Cluster mass profile reconstruction with size and flux magnification on the HST STAGES survey

3.6 MB·

by  Christopher A. J. Duncan

Mon.Not.R.Astron.Soc.000,1–22(2015) Printed12January2016 (MNLATEXstylefilev2.2)
Cluster mass profile reconstruction with size and flux magnification
on the HST STAGES survey
Christopher A. J. Duncan1(cid:63), Catherine Heymans1, Alan F. Heavens2, Benjamin Joachimi3
1ScottishUniversitiesPhysicsAlliance,InstituteforAstronomy,UniversityofEdinburgh,RoyalObservatory,BlackfordHill,Edinburgh,EH93HJ,UK
2ImperialCentreforInferenceandCosmology,DepartmentofPhysics,ImperialCollegeLondon,BlackettLaboratory,PrinceConsortRoad,London,SW72AZ,UK.
3DepartmentofPhysicsandAstronomy,UniversityCollegeLondon,GowerPlace,London,WC1E6BT,UK.
6
1 Accepted1January2016
0
2
ABSTRACT
n
Wepresentthefirstmeasurementofindividualclustermassestimatesusingweaklensingsize
a
J andfluxmagnification.UsingdatafromtheHST-STAGESsurveyoftheA901/902superclus-
8 terwedetectthefourknowngroupsinthesuperclusterathighsignificanceusingmagnifica-
tionalone.WediscusstheapplicationofafullyBayesianinferenceanalysis,andinvestigatea

] broadrangeofpotentialsystematicsintheapplicationofthemethod.Wecompareourresults
O
to a previous weak lensing shear analysis of the same field finding the recovered signal-to-
C noiseofourmagnification-onlyanalysistorangefrom45%to110%ofthesignal-to-noisein
. the shear-only analysis. On a case-by-case basis we find consistent magnification and shear
h
constraints on cluster virial radius, and finding that for the full sample, magnification con-
p
straintstobeafactor0.77±0.18lowerthantheshearmeasurements.
-
o
Keywords: gravitationallensing:weak-cosmology:darkmatter-dataanalysis-galaxies:
r
t clusters
s
a
[

1
1 INTRODUCTION with the shape distortion of distant sources and as a result much
v
3 time has been invested in developing the tools to accurately use
Galaxyclusterscomprisethelargestknowngravitationallybound
2 shear measurements. As an example, competitive analyses of the
objectsintheUniverse.Theycangiveinformationontheforma-
0 accuracyandprecisionofweaklensingobservablemeasurement,
tionofstructureandthecosmologicalmodelthroughknowledgeof
2 suchastheSTEP(Heymansetal.2006;Masseyetal.2007)and
theunderlyingdensityfield.Inordertointerpretgalaxyclustersin
0 GREAT(Bridleetal.2009;Kitchingetal.2010;Mandelbaumetal.
1. acosmologicalscenario,onemusthaveknowledgeoftheindivid- 2014) programs, have primarily focussed their attention on test-
ualmassesoftheclustersthatenterintothesample.Manydifferent
0 ingtheabilityofparticularalgorithmsinmeasuringsourceellip-
observablesarecommonlyusedasaproxyforclustermass,includ-
6 ticity with estimates of source size a secondary concern. Ideally,
ingclustermembercounts(clusterrichness)whichrelyondiscrete
1 onewouldliketoutilisethemaximumnumberofprobesinweak
observablesourcesastracersoftheunderlyingmatterdistribution,
: lensing analyses, as a means of reducing the statistical errors on
v or X-ray luminosity and temperature and the Sunyaev-Zeldovich
measurements for a given source sample, but also as a means of
i effectwhichutilisetheeffectsofhotgasinthevicinityoftheclus-
X mitigatingsystematicsineachindividualanalysis.
ter.Ineachofthesecases,onemustmakesimplifyingassumptions
r
a abouthowthesetracersfollowtheunderlyingdominantdarkmatter
distribution,andtakethisintoaccountwheninterpretingthemea- There has been a recent increasing trend to investigate the
surementasaproxyforthemassofthecluster.Fortheuseofclus- use of other weak lensing observables, including numerous con-
ter members in the optical this requires knowledge of the galaxy vincingdetectionsoffluctuationsinsourcecountsduetolensing
bias.ForX-rayderivedmassesonemustassumehydrostaticequi- byforegroundmatter,mostfrequentlydubbed‘fluxmagnification’
librium,althoughrecentstudiessuggestthatX-rayderivedmasses or‘magnificationbias’.Theseanalysesmeasuredangularcorrela-
may be biased low (Simet et al. 2015). By contrast, gravitational tionfunctionsbetweenradiallyseparatedbins(Myersetal.2003;
lensingusesmeasurementsofbackgroundgalaxysize,shapeorlu- Scrantonetal.2005;Hildebrandt,Waerbeke&Erben2009;Morri-
minositytoprobethelensingtotalmatterdistribution,andisinsen- sonetal.2012),andaroundstackedforegroundover-densitiesasa
sitivetothenatureofthelensingmatteritself. meansofmeasuringstackedmassprofiles(Fordetal.2014,2015;
Theuseofgravitationallensingmeasurementsasamethodof Baueretal.2011;Umetsuetal.2015;Hildebrandtetal.2011)or
mass reconstruction to date have predominantly dealt exclusively determining dust profiles (Me´nard et al. 2010; Hildebrandt et al.
2013).Ofparticularnote,theanalysesofHildebrandtetal.(2011,
2013) measure the mass profiles for high redshift lenses, using a
(cid:63) Email:[email protected] high redshift background source sample where shape determina-
(cid:13)c 2015RAS
2 C.Duncanetal.
tion would be expected to fail, and thus utilises one of the main function.Theyconcludedthathighresolutionspace-basedimaging
strengthsofanumber-countsmagnificationanalysis. isidealforasize-magnificationanalysis.
Contemporaneously,therehavebeenaseriesoftheoreticalin- Arecentobservationalapplicationoftheuseofthesizeand
vestigationsintotheuseofthemagnificationsignaltomeasurecos- magnitudemagnificationeffectistheapplicationinSchmidtetal.
mologicalparameters,throughtheclusteringofaphotometricsam- (2012)tostackedgrouplensingintheCOSMOSfield.Inthispa-
pleinDuncanetal.(2014);Joachimi&Bridle(2010);vanWaer- per, authors claim a detection of the magnification effect with a
bekeetal.(2010);vanWaerbeke(2010)oraspartofajointanaly- signal–to–noise ratio of ∼ 40% of the shear using a maximum-
sisusingaphotometricandspectroscopicsample(Gaztan˜agaetal. likelihoodestimatorbasedaroundtheassumptionoflog-normality
2012;Eriksen&Gaztanaga2015).Itisgenerallyfoundthatwhilst inthesizedistributionandGaussianityinthemagnitudedistribu-
magnificationaloneisuncompetitivewithshearwhenanunknown tion.Inthispaper,weinsteadapplyourmethodofmassestimation
galaxy bias must be simultaneously measured with the data, the using galaxy sizes and magnitudes to individual large clusters of
combinationofclusteringandshearcangiveasignificantincrease M =O(1014)M(cid:12)/hintheSTAGESsupercluster.
