Constraints on the shapes of galaxy dark matter haloes from weak gravitational lensing

25 Pages·2012·1.58 MB·English

A&A545,A71(2012) Astronomy
DOI:10.1051/0004-6361/201219295 &
(cid:2)c ESO2012 Astrophysics
Constraints on the shapes of galaxy dark matter haloes from weak
gravitational lensing
E.vanUitert1,3,H.Hoekstra1,T.Schrabback2,3,D.G.Gilbank4,M.D.Gladders5,andH.K.C.Yee6
1 LeidenObservatory,LeidenUniversity,NielsBohrweg2,2333CALeiden,TheNetherlands
e-mail:[email protected]
2 KavliInstituteforParticleAstrophysicsandCosmology,StanfordUniversity,382viaPuebloMall,Stanford,CA94305-4060,USA
3 Argelander-InstitutfürAstronomie,AufdemHügel71,53121Bonn,Germany
4 SouthAfricanAstronomicalObservatory,POBox9,Observatory7935,SouthAfrica
5 DepartmentofAstronomyandAstrophysics,UniversityofChicago,5640S.EllisAve.,Chicago,IL60637,USA
6 DepartmentofAstronomyandAstrophysics,UniversityofToronto,50St.GeorgeStreet,Toronto,Ontario,M5S3H4,Canada
Received28March2012/Accepted17June2012
ABSTRACT
Westudytheshapesofgalaxydarkmatterhaloesbymeasuringtheanisotropyoftheweakgravitationallensingsignalaroundgalaxies
inthesecondRed-sequenceClusterSurvey(RCS2).Wedeterminetheaverageshearanisotropywithinthevirialradiusforthreelens
samples: the “all” sample, which contains all galaxies with 19 < mr(cid:3) < 21.5, and the “red” and “blue” samples, whose lensing
signalsaredominatedbymassivelow-redshiftearly-typeandlate-typegalaxies,respectively.Tostudytheenvironmentaldependence
ofthelensingsignal,weseparateeachlenssampleintoanisolatedandclusteredpartandanalysethemseparately.Weaddressthe
impact of several complications on the halo ellipticity measurement, including PSF residual systematics in the shape catalogues,
multipledeflections,andtheclusteringoflenses.Weestimatethattheimpactoftheseissmallforourlensselections.Furthermore,
wemeasuretheazimuthaldependenceofthedistributionofphysicallyassociatedgalaxiesaroundthelenssamples.Wefindthatthese
satellitespreferentiallyresidenearthemajoraxisofthelenses,andconstraintheanglebetweenthemajoraxisofthelensandthe
averagelocationofthesatellitesto(cid:4)θ(cid:5)=43.7◦±0.3◦forthe“all”lenses,(cid:4)θ(cid:5)=41.7◦±0.5◦forthe“red”lensesand(cid:4)θ(cid:5)=42.0◦±1.4◦
forthe“blue”lenses.Wedonotdetectasignificantshearanisotropyfortheaverage“red”and“blue”lenses,althoughforthemost
elliptical“red”and“blue”galaxiesitismarginallypositiveandnegative,respectively.Forthe“all”sample,wefindthattheanisotropy
ofthegalaxy-masscross-correlationfunction(cid:4)f − f (cid:5) = 0.23±0.12,providingweaksupportfortheviewthattheaveragegalaxy
45
isembeddedin,andpreferentiallyalignedwith,atriaxialdarkmatterhalo.AssuminganellipticalNavarro-Frenk-Whiteprofile,we
findthattheratioofthedarkmatterhaloellipticityandthegalaxyellipticity f =e /e =1.50+1.03,whichforameanlensellipticity
h h g −1.01
of0.25correspondstoaprojectedhaloellipticityofe =0.38+0.26ifthehaloandthelensareperfectlyaligned.Forisolatedgalaxies
h −0.25
ofthe“all”sample,theaverageshearanisotropyincreasesto(cid:4)f − f (cid:5) = 0.51+0.26 and f = 4.73+2.17,whilstforclusteredgalaxies
45 −0.25 h −2.05
thesignalisconsistentwithzero.Theseconstraintsprovidelowerlimitsontheaveragedarkmatterhaloellipticity,asscatterinthe
relativepositionanglebetweenthegalaxiesandthedarkmatterhaloesisexpectedtoreducetheshearanisotropybyafactor∼2.
Keywords.gravitationallensing:weak–galaxies:halos
1. Introduction thegravitationalpotential.Onsmallscales(∼fewkpc),haloel-
lipticity estimates have been obtained throughthe combination
Overthelastfewdecadesacoherentcosmologicalparadigmhas of strong lensing and stellar dynamics (e.g. van de Ven et al.
developed,ΛCDM,whichprovidesaframeworkforthestudyof 2010; Dutton et al. 2011; Suyu et al. 2012), planetary nebulae
theformationandevolutionofstructureintheUniverse.N-body (e.g. Napolitano et al. 2011) and HI observations in late-type
simulationsthatarebasedonΛCDM predictthat(dark)matter galaxies (e.g. Banerjee & Jog 2008; O’Brien et al. 2010). On
haloescollapse such that their density profilesclosely followa largerscales,thedistributionofsatellitegalaxiesaroundcentrals
Navarro-Frenk-Whiteprofile(NFW;Navarroetal.1996),which hasbeenused(e.g.Bailinetal.2008),butsuchstudieshaveonly
isinexcellentagreementwithobservations.Anotherfundamen- providedconstraintsforrichsystemsthatmaynotberepresen-
talpredictionfromsimulationsisthatthehaloesaretriaxial(e.g. tativeforthetypicalgalaxyintheUniverse.
