰀁Timedelay󰀂=h−1×󰀁1month󰀂×󰀁imageseparationinarcsec󰀂2

×zlens×󰀁weakdependenceonzlens,zQSO,andcosmology󰀂(1)

×󰀁lens-mass-distributiondependentfactor󰀂

wherethelasttwofactorsareoforderunity.ToobtainH0usingthismethodonerequires

threetypesofinput:(i)theobservedtimedelay(s)betweenQSOimages,(ii)knowledge

ofthecosmology,and(iii)themassdistributioninthelensinggalaxy.Thefirstcanand

hasbeenmeasuredwithincreasingprecisionforabouteightsystemssofar.Thesecondis

notaseriousproblem,becausethedependenceoncosmologyisweakandtheuncertainty

duetoitiseasytoquantify;inthispaperwewillreferallresultstotheEinstein-deSitter

cosmology.TheuncertaintyinH0isdominatedbythethirditem;thenumberofusable

constraintsonthemassdistributioninthegalaxyarefew,whiletherangeofpossible

distributionsishuge.Thus,massdistributionisthemajorsourceofuncertainty.

Twodifferentpathscanbetakentocompensateforourlackofknowledgeaboutthe

galaxy.Oneistoassumeanexactparametricformforthegalaxymassdistributionandfit

theobservedlensingpropertiesasbestaspossible;theotheristotaketheimageproperties

asexact,andtrytoreconstructthegalaxymassmapasbestaspossible.Singleparametric

modelsfixthelasttermin(2)andthuscannotaccountfortheuncertaintyresultingfromit.

Blandford&Kundi´c(1996)advisethatevenifonefindsaparametricgalaxymodelwhich

isdynamicallypossibleandwhichreproducestheimagepropertieswithacceptablylowχ2,–4–

onestillhasto‘aggressivelyexploreallotherclassesofmodels’togetthetrueuncertainty

inH0.Toexplorethemodelspaceinasystematicfashiononeneedstousearepresentation

ofthegalaxymassdistributionthatisgeneralandnotrestrictedtoaparticularform.One

waywouldbetoexpandthemassdistributionusingasetofbasisfunctions,anotheristo

pixelatethegalaxyandtakeeachpixelasanindependentmasselement.Weintroduced

pixelatedmodelsinSaha&Williams(1997,hereafterSW97)butatthattimedidnothave

anystrategyforsearchingmodelspace.Wehavenowextendedthatworktoexplorethe

modelspacewiththegoalofestimatingtheuncertaintyinthederivedvalueofH0.

Theplanofthispaperisasfollows.InSection2wesummarizetheobservationalsituationwithregardtostronglylensedQSOs.InSection3wepresentthegenerallensing

formalismandpointoutafewpropertiesofthelensingequationsthatareusefulin

interpretingtheresultsofmodeling.Wealsoexplainthereasonsforconfiningouranalysis

toPG1115+080andB1608+656fornow.Sections4and5describeourmethodforderiving

H0andtestitonasyntheticsampleviaablindtest.Applicationtotherealsystemscan

befoundinSection6.Section7discussesourresults.

2.ObservedTime-DelayLenses

ThefirstpieceofinputforH0determinationisthemeasurementoftimedelays

betweenthevariousQSOimages.Atthepresenttime,tenmultiply-imagedQSOsalready

havemeasuredtimedelaysorarebeingmonitored:Q0957+561(Kundi´cetal.1997a),

PG1115+080(Schechteretal.1997,Barkana1997),B1608+656(Fassnachtetal.1999),

B0218+257(Biggsetal.1999),PKS1830-211(Lovelletal.1998),HE1104-1805(Wisotzki

etal.1998),B1030+074,B1600+434(Burudetal.1999),J1933+503,andRXJ0911+0551

(Hjorthetal.1999).Inthisworkwelimitourselvesto4-imagelenseswithknownsource

andlensredshiftsandaccuratetimedelaymeasurements;PG1115+080andB1608+656fit

thedescription.

PG1115(Weymannetal.1980)wasthesecondlenstobediscovered.Thesourceis

aradio-quietQSOatzs=1.722.Accuratepositionsfortheimagesweremeasuredby

Kristianetal.(1993);lightcurveswereanalyzedbySchechteretal.(1997),andtime

delaysderivedbySchechteretal.andBarkana(1997).Themainlensinggalaxyisan

outlyingmemberofasmallgalaxygroup,zl=0.311withanestimatedlineofsightvelocity

dispersionof270±70kms−1(Kundi´cetal.1997b).Asummaryofobservationalresultson

thissystemcanbefoundinSW97.

B1608wasdiscoveredintheCosmicLensAll-SkySurvey(Myersetal.1995,Myers

etal.1999).Thelensiseitheraperturbedsinglegalaxyoramerging/interactingpairof

galaxiessuperimposedintheplaneofthesky.Thesourceandlensredshiftsare1.394and

0.630respectively.ThetimedelayswererecentlyreportedbyFassnachtetal.(1999)based

onVLAobservationsspanning7months.Thetimedelaysweuseinthisworkareanearlier