inconstrainingpowerovertheshear-onlysignalthroughdegener- In section 2 we detail relevant weak lensing theory, and de-
acy lifting between the clustering, shear and galaxy-galaxy lens- tail a Bayesian method for determining cluster model parameters
ing.Further,Joachimi&Bridle(2010)foundthattheadditionof for a given lens from source size, magnitude and ellipticity mea-
existingnumberdensityinformationtoashearanalysisonapho- surementswhilstavoidingsomeofthesimplifyingassumptionsof
tometric sample can successfully counteract the loss of informa- previousanalyses.Wediscusshowajointanalysisusingallthree
tiondue tothemarginalisation overaflexibleintrinsic alignment observables could be combined in a self-consistent way. In Sec-
model. Such a combined analysis was adopted as part of the pri- tion3wedescribetheSTAGESdatasetandselectionofthesource
mary science driver in Euclid (Laureijs et al. 2011), however in sample.InSection4,themethodisappliedtomockcataloguesde-
Duncanetal.(2014)itwasshownthatsystematicuncertaintiesin signedtoreflectthemainfeaturesofthedata-set,andconclusions
themagnificationsignalcanleadtocatastrophicbiasesincosmo- are drawn on the ability to utilise the method to measure cluster
logicalmodelparameters. modelparametersondifferentmasslenses,andquantifytheeffect
oflimitationsinthedata-setandsimplifyingassumptions.Finally,
Similarly,therehasbeenarecentuptakeininvestigationsinto
inSection5themethodisappliedtotheSTAGESdataset,andre-
thedirectuseofsizeandmagnitudemeasurementstoinferlensing
sultsarepresentedfortheSTAGESclustersandcomparedtopre-
properties,eitherthroughacomparisoninstatisticsoflensedsam-
existingshearmeasurements.WeconcludeinSection6.
ples to unlensed samples (Heavens, Alsing & Jaffe 2013; Alsing
Throughout this paper we assume a flat fiducial cosmology
etal.2015b;Casaponsaetal.2013),orthroughtheuseofthefun-
withw=−1,Ω =0.3,Ω =0.7andh=0.7.Magnitudesare
damentalplanerelation(Huff&Graves2014;Sonnenfeld,Bertin M λ
givenintheABsystem.
&Lombardi2011;Bertin&Lombardi2006).Ineachcase,ma-
jorastrophysicalsystematics,similartointrinsicalignmentsfora
shear analysis, may be present through intrinsic size-density cor-
relations(Ciarlariello,Crittenden&Pace2015),orthecorrelation 2 THEORYANDMETHOD
betweenfundamentalplaneresidualsanddensity(Joachimi,Singh
2.1 WeakLensingTheory
&Mandelbaum2015).
In Heavens, Alsing & Jaffe (2013) it was demonstrated that Asphotonspropagatepastaforegroundmatterdensitycontrast,its
substantialgainscouldbemadeinthecombinationofsizemagni- path is deflected according to the Jacobian mapping between the
ficationwithshear,particularlywhennoisedominated,andnoted sourceplaneandtheobservedskyas
that the noise-free size measurement can be made to be uncorre- (cid:18) (cid:19)
1−g −g
latedtotheshearmeasurementprovidedthatthesizeismeasured A=(1−κ) 1 2 , (1)
−g 1+g
asthesquare-rootofameasuredsourcearea. 2 1
Rozo&Schmidt(2010)forecastanimprovementof∼ 50% inthelinearlimit.Theconvergence(κ)andcomplexreducedshear
inclustermassestimatesfromajointsize-magnification,cluster- (g=g1+ig2 =γ/[1−κ])varywithangularpositiononthesky
ing and shear analysis over shear-only. Eifler et al. (2014) found andarefunctionsofgravitationalpotentialofthelensandgeometry
that constraints on a set of cosmological parameters from a non- ofthelens-sourcesystem.Boththeconvergenceandtheshear(γ)
tomographic COSEBI shear analysis were significantly improved canberelatedtotheprojectedsurfacemassdensityofthelensing
withtheadditionofprojectedclusteringinformation,butthatthe matteras
furtherinclusionofdirectmagnificationdidnotgivesignificantfur-
κ(ξ) = Σ−1 Σ[ξ], (2)
therimprovement. Crit
γ(ξ) = Σ−1 [(cid:104)Σ(cid:105)(<ξ)−Σ(ξ)], (3)
InAlsingetal.(2015b)theauthorsforecastusingatheoreti- Crit
callymotivatedlinearalignmentandintrinsic-size-densitycorrela- whereξ isthedistancebetweenthesourceandlenscentreonthe
tionmodelthatthecombinationofsizeandmagnitudemagnifica- sourceplane,andthemeansurfacemassdensitywithinξisgiven
tionwithshearcangiveimprovementsindarkenergyparameters by(cid:104)Σ(cid:105)(<ξ).Thecriticalsurfacemassdensityisgivenby
of∼ 25 → 65%,whilstquantifyingthetypicaldispersiononthe
inferredconvergencefieldusinganintrinsicsize-magnitudedistri- Σ = c2 Ds , (4)
butionmeasuredwithCFHTLenS. Crit 4πGDdDds
InCasaponsaetal.(2013)itwasshownthroughtheuseofim- whereD andD istheangulardiameterdistancetothesourceand
s d
agesimulationsthatsizemeasurementsusinglensFit(Milleretal. lens,andD istheangulardiameterdistancebetweenthesource
ds
2007)couldestimatetheconvergencefieldinanunbiasedwaypro- andlens.
videdthesourcesamplewasselectedtobeaboveafluxsignal–to– Theconvergencedenotesanisotropicstretchingofthesource
noiseratioof10,andthegalaxiesarelargerthanthepointspread image, with a corresponding change in the observed size of the
(cid:13)c 2015RAS,MNRAS000,1–22
Massprofilereconstructionwithmagnification 3
source.AsaresultoftheapplicabilityofLiouville’sTheorem,this magnification is not unity, and can be avoided by calculating the
changeinsourcesizecorrespondsdirectlytoachangeintheob- fieldmeanoveralargeareaoronablankfield.Thirdly,wherea
servedfluxofthesource.Consequently,thelensedsizeandfluxof singlesourceisconsidered,orthesourcesampleischosenwithin
asourcecanberelatedtoitsunlensedquantitiesaccordingto flux of size ranges, any intrinsic size-luminosity relation must be
consideredtoaccountfortheflux-lensingofthesample,andtoen-
1
R = µ2R0, (5) surethattheestimatorcomparesmeansizesofequivalentsamples.
S = µS , (6) Finally,suchanestimatorgivesanestimatefortheaveragemag-
0
m = m +2.5log µ, (7) nificationfactorforthesourcesample.It’sphysicalinterpretation
0 10
is therefore only straight-forward where the sources are selected
whereR,S andmrepresentthesourcesize1,fluxandmagnitude locally, or on a region where they are expected to experience the
respectively, subscript “0” denotes intrinsic (or unlensed) quanti- same magnification, such as in an annulus around a spherically-
ties,andthelocalmagnificationfactorµisgivenby symmetriclensmassdistribution.