Dubinski& Carlberg1991;Allgoodetal. 2006),whichappear
Weakgravitationallensingdoesnotdependonthepresence
elliptical in projection. This prediction of dark matter haloes,
ofopticaltracersandiscapableofprovidingellipticityestimates
aswellasmanyothersconcerningtheevolutionoftheirshapes
on a large range of scales (between a few kpc to a few Mpc).
(e.g. Vera-Ciro et al. 2011), the effect of the central galaxy on
Therefore it is a powerfulobservationaltechnique to study the
the darkmatter halo shape (e.g.Kazantzidiset al. 2010;Abadi
ellipticity of dark matter haloes. In weak lensing the distortion
et al. 2010; Machado & Athanassoula 2010) and their depen-
oftheimagesoffaintbackgroundgalaxiesduetothedarkmat-
dence on environment (e.g. Wang et al. 2011), remain largely
ter potentialsof interveningstructures,the lenses, ismeasured.
untestedobservationally.
This has been used to determine halo masses (e.g. van Uitert
Directobservationalconstraintsonthehaloellipticitieshave et al. 2011) as well as the extent of haloes. If galaxies pref-
proventobedifficult,mainlyduetothelackofusefultracersof erentially align (or anti-align) with respect to the dark matter
ArticlepublishedbyEDPSciences A71,page1of25
A&A545,A71(2012)
haloesinwhichtheyareembedded,thelensingsignalbecomes that might have altered the observed shear anisotropy, and in
anisotropic.Thissignaturecanbeusedtoconstraintheelliptic- Sect. 4 we study the impact of two of them: multiple deflec-
ityofdarkmatterhaloesofgalaxies(Brainerd&Wright2000; tionandtheclusteringofthelenses.Theshearanisotropymea-
Natarajan&Refregier2000). surementsareshownandinterpretedinSect.5.Weconcludein
The core assumption in the weak-lensing-based halo ellip- Sect.6.ThroughoutthepaperweassumeaWMAP7cosmology
ticity studiesis thatthe orientationof galaxiesand darkmatter (Komatsu et al. 2011) with σ8 = 0.8, ΩΛ = 0.73, ΩM = 0.27,
haloesarecorrelated;iftheyarenot,theshearsignalisisotropic Ω = 0.046 and the dimensionlessHubble parameter h = 0.7.
b
andcannotbeusedtoconstraintheellipticityofthehaloes.The The errorsonthe measuredandderivedquantitiesin this work
relativealignmentbetweenthebaryonsandthedarkmatterhas generally show the 68% confidence interval, unless explicitly
been addressed in a large number of studies based on numer- statedotherwise.
ical simulations (e.g. van den Bosch et al. 2002, 2003; Bailin
etal.2005;Kangetal.2007;Bettetal.2010;Hahnetal.2010;
2. Lensinganalysis
Deasonetal.2011),instudiesbasedonthedistributionofsatel-
lite galaxies around centrals (Wang et al. 2008; Agustsson & ForourlensinganalysisweusetheimagingdatafromtheRCS2
Brainerd 2010) and in studies based on the ellipticity correla- (Gilbanket al. 2011).The RCS2 is a nearly900 squaredegree
tion function (Faltenbacher et al. 2009; Okumura et al. 2009). imaging survey in three bands (g(cid:3), r(cid:3) and z(cid:3)) carried out with
Thegeneralconsensusisthatalthoughthegalaxyanddarkmat- theCanada-France-HawaiiTelescope(CFHT)usingthe1square
terarealignedonaverage,thescatterinthedifferentialposition degreecameraMegaCam.Inthiswork,weusethe∼700square
angledistributionis large.Bett (2012)examineda broadrange degrees of the primary survey area. The remainder constitutes
of galaxy-halo alignment models by combining N-body simu- the“Wide”componentoftheCFHTLegacySurvey(CFHTLS)
lations with semi-analyticgalaxy formationmodels, and found whichwedonotconsiderhere.Weperformthelensinganalysis
thatformostofthemodelsunderconsideration,thestackedpro- onthe8minexposuresofther(cid:3)-band(r(cid:3) ∼24.3),whichisbest
jected axis ratio becomes close to unity. Consequently, the el- suitedforlensingwithamedianseeingliomf0.71(cid:3)(cid:3).
lipticity of darkmatter haloesmay be difficultto measure with
weaklensinginpractice.