µ=[det(A)]−1 =[(1−κ)2−γ2]−1. (8) Thispapermotivatesadeparturefromsuchaformalism,and
inthenextsectionwedetailaBayesianinterpretationofthemag-
Theactionofamagnificationfieldisthereforetoalterthesizeand nificationfieldsimilartothatdetailedinAlsingetal.(2015b),but
brightness of a lensed source, or locally shift the size-magnitude withanemphasisoninferringthemassmodelparametersforanin-
distributionforthesourcesample.Equivalently,onemayconsider dividuallensassumingknowledgeoftheintrinsicsize-magnitude
theactionofthemagnificationfieldasalocalshiftintheimposed andredshiftdistributionsofthesourcesample.Wediscussindetail
source size and flux limits of the analysis or data: together with theadvantageofsuchamethod,andextendittoincludeelliptici-
changesintheobservedpositionofthesources,thisformsthebasis ties,aswellasdiscussiontheapplicationofafulljointshearand
offlux-magnificationanalysesthroughclusteringstatistics. magnificationanalysiswithinthisframework.
Figure1showsasanexampletheactionofaconstantposi-
tiveconvergencefield(associatedwithalensingforegroundover-
density)onamodelsize-magnitudedistributioninthepresenceof 2.2.2 Ajointsizeandfluxmagnificationanalysis
abrightmagnitudelimit,andlargeandsmallsizelimits.Theaction
Considerasingleobservationofthesizeandmagnitude(R,m)ofa
ofsuchafieldistomaketheobservedsourceslargerandbrighter
lensedsource,fromwhichwewanttoplaceconstraintsonthemass
than their intrinsic values (blue crosses to red on the left panel),
profileofthelensingmedium.InBayesiannomenclature,wewish
consequentlylocallyremovingoraddingsourcestothesample(red
toconstructaposteriordistributionforasetofparameterswhich
andblueregionsintherightpanel).
definethelensingclustermassprofile(hereafterdenotedusingα)
fromanobservationoflensedquantities.ApplyingBayes’theorem,
2.2 BayesianMassProfileReconstruction thiscanbeformulatedas
2.2.1 Motivation p(R,m|α)p(α)
p(α|R,m)= ∝p(R,m|α)p(α). (10)
p(R,m)
In Heavens, Alsing & Jaffe (2013); Alsing et al. (2015b); Cas-
aponsa et al. (2013); Schmidt et al. (2012) the authors presented The likelihood [p(R,m|α)] describes the probability of making
theframeworkfortheuseofafrequentistestimatorbasedmethod suchanobservationgivenamodelforthelensingmassprofile,and
ofprobingthemagnificationfieldindifferingcontexts.Generally, priorknowledgeontheclustermassprofilemaybesetusingp(α).
insuchananalysis,oneconstructsanestimatorbasedonthemag- Fortheremainderofthisdiscussionweassumeaflatprior,andthe
nificationrelationsgiveninequations5to7.Forexample,forthe evidence[p(R,m)]istakenasanormalisingconstant,howeverthis
sizeinformationonecanconstructanestimatoras canbeeasilyrelaxed.
(cid:18) R (cid:19)2 The likelihood can be related to intrinsic quantities by
µˆ= (9) marginalisingoverthesequantitiesasnuisanceparameters
(cid:104)R(cid:105)
field (cid:90)
where the numerator corresponds to the size of the source or the p(R,m|α) = dm0 dR0 dzp(R,m|α,R0,m0,z)
meanofalocallyselectedsourcesample,andthedenominatorcor-
× p(R ,m ,z|α), (11)
respondstothemeansizeoverthewholefield,assumedtobean 0 0
(cid:90)
unbiasedestimatorofthemeanofthedistributionofintrinsicsizes = dm dR dzp(R,m|α,R ,m ,z)
0 0 0 0
forthesampleconsidered.
Theuseofsuchanestimatorrequiresspecialcare.Firstly,one × p(R0,m0|α)p(z|R0,m0,α). (12)
musttakeintoaccountthepresenceofsizeofflux/magnitudecuts
By integrating over an assumed redshift distribution where the
requires an alteration of the relations in equations 5 to 7 using
sourceredshiftisnotknown,themethodautomaticallytakesinto
magnification ‘responsivity’ factors to account for sources being
accountthepossibilitythesourceliesradiallyclose-toorinfrontof
boostedoutsidetheselimits,andthesefactorsmustbethemselves
thelens.Theintrinsicsize,magnitudeandredshift(R ,m ,z)of
estimated from the data (see Alsing et al. 2015b; Schmidt et al. 0 0
thesourcearetakentobeindependentofthelensingforegroundso
2012, forfurtherdiscussion).Secondly,theestimatorreliesonthe
p(R ,m |α)→p(R ,m )andp(z|R ,m ,α)→p(z|R ,m ).
assumptionthatthefieldmean((cid:104)R(cid:105) inthisexample)isrepre- 0 0 0 0 0 0 0 0
field By enforcing this simplification, one assumes that there are no
sentativeoftheunlensedmeanofthesourcesample.Thiscanoccur
intrinsicsize-,magnitude-norredshift-densitycorrelationswhich
whenthe‘field’sampleischosenoveranareawheretheaverage
could cause a general change in size or magnitude of a popula-
tionofsourcesphysicallyclosetothelens.Thisassumptionshould
1 Thesourcesizeistypicallydefinedasthesquare-rootoftheareaofthe giveaccurateresultsifthesourcesampleisselectedtoberadially
source distantfromthelenssothatthelensingeffectdominates,however
(cid:13)c 2015RAS,MNRAS000,1–22
4 C.Duncanetal.
= 0.3
20
15
e
z
Si
el
x10
Pi
5
26 24 22 20 18 16 26 24 22 20 18 16
Apparent Magnitude Apparent Magnitude
Figure1.IllustrativefigureshowingtheeffectoflensingonabodyofsourceswhosesizesandmagnitudesaresampledfromamultivariateGaussian.Blue
crossescorrespondtounlensedsources,whilstredcrosses(leftpanelonly)showlensedcounterpartsafteraconstantconvergencefieldofκ=0.3isapplied.
Dashedlines(andarrows)showthedirectionofshiftinthesize-magnitudeparameterplanceaftertheapplicationoftheconvergencefield.Horizontaland
verticallinesshowlimitsontheobservedsourcesizeandmagnituderespectively.Therightpanelshowstheequivalentlocalchangeinenforcedsourcesize
andmagnitudelimits:redareasshowregionsofparameterspacenowunobservableonthelensedpatchofsky,whilstgreenareasshowregionsonlyobservable
duetotheactionofthelocalconvergencefield.Sourcesinthered(green)patcharethereforeremoved(added)totheobservedsourcesample.
such separation is not always possible. The implications of such theform
coonreremlaatyionnostiestthaaktegnivtoenbeaosuuittwabitlhetmheodsceolpfoerotfhtihsirsewlaotirokn,,hoonweecvaenr p(R,m|α)=(cid:90) dzµ−12p[R0,m0|z](cid:16)µ−12R,m+2.5log10µ(cid:17)
naturally incorporate this model into the intrinsic size-magnitude
relationbykeepingtheαdependenceofthistermexplicit. × p[z|m0,R0](z|m+2.5log10µ,µ−12R),
Wheretheintrinsicsize,magnitudeandredshiftofthesource (14)
are known, the final line of equation 12 is described by a prod-
wherethenotationp (y)denotestheprobabilitydensityfunction
uct of Dirac Delta functions centred on these values. In this case [x]
of x evaluated at x = y. The likelihood for each galaxy is then
themagnificationfactorassociatedwiththatlens-sourcesystemis
constructed by sampling the intrinsic size-magnitude distribution
well known. In practice, such quantities are not observable, and
alonga‘de-lensing’line,i.e.takingtheprobabilitythatthesource
onemayinsteadmarginaliseoverthedistributionoftrueproperties
hasanintrinsicsizeandmagnitudegivenbyitsmeasuredquantities
conditioned on observed values. This distribution must be repre-
correctedforthemodelledlocalmagnificationfieldgivenbycluster
sentativeofthesourcesampleconsidered,andthereforeaccurately
parametersα.Asimilarresultisgiveninequation9ofAlsingetal.
reflecttheselectioncriteriainproducingthesourcesamplebeing
(2015b) where the likelihood is constructed for the convergence
consideredtoensureparametervaluesareunbiased:forexample,
assumingthelinearisationofthelensingrelations.
wherethesampleisconsideredinatomographicredshiftbin,the
Theposterioronlensmassprofileparameterscanthenbecon-
redshiftdistributionshouldreflectthischoice.Theextensiontoto-
structedforasinglesourcebyreapplicationofBayes’Theorem(as
mographicsamplesistrivial,howeverthiscomeswiththecaveat
inequation10),andjointconstraintsusingthewholesourcesam-
thattheformalismpresentedhereassumesthattheredshiftdistribu-
ple can be obtained by multiplying single-source likelihoods (or
tionisthatofthetrueredshiftforthesample:whereanuncertainty
summinglog-likelihoods)intheusualway.
isassociatedwiththemeasuredredshift,thiscanbeincorporated
by integrating over a latent variable (discussed further in section
2.2.6).
2.2.3 Normalisationofthelikelihood
Intheabsenceofmeasurementnoise,theformerterminequa-
tion12containsinformationonthelensingofthesourceandcan Ifthesourcesampleischosenusingsomeselectionbasedonpa-
bedeterminedusingtherelationsgiveninEquations5to6as rametersalteredbythemagnificationfield(e.g.size,magnitudeor
fluxsignal–to–noise)thismustbetakenintoaccountintheevalua-
tionofthelikelihoodtoavoidinaccurateparametermeasurements.
1
p(R,m|α,R0,m0,z) = δD(R−R0µ2[α,ξ,z]) (13) Insuchacase,theapplicationofanon-zeromagnificationfactor
× δD(m−m0+2.5log10{µ[α,ξ,z]}), willshiftthetrueunderlyingintrinsicsize-magnitudedistribution
inthesizeandmagnitudeplanes,alteringthenormalisationofthe
likelihood(seeFigure1).Wherehardsizeandmagnitudecutsare
whereξdenotesthephysicaltransverseseparationofthelensand
usedthelikelihoodmustbenormalisedsuchthat
sourceandissuppressedfortheremainderofthistextforclarity.
Usingachangeinvariables,themarginalisationovertheintrinsic (cid:90) mu (cid:90) Ru
dm dRp(R,m|α)=1, (15)
sizeandmagnitudecanbecarriedoutsothatthelikelihoodtakes
ml Rl
(cid:13)c 2015RAS,MNRAS000,1–22
Massprofilereconstructionwithmagnification 5
where the integrals are understood to extend over lensed quanti- measuredsizeandmagnitudeandtheirunlensedcounterparts,the
ties,betweenlowerandupperlimitsdenotedbysubscriptlandu magnification-dependentnatureofthenormalisationcanbemade
respectively. By substituting the form of the likelihood in equa- moreexplicit:
tion 14 and assuming an deterministic relationship between the
(cid:90) dzµ−12 (cid:90) mudm(cid:90) RudRp[R0,m0|z](cid:16)µ21R,m+2.5log10{µ}(cid:17)p[z|m0,R0](cid:16)z|m+2.5log10{µ},µ12R(cid:17),
ml Rl
(cid:90) (cid:90) mu+2.5log10{µ} (cid:90) µ−21Ru
= dz dm dR p(R ,m )p(z|m ,R )=1.
ml+2.5log10{µ} 0 µ−12Rl 0 0 0 0 0
Thenormalisationvarieswithmagnificationfactor,andconsequently integratingoverintrinsicquantitiesasnuisanceparameters
with the set of cluster mass profile parameters (α) for a given (cid:90)
source. In contrast to the case where no cuts are applied, such a p(R,m,e|α) = dzdR0 dm0 d2e0 p(R,m,e|α,R0,m0,e0,z)
normalisation will change the shape of the recovered likelihood,
× p(R ,m ,e )p(z|R ,m ,e ), (20)
and thus neglecting this effect will bias recovered cluster profile 0 0 0 0 0 0
parameters. whereedenotesthesetof bothellipticitycomponentsinagiven
Here,wehaveconsideredonlyhardcutsonthedata,however co-ordinate frame. As before, the second term gives the redshift
inrealityitmayoftenbethecasethatasmoothselectionfunction distribution of the population from which the source was a sam-
isappliedtothedata.Suchacaseisconsideredinmoredetailin pled,andanyredshiftdependenceoftheintrinsicellipticity,sizeor
Alsing et al. (2015b) and can be easily extended to the analysis magnitudecanbeincorporatedintothisterm.Thefirstterminthis
presentedherewheretheformoftheselectionfunctionisknown. equationgivestherelationbetweentheobservedquantitiesandthe
intrinsicquantities,whichisassumedtobedeterministicandsolely
duetolensinginthelimitofnegligiblemeasurementerrors
2.2.4 Analysisusingsizesormagnitudesonly
Whereonlyreliablemagnitudeinformationisavailable,posteriors p(R,m,e|α,R0,m0,e0,z)=δD(R−R0µ12[α,ξ,z])
maybeproducedbymarginalisingthelikelihoodgiveninequation
×δ (m−m +2.5log {µ[α,ξ,z]})
14overthefullrangeofsourcesizesconsideredinthesample D 0 10
×δ (e−E[e ,g]), (21)
D 0
p(m|α)=(cid:90) dz(cid:90) µ−21Ru dR0p (R ,m+2.5log {µ}) where E denotes the action of the lensing reduced shear on
µ−21Rl µ12 [R0,m0] 0 10 the nuisance intrinsic ellipticity parameter considered, such that
E−1(e,g) = e andE(e ,g) = e.Intheweaklensinglimit,the
×p (z|m+2.5log {µ},R ), (16) 0 0
[z|m0,R0] 10 0 observedellipticitymayberelatedtotheintrinsicellipticityofthe
and sourceandtheappliedshearfieldbywayofaTaylorExpansion
(cid:90) mu+2.5log10{µ}dm0 p(m0|α)=1. (17) eα(e0,g)=eα0 + ∂∂eeαγβ+O(|γ|2)=eα0 +Pαγβγβ, (22)
ml+2.5log10{µ} β
wherethecoefficientofthelineartermisfrequentlyreferredtoas
In the final relation, we have again assumed a deterministic,
the ‘shear responsivity’, and details how the measured ellipticity
lensing-only relation between observed magnitude and intrinsic
respondstotheappliedshearfield,andEinsteinsummationisas-
magnitude, to make the magnification-factor-dependent nature of
sumed.Intheparlanceusedhere,thiscanbeexpressedas
thenormalisationexplicit.
Similarly,asize-onlylikelihoodmaybeformedbymarginal- E =eα+Pγ γ
α 0 αβ β
isingoverthelensedmagnitude,giving
E−1 =eα−Pγ γ .