2.1.Datareduction
Knowledgeoftherelativealignmentdistributionisnotonly
crucial for halo ellipticity studies, but also for studies of the The photometric calibration of the RCS2 is described in detail
intrinsic alignmentsof galaxies. Numericalsimulations predict in Gilbank et al. (2011). The magnitudes are calibrated using
that the shapes of neighbouring dark matter haloes are corre- thecoloursofthestellarlocusandtheoverlappingTwo-Micron
lated(e.g.Splinteretal. 1997;Croft& Metzler2000;Heavens All-Sky Survey (2MASS), and have an accuracy better than
etal.2000;Leeetal.2008).Theshapesofgalaxiesthatformin- 0.03magin eachbandcomparedto theSDSS. Thecreationof
sidethesehaloesmaythereforebeintrinsicallyalignedaswell. the galaxyshape cataloguesis describedin detail in van Uitert
Measuringthiseffectisinterestingasitprovidesconstraintson etal.(2011).Wereferreaderstothatpaperformoredetail,and
structure formation. Also, the lensing properties of the large- presenthereashortsummaryofthemostimportantsteps.
scale structure in the Universe, known as cosmic shear, are WeretrievetheElixir1 processedimagesfromtheCanadian
affectedbyintrinsicalignments,andbenefitfromacarefulchar- Astronomy Data Centre (CADC) archive2. We use the THELI
acterizationoftheeffect.Intrinsicalignmentsarestudiedobser- pipeline (Erben et al. 2005, 2009) to subtract the image back-
vationally by correlating the ellipticities of galaxies as a func- grounds, create weight maps that we use in the object detec-
tionofseparation;misalignmentscansignificantlyreducethese tion phase, and to identify satellite and asteroid trails. To de-
ellipticitycorrelationfunctions(e.g.Heymansetal.2004). tect the objects in the images, we use SExtractor (Bertin &
Todate,onlythreeobservationalweaklensingstudieshave Arnouts 1996). The stars that are used to model the PSF vari-
detected the anisotropy of the lensing signal (Hoekstra et al. ation across the image are selected using size-magnitude dia-
2004;Mandelbaumetal.2006a;Parkeretal.2007).Thesestud- grams. All objects larger than 1.2 times the local size of the
ieshaveprovidedonlytentativesupportfortheexistenceofel- PSF are identified as galaxies. We measure the shapes of the
liptical dark matter haloes, as they were limited by either their galaxies with the KSB method (Kaiser et al. 1995;Luppino &
surveysizeandlackofcolourinformation(Hoekstraetal.2004; Kaiser 1997; Hoekstra et al. 1998), using the implementation
Parkeretal.2007)ortheirdepth(Mandelbaumetal.2006a).To describedbyHoekstraetal.(1998,2000).Thisimplementation
improveonthese constraints,weuse the Red-sequenceCluster hasbeentestedonsimulatedimagesaspartoftheShearTesting
Survey2(RCS2; Gilbanketal.2011).Covering900squarede- Programmes(STEP)(the“HH”methodinHeymansetal.2006;
greeintheg(cid:3)r(cid:3)z(cid:3)-bands,alimitingmagnitudeofr(cid:3) ∼24.3and andMasseyetal.2007),andthesetestshaveshownthatitreli-
amedianseeingof0.7(cid:3)(cid:3),thissurveyisverywellsluimitedforlens- ablymeasuresthe unconvolvedshapesofgalaxiesfora variety
ing studies (see van Uitert et al. 2011). Using the colours we ofPSFs.Finally,thesourceellipticitiesarecorrectedforcamera
selectmassiveluminousforegroundgalaxiesatlowredshifts.To shear,whichoriginatesfromslightnon-linearitiesinthecamera
investigatewhethertheformationhistoriesandenvironmentaf- optics. Theresultingshape catalogueof the RCS2 containsthe
fecttheaveragehaloellipticityofgalaxies,thelensesaresepa- ellipticitiesof2.2×107galaxies.Amoredetaileddiscussionof
ratedbygalaxytypeandenvironment,andthesignalsarestudied theanalysiscanbefoundinvanUitertetal.(2011).
separately.
The structure of this paper is as follows. We describe the
lensinganalysis, includingthe data reductionofthe RCS2 sur- 2.2.Lenses
vey,thelensselectionandthedefinitionoftheshearanisotropy
To study the halo ellipticity of galaxies, we measure the shear
estimators, in Sect. 2. We present measurements using a sim-
anisotropy of three lens samples. The first sample contains all
ple shear anisotropy estimator in Sect. 3, and use it to study
thepotentialimpactofpointspreadfunction(PSF)residualsys- 1 http://www.cfht.hawaii.edu/Instruments/Elixir/
tematics in the shape catalogues. Various complications exist 2 http://www1.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/cadc/
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E.vanUitertetal.:Constraintsontheshapesofdarkmatterhaloes
Table1.Propertiesofthelenssamples.
Sample Nlens (cid:4)z(cid:5) (cid:4)Lr(cid:3)(cid:5) (cid:4)|eg|(cid:5) fiso M200 r200 rs
[1010h7−02 L(cid:8)] [1010h−701M(cid:8)] [h−701kpc] [h−701kpc]
All 1681826 0.31 2.86 0.25 0.20 21.1+0.5 150+1 23.9+0.2
−1.4 −3 −0.5
Red 136196 0.43 8.91 0.20 0.41 138±8 280±5 54.6+1.1
−1.0
Blue 147079 0.31 4.68 0.26 0.55 44.4+3.3 192±5 32.7+0.8
−3.8 −0.9
Notes.Columns:numberoflenses,meanredshift,meanluminosity,meanellipticity,fractionoflensesthatareisolated,virialmass,virialradius,
andscaleradius.
galaxieswith19<mr(cid:3) <21.5,andisreferredtoasthe“all”sam- of massive late-type lenses. Finally, we note that ∼70% of the
ple.Thissampleconsistsofdifferenttypesofgalaxiesthatcover “all”sampleareconsideredbluebasedontheiru(cid:3)−r(cid:3)colours.
a broad range in luminosity and redshift. The shear anisotropy Tostudythesecondobjectiveofthelensselection,i.e.tose-
measurement of this sample enables us to determine whether lectmassiveandbrightlow-redshiftlenses,weapplythecolour
galaxiesareonaveragealignedwiththeirdarkmatterhaloes. cuts to the CFHTLS W1 photometric catalogue, and show the
The formation history of galaxies differs between galaxy distribution of absolute magnitudes and photometric redshifts
types, and consequentlythe relation betweenbaryonsand dark of the lens samples in Fig. 1. We find that the “red” lens sam-
matter may differ too. Therefore, the average dark matter halo ple consists of galaxies with absolute magnitudes in the range
shapes,andtheorientationofgalaxieswithinthesehaloes,might −24 < M < −22,andmostwithredshiftsbetween0.3and0.6.
r
dependongalaxytype.Toexaminethis,weseparatethelenses Thegalaxiesfromthe“blue”samplehaveabsolutemagnitudes
asafunctionoftheirtype. in the range −24 < M < −20, and are located at redshifts
r
Various selection criteria have been employed to separate between 0.1 and 0.6. For the blue galaxies, we cannot define
early-type from late-type galaxies. In most cases, galaxies are a criterion that exclusively selects luminouslenses in a narrow
either selected based on the slope of their brightness profiles redshiftrange,basedontheg(cid:3)r(cid:3)z(cid:3)magnitudesalone.Finally,the
(Mandelbaum et al. 2006b; van Uitert et al. 2011), or on their “all” sample has the broadest luminosity and redshift distribu-
colours (Mandelbaum et al. 2006a; Hoekstra et al. 2005). To tion. It is possible to narrow down the redshift range by dis-
study how these selection criteria relate, Mandelbaum et al. carding the lenses with the largest apparent magnitudes from
(2006a)comparetheselectionbasedontheirSDSSu−rmodel eachsample.Wechoosenotto,however,becausethislowersthe
colour to the selection based on the frac_dev parameter3, and signal-to-noiseofthelensingmeasurement,whichconsequently
findthattheassignedgalaxytypesagreefor90%ofthegalaxies. broadenstheconstraintsontheaveragehaloellipticity.
We choose to separate the galaxy types based on their Note that due to the lack of a very blue observing band in
colours, as the g(cid:3)-, r(cid:3)- and z(cid:3)-band colours are readily avail- the CFHTLS, the photometric redshifts below 0.2 are biased
able for all galaxies in the RCS2. The aim of the separation high(Hildebrandtet al. 2012).As a consequence,a fractionof
is two-fold: to make a clean separation between the red qui- the galaxies of the “blue” lens sample may have been shifted
escent galaxies which typically exhibit early-type morpholo- tohigherredshifts,andthuslargerluminosities.Themeanred-
giesandbluestar-forminggalaxiesthattypicallyhavelate-type shift and luminosityof the sample may thereforebe somewhat
morphologies,andtoselectmassivelensesatlowredshiftstoop- smaller thanthe valuesquotedin Table 1, andthe distributions
timize the lensing signal-to-noise,and minimize potential con- showninFig.1areonlyindicative.
tributions from multiple deflections (see Sect. 4.1). To deter-
Sincethedarkmatterhaloellipticityismeasuredrelativeto
mine where these massive low-redshift galaxies reside in the
theellipticity ofthegalaxy,itis interestingtoexaminethe dis-
colour-magnitude plane, we use the photometric redshift cata-
tribution of the latter. In Fig. 2, we show the ellipticity distri-
loguesoftheCFHTLSWidefromtheCFHTLenScollaboration
bution of the lens samples; the mean galaxy ellipticity of each
(Hildebrandtetal.2012),anddefineourboxesaccordingly;de-
sample is given in Table 1. The ellipticity distributions of the
tails of the selection of the “red” and “blue” lens sample are
“all” and “blue” sample are comparable, and are broader than
describedin AppendixA. Note thatthese lenssamplesoverlap
the“red”sampleone,becausethe“all”and“blue”samplehave
with the “all” sample, but not with each other. Details of the aconsiderablefractionofdiscgalaxies.Thedifferencesbetween
samplesaregiveninTable1.
the ellipticity distributions have consequences for the weight-
To study how well we can separate early-types from late-
ingschemeofthelensinganisotropymeasurements,aswewill
types, we compare our selection to previously employed sep-
discuss in Sect. 2.3. In the analysis, we only use galaxies with
aration criteria. Details of the comparison can be found in
0.05 < e < 0.8, which excludes round lenses that do not
Appendix A. We find that the “red” sample is very similar to g
have a well-defined position angle, and very elliptical galaxies
the selection based on the u(cid:3) − r(cid:3) colour, whilst ∼58% of the whoseshapesarepotentiallyaffectedbyneighboursand/orcos-
“blue”sampleareactuallyredaccordingtotheiru(cid:3)−r(cid:3) colour.
micrays.