α αβ β
p(R|α) = (cid:90) dzµ−21 (cid:90) mu+2.5log10{µ}dm0 (18) Similar expressions can be determined where the weak lensing
ml+2.5log10{µ} limithasnotbeenapplied,asinSeitz&Schneider(1995,1997).
× p[R0,m0](cid:16)µ−12R,m0(cid:17)p[z|m0,R0](z|m0,µ−21R), Usingtheseexpressions,thelikelihoodisthengivenby
with p(R,m,e|α) = (cid:90) dz(cid:32)(cid:89)2 ∂E−1(cid:33) p(µ−12R,m+2.5log µ,E−1)
∂e 10
i
(cid:90) µ−1/2Ru i=1
dR0 p(R0|α)=1. (19) × p(z|µ−12R,m+2.5log µ,E−1). (23)
µ−1/2Rl 10
Whenellipticitymeasurementsonlyareconsidered,thiscanbere-
ducedto
2.2.5 Extensiontoellipticities
Whereellipticityinformationisalsoavailable,theaboveformalism p(e|α)=(cid:90) dz(cid:32)(cid:89)2 ∂E−1(cid:33) p(E−1)p(z|E−1) (24)
∂e
i
canbeextendedtoconstructajointshearandmagnificationanal- i=1
ysisofthelensmassprofile.Thelikelihoodcanbeconstructedby andtheposteriorforthesourcesampleconstructedasbefore.
(cid:13)c 2015RAS,MNRAS000,1–22
6 C.Duncanetal.
2.2.6 Includingmeasurementnoise 2.2.7 AdvantagesandCaveats
Sofar, wehaveconsideredthe case wherethedatais considered We have motivated a way to produce full posterior distributions
exact,howeverinrealitythedatawillconsistofnoisyestimators of cluster parameters based on the assumption of an underlying
ofthetrueunderlyingquantityofinterest.Inthiscase,therelation mass profile model which can be related to lensing observables,
betweentheobservedsize,magnitude,ellipticityorredshiftandthe andaprioriknowledgeoftheintrinsicsize-magnitudedistribution.
intrinsicvaluesassociatedwiththesourcegalaxiesisnotlongera The main strengths in utilising such a technique lies in the flexi-
deterministicrelationshipdependentonlyonthelensingmass,and bility of the method: complications and extensions can be easily
therelationsgiveninequation21and13nolongerhold. addedthroughexplicitmarginalisationoflatentvariablesprovided
Noiseinthedatacanbeintegratedwithinthisformalismby theycanberelatedtotheobservablesandintrinsicquantities,and
marginalisingoveralatentvariablewhichdenotesthetruelensed thisisdone explicitlyinthemarginalisation overanapriori red-
quantity.Forthesourcesize,magnitudeandellipticity,thisrequires shift distribution. In contrast to the frequentist analysis described
that in section 2, in this formalism any intrinsic correlation between
the size and magnitude measures is encompassed in the intrinsic
(cid:90) size-magnitudedistribution,negatingtheneedforanycorrection.
p(R,m,e|α,R ,m ,e ,z)= dmˆ dRˆ d2eˆp(R,m,e|Rˆ,mˆ,eˆ)
0 0 0 Further, the method can be applied to produce lens mass profile
×p(Rˆ,mˆ,eˆ|α,R ,m ,e ,z), constraintsforeachsourceindividually,simplifyingtheinterpreta-
0 0 0
tionofthemeasurementsforachosensampleofsources.
(25)
The use of a priori distributions means that the method can
beeasilyimplementedusingwell-motivatedtheoreticalmodels,or
where variables with a hat denote latent variables which are
usingmeasurementsfromthedatawhereavailable.Assuch,theap-
marginalisedover.Inthisrelation,thep(R,m,e|Rˆ,mˆ,eˆ)therefore
plicationcanbeentirelyself-consistent.However,wherethemodel
reflectstheuncertaintyinthemeasureddata,andthelatterrelation
ismeasuredfromdata,onemustbeawarethatnoiseorsystematic
givestheusuallensingrelations(givenbyequation21)).
uncertainties in the measurements can enter the analysis through
Similarly,uncertaintyintheredshiftestimatecanbeabsorbed
theireffectontheaprioridistributionsthemselves.Wherethisis
intotheanalysistaking
thecase,onlysystematicerrorsinthemeasuredintrinsicquantities
which vary in a spatially dependent way will be problematic, as
(cid:90)
constantoffsetsacrossthewholefieldwillcauseaidenticalshift
p(z|R ,m ,e )→ dzˆp(z|zˆ)p(zˆ|R ,m ,e ). (26)
0 0 0 0 0 0 inboththeaprioridistributionsandthesample,providedtheyare
bothequallyaffected.Thiswillthereforenotaffecttherecovered
mass profile parameter interpretation. Noise in the measured dis-
Wherethemeasurementnoiseisadditiveonthequantityofinter-
tributionscanbedealtwithbysmoothing,orfittingatheoretically
est,eachofthesecasesconsiderstheconvolutionofthenoise-free
motivated model to the data. Similarly, the intrinsic distributions
likelihoodwithadistributiondescribingtheuncertaintyonthepa-
shouldbeconstructedfromasamplewhichisrepresentativeofthe
rameterofinterest,wherethewidthofthedistributionvarieswith
de-lensedsourcesample.Thiscanbedonebyconstructingthedis-
eachsource.Assuch,theapplicationofsuchamarginalisationin
tribution across a large area, where the average magnification is
brute force will extend the run-time of the likelihood evaluation
by a factor of N for each noisy redshift, size or magnitude esti- unity,orusingasampleoffieldgalaxies.
matorpersource,whereN describesthenumberoftimesthatthe Aparticularadvantageoftheuseofthismethodisthefactthat
noise-freelikelihoodmustbesampledtoensureconvergenceofthe posteriorscanbeconstructedindividuallyforeachsourcegalaxy,
convolution.Wherethenoise-freelikelihoodisexpensivetocalcu- andindividuallyforeachforegroundlensbeforefurthercombina-
late(forexampleduetoalargesourcesample,ortherequirement tion.Assuch,forananalysiswhichaimstomaximisesignal–to–
tomarginaliseovermanylatentvariables),thismayresultinapro- noisebystackinglenses,theapplicationofthismethodallowsone
hibitivelylongruntimewhichrequiresmoreadvancedtechniques tofitthechosenmassprofilemodeltoeachlensindividuallyand
toovercome. producemodelparameterconstraintsforthelenssamplebycom-
Alimitationintheextensiontosuchamarginalisationliesin biningtheselikelihoods.Thisthereforeavoidstheneedtofitamass
the fact that the noise-free likelihood has a high dimensionality, profile to the stacked measurement, whose shape can be affected
asitdependsonthesourcepositionandclustermodelparameters bysystematicsineachindividuallensmeasurement.Anexample
aswellaslatentsize,magnitude,andredshift,sothattheevalua- wouldbeinsmearingouttheprofiletowardsthecentrecausedby
tionofthenoise-freelikelihoodonagridwhichcanbeappliedto mis-centeringoneachlensofthestack.Withourapproach,theun-
allsources(removingthisasabottle-neck)isintractable.Alterna- certaincentroidcanbetakenasafreeparameterinthefitforeach
tively, the dependancy on cluster model parameters, source posi- individualclusterinthesample.
tionandredshiftcanbeabsorbedintothelocalmagnificationfac- In the application of this method, we choose to work with
tor, thus significantly reducing the dimensionality of the problem full recovered model parameter PDFs until the final stage where
and allowing the measurement-noise-free likelihood to be evalu- a maximum-posterior estimator is used to visualise the results in
atedonagridoflatentlensedsize,magnitudeandlocalmagnifica- different contexts. Doing so increases the run-time over the case
tionfactor(andredshiftifunknown)whichcanbereferencedfor wherestatisticsareformedfromfrequentistestimators.Especially
eachclustermodelparameterchoiceandsourceconsidered.Whilst in the case where multiple latent variables are marginalised over,
theevaluationofsuchagridisexpensivewheretheevaluationof this can be computationally expensive, however we note that re-
thelikelihoodpersourceisalsoexpensive,suchacasecouldspeed- cent work in advanced statistical techniques such as Hierarchical
uptheapplicationwhenappliedtolargesourcesamplessincethe BayesianInference(e.g.Alsingetal.2015a;Schneideretal.2015)
convolutionitselfisfastusingFFT,givingarun-timescalingfaster andadvancedsamplingmethodscangosomewaytoreducingthe
thanthatdetailedhere. necessaryrun-time.