Most of these contaminants of the “blue” sample are not mas-
Theellipticityofdarkmatterhaloesmaydependontheen-
sive, and actually dilute the lensing signal. The purity of the
vironmentofagalaxy.Wethereforedividethelenssamplesfur-
“blue”samplecouldbeimprovedbyshiftingtheselectionboxes
therintoisolatedandclusteredones,andstudythelensingsig-
tobluercolours,butthisattheexpenseofremovingthemajority
nalseparately.Aswelackredshiftsforallthegalaxies,wehave
3 The frac_dev parameter is determined by simultaneously fit- to use an isolation criterionbased on projectedangularsepara-
ting frac_deV times the best-fitting de Vaucouleur profile plus tions: if the lens has a neighbouringgalaxywithin a fixed pro-
(1-frac_deV) times the best-fitting exponential profile to an object’s jected separation that is brighter (in apparent magnitude) than
brightnessprofile. thelens,itisselectedfortheclusteredsample.Ifthelensisthe
A71,page3of25
A&A545,A71(2012)
Fig.1.Numberoflensesasafunctionofabsolutemagnitudea)andredshiftb)forthethreelenssamples,obtainedbyapplyingidenticalcutsto
theCFHTLSW1photometricredshiftcataloguefromtheCFHTLenScollaboration(Hildebrandtetal.2012).The“all”sample(blacksolidlines)
hasthebroadestdistributions,andcoversabsoluter(cid:3)-bandmagnitudesbetween−18and−24,andredshiftsbetween0and0.6.Theluminosities
ofthe“blue”sample(bluedottedlines)areintherange−24 < M < −20,withredshifts0.15 < z < 0.6.The“red”sample(purpledot-dashed
r
lines)hasthenarrowestdistributions,withluminosities−24< M <−22andredshifts0.3<z<0.6.
r
environmentselectionbasedonapparentmagnitudescannotbe
verypure;afractionofthelensesfromtheisolatedsamplemay
still be the brightest galaxy in a cluster. Some of the lenses of
the clustered sample may in reality be isolated, but have been
selected for the clustered sample due to the presence of bright
foregroundgalaxies.However,thedifferencebetweenthelarge-
scalelensingsignaloftheisolatedandtheclusteredsamplein-
dicates that our selection criterion works reasonably well. The
fraction of the lens sample that is isolated, f , is indicated in
iso
Table1.
2.3.Shearanisotropy
The lensing signal is quantified by the tangential shear, γ,
t
around the lenses as a function of projected separation. As the
distortionsaresmallcomparedtotheshapenoise,thetangential
shear needsto be azimuthallyaveragedovera large numberof
lens-sourcepairs:
ΔΣ(r)
(cid:4)γ(cid:5)(r)= , (1)
t Σ
crit
whereΔΣ(r) = Σ¯(<r)−Σ¯(r)isthedifferencebetweenthemean
projectedsurfacedensityenclosedbyr andthemeanprojected
Fig.2.Ellipticitydistributionoftheg(cid:3)r(cid:3)z(cid:3)-colourselectedlenssamples. surface density at a radius r, and Σ is the critical surface
crit
The dashed lines indicate the ellipticity cuts we apply to exclude the
density:
roundest andmost ellipticallenses. Theellipticitydistributionsof the
“all”andthe“blue”samplearesimilar,butthe“red”samplecontains
c2 D
relativelymoreroundgalaxies. Σ = s , (2)
crit 4πGDD
l ls
withD,D andD theangulardiameterdistancetothelens,the
l s ls
brightestobject,itisselectedfortheisolatedsample.Wetestvar- source,andbetweenthelensandthesourcerespectively.Since
iousvaluesforthefixedminimumseparation,andcomparethe we lack redshifts, we select galaxies with 22 < mr(cid:3) < 24 and
tangential shear at large scales in Appendix B. Based on these areliableshapeestimateassources.Weobtaintheapproximate
results,weuseaminimumseparationof1arcmin.Notethatan sourceredshiftdistributionbyapplyingidenticalmagnitudecuts
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E.vanUitertetal.:Constraintsontheshapesofdarkmatterhaloes
A value of f that is significantly larger (smaller) than
mm
unity at small scales indicates that the dark matter haloes are
(anti-)alignedwiththegalaxies.Systematiccontributionstothe
shear,however,maybiastheanisotropyofthelensingsignal.If
the systematic shear is fairly constant on the scales where we
measure the signal, it can be removed following Mandelbaum
etal.(2006a).Inthisapproach,thecrossshearcomponentcom-
puted in the regionsthat are rotated by 45◦ with respect to the
major/minoraxes (region C and D in Fig. 3), γ×,C−D ≡ (γ×,C −
γ×,D)/2,is subtracted fromthe tangentialshear. Spuriousshear
Fig.3.Schematicofalensgalaxy.Thetangentialshearismeasuredin signalscontributeequallytoγt,A,γt,B andγ×,C−D,andarethere-
regions A and B, the cross shear ismeasured in regions C and D. The foreremoved.Thecorrectedratiothenbecomes:
crossshearissubtractedfromthetangentialsheartocorrectforsystem-
aticcontributionstotheshear. fcorr(r)= γt,B(r)+γ×,C−D(r)· (5)
mm γt,A(r)−γ×,C−D(r)
tSourtvheeyp”hfioetoldmse(trIilcberretdesthiaftl.c2a0ta0l6o)g,uaensdoffinthdeaCFmHedTiLanS s“oDuerecpe Ifγ×,C−(cid:3)D iszero√,theerrorson fmcomrr approximatelyincreasebya
redshift of zs = 0.74. This redshift distribution is not exactly factor 1+1/ 2;ifγ×,C−D isnon-zero,however,theerrorsof
identical to the one of the sources due to the additional shape fcorr caneitherbecomelargerorsmallerthanthoseof f .