(cid:13)c 2015RAS,MNRAS000,1–22
Massprofilereconstructionwithmagnification 7
3 THEHSTSTAGESSURVEY along the line of sight using the analytic relations of Wright &
Brainerd(2000).Thebasemodelprofileisafunctionoffourpa-
The Space Telescope A901/902 Galaxy Evolution Survey
rameters,namelythepositionofthecentreoftheprofile(centroid),
(STAGES),(Grayetal.2009)utilisedtheF606WfilteroftheAd-
theredshiftofthelens,thevirialradius/virialmassandtheconcen-
vancedCameraforSurveys(ACS)oftheHubbleSpaceTelescope
tration.WiththeexceptionofCB1atz=0.46(Tayloretal.2004)
(HST)toimageaquartersquaredegreecentredontheA901/2su-
alllensesareplacedatafixedredshiftz=0.165(Grayetal.2009).
percluster. The supercluster is made up of four structures at red-
Followingtheshearanalysis,weusethemass-concentrationrela-
shiftz =0.165,A901aandA901binthenorthandA902andthe
tionofDolagetal.(2004),andtaketheclustercentrepositionsto
SWgroupinthesouth.Inaddition,thereisabackgroundcluster
bethosequotedinHeymansetal.(2008).Asaresult,theNFWfit
(CB1) seen in projection with A902 at redshift z = 0.46, deter-
isafunctiononlyofthevirialmass/virialradius.Whilstwenote
minedwiththeapplicationofa3DlensinganalysisinTayloretal.
thatmorerecentmass-concentrationrelationsexist,andemphasise
(2004).STAGESimagesarecomplementedbyopticalimagingus-
that the centroid position and concentration could be simultane-
ingCOMBO-17(Wolfetal.2003)withfivebroadbandsandtwelve
ouslyfittedusingthismethod,theover-ridingaimofthisanalysis
narrow bands, and which provides high quality photometric red-
istocompareclusterprofileestimatesbetweentheshearandmag-
shifts,withtheprecisionσ ∼0.02(1+z)forabout∼10%ofthe
z nificationanalyses,andsowechoosetosetuptheanalysisusing
brightestgalaxies(R <24)intheSTAGESsample.Wolfetal.
Vega thesameassumptionsasHeymansetal.(2008)tofacilitatecom-
(2004)recommendsthatthelimitofR < 24isappliedinor-
Vega parison.
dertokeepthephotometricredshifterrorscatteratlessthan7%.In
Hildebrandt,Wolf&Ben´ıtez(2008),ananalysisoftheCOMBO-
17 data in the magnitude range 23 < RVega < 24, showed that 3.2 SourceSelection
excluding the narrow band data causes the redshift scatter to in-
WeanalysisthesourcecatalogueusedintheanalysisofHeymans
creaseby30%andthecatastrophicoutlierratetoincreaseby20%.
et al. (2008) (hereafter referred to as H08), matched to the pub-
Thisshowstheimportanceofthenarrow-bandinformationinac-
licly available STAGES catalogue Gray et al. (2009) (hereafter
curate redshift estimation, and also suggests that there would be
G09).TheH08catalogueconsistsof79,366sources(n =96.5
littlegaininusingonlybroad-bandinformationtoextendthepho- gal
sourcespersquarearcminute)withSExtractor(Bertin&Arnouts
tometric redshift range beyond R = 24. STAGES provides
Vega
deep(m (cid:46)27.5),highresolutionHSTimagesof∼70,000 1996)MAG BESTmagnitudeinformation,whilsttheG09source
F606W
catalogue with a higher detection threshold contains a total of
extendedsources,fromwhichalargesamplesetofrobustgalaxy
46,471sources(n =56.5)withSExtractorandGALFIT(Peng
shapes,sizesandfluxescanbeobtained.Themaskedobservational gal
etal.2002)sizeandmagnitudemeasures,aswellasCOMBO-17
footprint of the survey covers ∼ 0.22 square degrees, giving a
globalnumberdensityofsourcesof∼ 85 gal/arcminute2 using redshiftestimationfor10,790sourcesaftermatching.
Aninvestigationintosizemeasurementwithquadrupolemo-
the whole sample of extended sources. Observations were taken
ments (detailed further in Appendix A) found that model-fitting
withinasmallobservationaltimeframe,withgreaterthan50%of
methodsprovideamoreaccuratesizedeterminationforlowsurface
thetilesobservedinonefive-dayperiod,andover90%within21
brightnesssourcesincomparisontonon-parametricmeasuressuch
days,whilstseventileswereobservedsixmonthslater,minimising
asquadrupolemomentsoraperturesizeswhichcannotdistinguish
temporal(andthereforespatial)variationinthepointspreadfunc-
betweenfaint,largegalaxiesandbright,smallgalaxies.Thefinal
tionacrossthefield.Themosaicof80ACStileswhichconstitutes
sourcesamplethereforeconsistsoftheSExtractoraperturemagni-
theSTAGESfieldisshowninHeymansetal.(2008),groupedin
tudeinformation(MAG BEST)asgivenintheH08catalogue,cho-
colourbyobservationperiod.
sentogivethelargestsourcesamplewithmagnitudeinformation,
The application of the analysis outline in section 2.2 to the
withGALFIThalf-lightradiusdeterminedusingtheGALAPAGOS
STAGESdataprovidesauniquesetofidiosyncraticcomplications.
datapipelineandskysubtractionBardenetal.(2012)usedasthe
Thebiggestcomplicationisthelackofredshiftinformationforap-
sourcesizewhereavailable(see Heymansetal.2008;Grayetal.
proximately90%ofthesourcesample.Thisaffectstheanalysisas
2009, for further details on the source catalogue and the source
presentedintwoways:firstly,thelackofmulti-bandphotometry
magnitudeandsizedetermination.).Itisassumedthatgalacticdust
for this sample complicates the removal of cluster members, de-
variationacrossthefieldisnegligibleduetothesmallsurveyarea.
tailedfurtherinthenextsection;andsecondly,withoutredshiftin-
Thea-priorisizeandmagnitudedistributionsareconstructed
formationwemustmarginaliseoveranaprioriredshiftdistribution
from the full catalogue after masking conservative 3 arc-minute
forthesourcestoconvertlensingobservablestoclustermasspro-
apertures around the brightest central galaxy (BCG) cluster cen-
fileparameters.FollowingtheshearapplicationinHeymansetal.
trestoremoveclustermembers.Themeasureddistributionisgiven
(2008),wemodeltheredshiftdistributionas
inFigure2,andformstheaprioridistributionforthisanalysis.The
p(z|m )= 3 (cid:18) z (cid:19)2e−(z/z0)1.5, (27) bottompanelofFigure2showsthemarginalisedmagnitudedistri-
F606W 2z0Γ(2) z0 butionbetweentheH08andG09catalogues.Onecanseethatthe
marginalisedmagnitudedistributionfortheH08catalogueextends
with z = z /1.412, and using the median-redshift magni-
0 median
tofaintermagnitudesthanthepublicG09catalogue,reflectingthe
tuderelationofSchrabbacketal.(2007)
differentselectionofthesourcesample,wheretheH08catalogue
zmedian =0.29[mF606W−22]+0.31. (28) includes smaller and faintersources used in the shear analysis of
H08.