mm mm
parametercuts appliedto the sourcesample, which are weakly Alternatively, we can assume that the differential surface
dependenton apparentmagnitude,but the difference is neglig- density distribution can be described by an isotropic part plus
ble. To convert the tangential shear to ΔΣ, we use the average anazimuthallyvaryingpart(Mandelbaumetal.2006a):
criticalsurfacedensitythatisdeterminedbyintegratingoverthe
ΔΣ (r)=ΔΣ (r)[1+2fe cos(2Δθ)], (6)
sourceredshiftdistribution: model iso g
(cid:2)
c2 1 ∞ D where e is the observedellipticity of the lens, Δθ is the angle
(cid:4)Σ (cid:5) = dz p(z ) s ; g
crit 4πGA s s DD from the major axis, and f is the ratio of the amplitude of the
(cid:2) ∞ norm zl l ls anisotropyofthelensingsignalandtheellipticityofthegalaxy,
A = dz p(z ), (3) whichistheparameterwewanttodetermine.Mandelbaumetal.
norm s s
0 (2006a)showthattheazimuthallyvaryingpartisgivenby:
(cid:4)
withp(zs)theredshiftdistributionofthesources,andzlthemean wΔΣe cos(2Δθ)
rmeedashsuifrteotfhtehecrloensssssahmeapr,leγu×s,etdhetocdoemtepromniennetDolfatnhdeDshlse.aWr ienatlhsoe fΔΣiso(r)= 2i(cid:4)iiwiei2g,igc,ios2(2Δθii) , (7)
directionof45◦ fromthelens-sourceseparationvector.Theaz-
withitheindexofthelens-sourcepairs,w theweightappliedto
imuthallyaveragedcrossshearsignalshouldvanishsincegrav- i
theellipticityestimateofeachsourcegalaxy,whichiscalculated
itational lensing does not produce it. If this signal is non-zero,
from the shape noise, and e the ellipticity of the lens. This
however, it indicates the presence of systematics in the shape g,i
ellipticityisalsodeterminedusingtheKSBmethod,anditisa
catalogues.Asthe lensesarelargeandtheirlightmaycontam-
measure of (1−R2)/(1+R2) with R the axis ratio (R ≤ 1) if
inate the lensing signal near the lenses, we only consider the
the lens has elliptical isophotes. To remove contributionsfrom
signalonscaleslargerthan0.1arcminforlenseswithmr(cid:3) > 19,
systematicshear,wealsomeasure
andscaleslargerthan0.2arcminforlenseswithmr(cid:3) <19.These (cid:4)
criteriaarebasedonthereductionofthesourcenumberdensity wΔΣ e cos(2Δθ +π/2)
near the lenses, as discussed in Appendix D. Hence the small- f ΔΣ (r)= i (cid:4)i i,45 g,i i , (8)
45 iso 2 we2 cos2(2Δθ +π/2)
est scales we probe is 28 kpc for the “all” and “blue” sample, i i g,i i
and 34 kpc for the “red” sample at the mean lens redshift. To
whereΣ istheprojectedsurfacedensitymeasuredbyrotating
removecontributionsofsystematicshear(from,e.g.,theimage thesourci,4e5galaxiesby45◦.Thesystematicshearcorrectedhalo
masks),wesubtractthesignalcomputedaroundrandompoints
ellipticityestimatoristhengivenby(f − f )ΔΣ (r).Theaver-
fromthesignalcomputedaroundthereallenses(seevanUitert 45 iso
agevaluesof f , fcorrand(f−f )withinacertainrangeofpro-
etal.2011). mm mm 45
jectedseparationsaredeterminedbycalculatingtheratiooftwo
Thelensingsignalaroundtriaxialdarkmatterhaloeshasan
measurements for each radial bin, and subsequently averaging
azimuthal dependence.If galaxies are preferentially aligned or
oriented at a 90◦ angle (anti-aligned) with respect to the dark thatratiowithintherangeofinterest.Weassumethattheerrors
ofeachmeasurementareGaussian.Consequently,theprobabil-
matterdistribution,thelensingsignalalongthegalaxies’major
itydistributionoftheratioisasymmetric,whichwehavetoac-
axisisrespectivelylargerorsmallerthanalongthe minoraxis,
countfor.Wedescribehowtocalculatethemeanandtheerrors
andthisdependencecanbedetermined.
oftheratioforaradialbin,andhowtoaveragethatratiowithina
To measurethe anisotropyin the signal, we first follow the
certainrangeofprojectedseparations,inAppendixC.Notethat
approach used by Parker et al. (2007). For each lens, the tan-
to convert f, the anisotropy in the shear field, to f = e /e ,
gential shear is measured separately using the sources that lie h h g
within45◦ofthesemi-majoraxis(γ ),andusingthosethatlie the ratio of the ellipticity of the dark matter halo and the el-
within 45◦ of the semi-minor axis (tγ,B ) (indicated by B and A lipticity of the galaxy, we have to adopt a density profile (e.g.