Intheapplicationofthemethod,theapriorisize-magnitude
3.1 MassProfileModelling
distributionisconstructedandsmoothedusingKernelDensityEs-
We model the mass profile of the lensing clusters as spherically timation(KDE),usingabivariate-Gaussiansmoothingwindowin
symmetric NFW profiles (Navarro, Frenk & White 1997), and size and magnitude, with covariance equal to 0.01 times the co-
relate the profile parameters to lensing parameters by projecting variance of the data sample. KDE-smoothed apparent magnitude
(cid:13)c 2015RAS,MNRAS000,1–22
8 C.Duncanetal.
6.00
5.00
4.00
x
pi 3.00
R
n
l 2.00
1.00
0.00
-1.00
9000 0 1000 2000 3000 4000 5000
All
8000 Size Sample
7000
6000
5000
4000
3000
2000
1000
0
20 21 22 23 24 25 26 27
m
F606W
Figure2.Thejointsizemagnitude(upperleft)andmarginalisedsize(upperright)andMF606Wmagnitude(lower)distributionsusedtofortheapriori
size-magnitudedistribution.BlacksolidlinesshowthedistributionsobtainedfromthesourcesinthematchedH08andG09catalogues,whilstthebluedashed
lineshowsthemagnitudedistributionfortheH08catalogueonly
andsizedistributionsconstructedinthismannercomparewellto accountedforfrommagnificationbiasalone.Thissuggeststhatthe
histogramsofthesamequantities. appliedbrightmagnitudeandredshiftcutsareinsufficienttofully
The inadvertent inclusion of cluster members in the source removeclustermembers.
sample can introduce a bias in the derived cluster model param- Thetoppanelsoffigure3showsthedifferencebetweenthe
eters, as they are mistakenly interpreted as lensed sources in the meanmagnitudeinradialbinsaroundtheBCGofeachclusterto
analysis. This is a particular problem in the application to the the field mean (after the masking of the four clusters) as a func-
STAGESdataset,asCOMBO-17redshiftinformationisonlyavail- tionofvaryingthefaintlimitingmagnitudeofthesourcesample.
able for ∼ 10% of the sample, meaning a simple redshift cut is Each magnitude difference can be related to the average magni-
unlikely to remove all cluster members from the sample. We cut ficationfactorforsourceswithinthatannulus,howeveronemust
sourcewithz < 0.2whereredshiftinformationisavailable,and notethatthismeasurehasnotbeencorrectedfortheapplicationof
sourcesbrighterthanm=23,correspondingtoamedianredshift sizeandmagnitudecuts,andisthereforenotanunbiasedestimate
ofz = 0.6inthemedian-redshift-magnituderelationofequation of the cluster mass. However, the use of this estimate can give a
28, following H08. The lower panels of Figure 3 show the num- usefuldiagnosticonthebehaviourofthesignalaroundthecluster
berdensitycontrastofsourcesinannularbinsaroundtheBCGfor centre.Onecanseethat,forA901a,A902andSWthemagnitude
eachofthefourmainclustersconsidered,aftertheapplicationof differenceinradialbinsiswellbehavedatlargeradii,withagen-
redshiftandmagnitudecutsonthesample.Onecanseethateven eral trend towards more negative values as the faint limit used is
aftertheapplicationofsuchcuts,thenumberdensityofsourcesis relaxed. This behaviour may be attributed to the lack of correc-
higherthanthefieldaveragetowardsthecentreofthecluster,most tion for the use of a magnitude cut: where a global faint cut is
noticeablyforA901bandSW,withanamplitudelargerthancanbe applied, the mean measured around a magnification field will be
(cid:13)c 2015RAS,MNRAS000,1–22
Massprofilereconstructionwithmagnification 9
underestimated.Bycontrast,forA901b,theuseofabrighterfaint ofanapplicationofthemethodtoHSTdata,andtoquantifyany
cutshowstheoppositetrend,andweseethatforA901bthemagni- inherentbiasesintheanalysis.
tudedifferenceusingthem < 26sampleisdiscrepantwithmore
relaxedcuts.ThisindicatesthatthesignalaroundA901bissensi-
4.1 MockCatalogueConstruction
tivetothelimitingmagnitude,andprovidesaflagtothereliability
ofthemagnitudeestimationofthefaintsourcesinthatregion.We MockcataloguesareconstructedtomimictheSTAGESdatasetus-
notethatA901bshowsthelargestextendedX-rayemissiononthe ingthefollowingprocess:
STAGES field, and consequently the reliability of the magnitude
(i) Galaxiesarerandomlypositionedinthemocksurveyfield.
determination of the faint sources could be compromised by the
(ii) Each mock galaxy is assigned an intrinsic magnitude, size
presenceofunaccounted-forintra-clusterlighterroneouslyadding
andsignal–to–noiseratiorandomlysampledsimultaneouslyfrom
fluxtothegalaxiesbehindA901b.Asaresult,thesourceschosen
the STAGES catalogue. This preserves the form of the size and
around A901b are taken to be those which satisfy m < 26 such
magnitude distributions in the STAGES field, including any size,
thattheextraintra-clusterlightissub-dominanttothegalaxyflux.
magnitudeandsignal–to–noiseratiocorrelation.Weconsidertwo
Inthiscase,thesampleofsourcesaroundA901bareconsideredas
samples here: the “GALFIT sample” samples GALFIT sizes and
aseparatesampletotheremainingsample,andtheapplicationofa
SExtractormagnitudesdirectlyfromtheMastercatalogue,andas
strictermagnitudecutrequiresthattheposteriorsobtainedforeach
suchconsidersthecasewhereasubsetoftheSTAGESsourceshave
ofthesegalaxiesmustbecorrectlynormalisedtoaccountforthis.
valid size measurements, and therefore most closely reflects the
Motivated by the trends described here, we therefore apply
core cuts on the sample of 1.2(cid:48),1.2(cid:48),0.5(cid:48), and 0.9(cid:48) around the application to the STAGES field; the “All Sizes” sample samples
quadrupolemeasuredsizes(seeAppendixA)andSExtractormag-
A901a, A901b, A902and SW BCGsrespectively (shown as dot-
nitudes from the H08 catalogue, and considers the idealised case
dashed vertical lines in Figure 3). Sources are selected in 3 arc-
whereallsourceshavevalidsizemeasurements.
minute apertures around the cluster BCG, taken from Table 1 of
(iii) Eachgalaxywithm > 23isassignedaredshiftrandomly
H08.InthecaseofA901bthenumberover-densityofsourcesex-
sampledfromaredshiftdistributiongivenbyequation27withme-
tends across the whole angular scale considered here suggesting
dian redshift given by the median-redshift-magnitude relation of
that cluster member contamination may persevere in spite of the
Schrabbacketal.(2007)measuredontheGOODSfield.
applicationofsourceremovalwithinthisaperture,howeverstricter
(iv) Unlensed distributions are output, where all redshifts are
cutswillremoveprogressivelymoreofthesourcesample,andwill
discarded for the unlensed STAGES mock catalogue, and where
leaveonlythosesourcesfurthestfromtheclustercentrewhichare
a mock “COMBO” subset of galaxies is constructed by ran-
leastlensedandwhoselensingparameterdeterminationisexpected
domlysamplingasub-setof10%ofthefullSTAGESmock.The
tobemostnoisy.