t,A f/f =0.25forasingularisothermalellipsoid,seeMandelbaum
in Fig. 3, respectively). The ratio of the shears captures the h
etal.2006a).
anisotropyofthesignal:
ItisclearfromFig.2thattheellipticitydistributionsofthe
γ (r) red and blue lens samples are different. It is unclear, however,
f (r)= t,B · (4)
mm γ (r) whethertheunderlyingellipticitydistributionofthedarkmatter
t,A
A71,page5of25
A&A545,A71(2012)
haloesdiffersaswell.Iftheunderlyingdistributionissimilarfor and is only larger than the SIE signal on very small scales. If
both samples, the projected dark matter halo ellipticity cannot noredshiftinformationisavailableforthelenses,therapidde-
depend linearly on the galaxy ellipticity. Hence Eq. (6) might cline of the shear anisotropy is particularlydisadvantageousas
notbeoptimal,andcoulddependdifferentlyone .Wetherefore thesignalcanonlybeaveragedasafunctionofangularsepara-
g
generaliseEq.(7)to tion.Consequently,theanisotropysignalissmearedout,making
(cid:4) it harder to detect. Finally, if the galaxy and the halo are mis-
wΔΣeα cos(2Δθ)
fΔΣ (r)= A i(cid:4)i i g,i i ; (9) aligned,thesignaldecreasesevenfurther.Theseconsiderations
iso 2 we2αcos2(2Δθ) showthatweneedverylargelenssamplestoachievesufficient
i i g,i i
signal-to-noisetoenableadetection,anditmotivatesourchoice
A= Σie2gα,i Σieg,i, (10) toselectbroadlenssamples.
Σeα Σe2
i g,i i g,i 2.4.Contaminationcorrection
and calculate it for differentvalues of α. Equation (8) changes
Afractionofoursourcegalaxiesarephysicallyassociatedwith
similarly.ThefactorAinEq.(9)scaleseachmeasurementof f
to the “standard” of α = 1 as used in Mandelbaum et al. thelenses.Theycannotberemovedfromthesourcesamplebe-
(2006a), which eases a comparison of f for different values causewelackredshifts.Sincethesegalaxiesarenotlensed,but
are included in calculating the average lensing signal, they di-
of α. The optimal weight results in the best signal-to-noise of
lutethesignal.Tocorrectforthisdilution,weboostthelensing
themeasurement.
The different halo ellipticity estimators can in principle be signalwith a boostfactor,i.e. the excesssourcegalaxydensity
ratio aroundthe lenses, 1+ f (r). Thisis the ratio of the local
used to study the relation between the ellipticity of the galaxy cg
total(satellites+sourcegalaxies)numberdensityandtheaver-
andtheellipticityoftheirdarkmatterhosts.Inparticular,Eq.(5)
agesourcegalaxynumberdensity.Thiscorrectionassumesthat
is defined such that it dependson the average darkmatter halo
the satellite galaxiesare randomlyoriented.If the satellites are
ellipticity,whilstEq.(9)issensitivetotherelationbetweenthe
preferentiallyradiallyalignedtothelens,thecontaminationcor-
galaxyellipticityandthedarkmatterellipticity.Hencebycom-
paring the fΔΣ (r) for different values of α, we gain insight rectionfortheazimuthallyaveragedtangentialshearwillbetoo
iso
low.Iftheradialalignmentofthephysicallyassociatedgalaxies
in the relation between the ellipticity of the galaxies and their
hasanazimuthaldependence,theshearanisotropycaneitherbe
darkmatterhaloes.Notethatasanalternative,wecouldweight
biasedhighorlow.
Eq.(5)withthelensellipticity.
This type of intrinsic alignment has been studied with
Itisusefultoassessthesignal-to-noiseweexpecttoobtain
seemingly different results; some authors (e.g. Agustsson &
for the shear anisotropy measurement compared to the signal-
Brainerd 2006; Faltenbacher et al. 2007) who determined the
to-noiseofthetangentialshearitself.Forthispurpose,wewrite
galaxyorientationusingtheisophotalpositionangles,haveob-
Eq.(6)initsmostbasicform:
servedastrongeralignmentthanothers(e.g.Hirataetal.2004;
ΔΣ (r)=ΔΣ (r)[1+ f¯cos(2Δθ)], (11) Mandelbaumetal.2005a)whousedgalaxymoments.Thisdis-
model iso
crepancy was attributed by Siverd et al. (2009) and Hao et al.
whichhasthefollowingsolutionfortheanisotropicpart: (2011) to the different definitions of the position angle of a
(cid:4)
wΔΣ cos(2Δθ) galaxy; the favoured explanation is that light from the central
f¯ΔΣ = (cid:4)i i i i · (12) galaxy contaminatesthe light from the satellites, which affects
iso w cos2(2Δθ)
i i i theisophotalpositionanglemorethanthegalaxymomentsone.