COMBOmockswillthereforevaryqualitativelyfromtheobserved
Theapplicationofthemaskaroundtheclustercoreprovides
COMBO-17sub-sampleofSTAGESgalaxieswithredshiftinfor-
anaturalminimumphysicallengthscaleonsource-clustersepara-
mation: in the observations, redshifts are obtained only for the
tion, as the removal of the cone around the centre of the cluster
brightestgalaxies,whilstnomagnitudecutsareappliedinthecon-
meansthatnosourcecanbecloserthanthephysicaldistancebe-
struction of the COMBO mock catalogue; as such the mock will
tweentheclustercentreandtheedgeofthemaskedregiononthe
have an overall larger median redshift than the observations. The
clusterredshiftplane.Aswellaslimitingclustermembers,sucha
COMBO mock catalogues considered here are constructed with
cuthasthefurtheradvantageofreducingtheeffectofanyintrinsic
thepurposeoftestingthesensitivityofthemethodtothechange
size-ormagnitude-densitycorrelationsinadditiontotheredshift
innumbercountsandredshiftknowledgethatresultsfromtheap-
andmagnitudecutsappliedtolimitthepresenceofsourcesradi-
plication of the method to the sub-set of STAGES galaxies with
allyclosetothelens.
COMBO-17 redshift information, and are not constructed to be
Theapplicationofsuchcutsremovesasignificantfractionof
fullyrepresentativeofthatsample.
thesourcesforwhichthelensingsignalwillbestrongest,remov-
(v) Eachgalaxyhasitssizeandmagnitudealteredaccordingto
ing402,515,70and282galaxiesfromthesourcesamplearound
thelensingrelations giveninequations 5and7respectively. The
eachclusterrespectively,withafurther1194faintsourcesremoved
weaklensinglimitisthereforenotenforcedforthemagnification
aroundA901baftertheapplicationofafaintcutofm < 26.The
relations. Each galaxy is assigned a local magnification due to a
needtoapplysuchstrictcorecutsshouldbeconsideredaparticular
set of foreground clusters, modelled as NFW profiles, where the
limitationofthedata-setused,andclustermodelparametervalues
redshiftinformationfromthepreviousstepisretainedandusedto
wouldbeconstrainedtohighersignificanceinadata-setwithmore
evaluateΣ (equation4)foreachgalaxy.Eachlensingclusteris
completeredshiftinformationbyallowingthesampletobesuffi- Crit
placedataredshiftofz =0.165,whichisthemeasuredredshift
cientlycleanedofsourcesclosetothelenswithouttheapplication lens
of the four largest STAGES clusters. Where only a single mock
ofconservativeblanketcuts,orallowingtheapplicationofamodel
cluster is considered, the cluster is placed with its centre on the
toaccountforthepresenceofunlensedclustermembersorintrinsic
BCGoftheA901acluster.Nolimitationsonthesizeofthemagni-
sizeandmagnitude-densitycorrelations.Alternativestothesource
ficationfactorareenforced.Therareoccasionalsourcewhichlies
selectioncriteriaherewhichavoidtheremovalofsourcesfromthe
withinthecausticofthecluster,andthereforeexperienceanega-
sampleareconsideredinAppendixB.
tivemagnificationequivalenttoaflipinparity,areremovedfrom
thesample.
Unless otherwise stated, the intrinsic size-magnitude distri-
bution is constructed from the unlensed catalogue, using the full
4 APPLICATIONTOMOCKS
STAGES dataset even when the COMBO redshift subsample is
In this section, the method described in Section 2.2 is applied to considered,toreducenoise.Nosize-redshiftrelationisenforced,
mockcatalogues,toascertainthelevelofstatisticalerrorexpected howeveraredshift-magnitudedependenceisenforcedbysampling
(cid:13)c 2015RAS,MNRAS000,1–22
10 C.Duncanetal.
0.1 AAAA999900001111aaaa AAAA999900001111bbbb
d
fiel fi0.0
m
›
− 0.1
)
θ
(
m fi0.2
›
0.3
2.0
1.5
n fi
n(θ)−nfi›1›fi.0
›
0.5
0.0
0.1 AAAA999900002222 SSSSWWWW
d
fiel fi0.0
m
›
− 0.1
)
θ
m( fi0.2 mm<<2256..50
›
m<27.0
0.3 m<28.0
2.0
1.5
n fi
n(θ)−nfi›1›fi.0
›
0.5
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
θ [arcmin] θ [arcmin]
Figure3.Plotshowingthedifferencebetweenthemeanmagnitudeandfractionalnumberover-densityofsourcesinradialbinsfromtheBCGagainstthe
fieldmean,forA901a(topleft),A901b(top-right),A902(bottom-left)andSW(bottom-right),asafunctionoflimitingfaintmagnitude.
source redshift using the median-redshift-magnitude relation of sistency.Nocutsareimposedonsourcesize,noronfaintmagni-
Schrabbacketal.(2007)inpoint(iii). tudes. Source sizes and magnitudes have negligible measurement
Mock clusters are modelled as spherically symmetric NFW error,andassuchthemethoddetailedinsection2.2canbeapplied
profiles,wheretheΛ-CDMmass-concentrationrelationofDolag exactly.Thisapplicationthereforeconstitutesanidealisedcase,and
etal.(2004)isenforced:thusthemodelassumedforthemasspro- one must note that the application of size cuts, PSF confusion or
fileparameterrecoveryisexactintheapplicationtothemocksam- measurementerrormaycauseadecreaseintheconstrainingpower
ple. oftheanalysis.
Posteriors are evaluated on the virial radius by default, and
posteriors on the virial mass determined from these results using
4.2 ApplicationofMethod conservationofprobability:
Theapplicationofthemethodischosentomatchitslateruseonthe p(M )∝ p(r200) ∝ p(r200), (29)
STAGESfield.Resultsareshownusingamaskof0.5(cid:48)aroundeach 200 r2200 M23
200
clusterBCG:whilsttheuseofacoremaskisunnecessaryforthe
idealisedcasespresentedhere,thismaskingoftheclustercentreis whereM ∝ r3 wasassumed.Asaresult,evenwheremass
200 200
thesmallestofthecorecutsusedintheapplicationtotheSTAGES constraintsarepresented,aflatprioronvirialradiushasbeenas-
fieldtoremoveclustercontaminants,andisincludedhereforcon- sumed:thistranslatestoaprioronvirialmasswhichdown-weights
(cid:13)c 2015RAS,MNRAS000,1–22