Aswe measuretheshapesofsourcegalaxiesusinggalaxymo-
Ifthedarkmatterhaloisdescribedbyasingularisothermalel-
ments,weexpectthatintrinsicalignmenthasaminorimpactat
lipsoid (SIE; see Mandelbaum et al. 2006a), and if the galaxy
isperfectlyalignedwiththehalo,we find f¯ = e /2.Hencethe mostandcanbeignored.
h
To study whetherthe distribution of source galaxieshas an
anisotropic signal is a factor e /2 lower than the isotropic sig-
h
nal. To assess the relative size of the errorof f¯ΔΣ compared azimuthaldependence,weperformtheanalysisseparatelyusing
iso
toΣiso,weinsertEq.(11)intoEq.(12),defineanewweig(cid:6)h(cid:4)tw(cid:5)i ≡ the galaxies residing within 45 degrees of the major axis, and
within 45 degrees of the minor axis. On small scales, the ex-
wicos2(2Δθi),anddeterminetheerrorusingσf¯ΔΣiso =1/ iw(cid:5)i. tendedlightofbrightlensesleadstoerroneousskybackground
Sin(cid:6)ce(cid:4)wiandcos2(2Δθi)ar√euncorrelated,itfollowstha(cid:6)t(cid:4)σf¯ΔΣiso = estimates, which causes a local deficiency in the source num-
1/ iwi(cid:4)cos2(2Δθ)(cid:5) = 2σΔΣiso, with σΔΣiso = 1/ √ iwi the ber density. This deficiency is different along the major axis
error on ΔΣ . Hence the error of f¯ΔΣ is a factor 2 larger and minor axis, which could bias the correction we make to
iso iso
thantheerrorofΔΣ .Consequently,thesignal-to-noiseofthe account for physically associated galaxies in the source sam-
anisotropic part of thisoe lensing signal, (S/N) , is related to the ple.Todeterminewhichscalesareaffected,westudythesource
ani
signal-to-noiseoftheisotropicpart,(S/N) ,as: number density around galaxies as a function of their bright-
iso
(cid:7) (cid:8) ness and ellipticity. The results are shown in Appendix D. For
(S/N) = 0√.15 eh (S/N) . (13) galaxieswithmr(cid:3) <19,wefindalargerdeficiencyalongthema-
ani 2 0.3 iso jor axis onprojectedscales smaller than0.2 arcmin;forgalax-
ies with mr(cid:3) > 19, the deficiency is larger on projected scales
In the best-case scenario, the expected signal-to-noise of the smaller than 0.1 arcmin. Therefore, we only use scales larger
shearanisotropyisanorderofmagnitudelowerthanthesignal- than 0.1 arcmin for galaxies with mr(cid:3) > 19, and scales larger
to-noiseoftheazimuthallyaveragedshear.Applyingthecorrec- than 0.2 arcmin for galaxies with mr(cid:3) < 19. The overdensities
tion to remove systematic contributions√increases the errors of around the lens samples are shown in Fig. 4. We find that the
theshearanisotropybyanotherfactorof 2.Ifthedarkmatteris source sample is only mildly contaminatedby physicallyasso-
describedbyanellipticalNFW,thesignaldecreasesrapidlywith ciatedgalaxies,astheoverdensitiesreachamaximumexcessof
increasingseparation (see Fig. 2 of Mandelbaumet al. 2006a), only 30%for the “red” lenses at the smallest projectedsepara-
A71,page6of25
E.vanUitertetal.:Constraintsontheshapesofdarkmatterhaloes
with1+f (r,Δθ)theazimuthallyvaryingexcessgalaxydensity
cg
ratio, and Δ(cid:2)Σ the unboosted lensing signal. We assume that
iso
1+ f (r,Δθ) has a similar azimuthal dependenceas the shear,
cg
andcanbedescribedby
1+ fcg(r,Δθ)= Niso(r)+2NΔθ(r)eαgcos(2Δθ), (15)
with α the exponent of the ellipticity used to weigh the
shear measurement, N the azimuthally averaged boost factor
iso
andNΔθ theamplitudeoftheanisotropy.UsingaTaylorexpan-
sion,wefindthattofirstorder
f(r)= f(cid:10)(r)+ feff(r), (16)
with feff(r) = ANΔθ(r)/Niso(r). To determine feff(r), we mea-
sure both the angle-averaged boost factor, N (r) = N /N ,
iso LS LR
where N denotes the number of lens-source pairs and N
LS LR
the number of pairs of lenses with(cid:4)random sources, and the
azimuthally varying part, ξΔθ(r) = LSeαgcos(2Δθ)/NLR. For
the adoptedmodelof the excessgalaxydensity ratio this gives
Niso(r) = (cid:4)1+ fcg(r)(cid:5)Δθ, which is averaged over the angle, and
ξΔθ =2NΔθ(r)e2gα.Thesemeasurementsarecombinedtogive
dFiisgt.a4n.ceExtocetshsesloeunrscees.gTahlaexgyredeennssiqtyuarraetsio(balsueatfruianncgtiloens)oinfdpircoajteecttehde feff(r)= A(cid:4)1+ fcξgΔ(θr()r(cid:5))Δθ(cid:4)e2gα(cid:5)· (17)
excess density ratio measured using sources within 45 degrees of the Wedeterminetheaveragevalueof feff(r)withinthevirialradius,
major(minor)axis.Thearrowsindicatethelocationofthevirialradius andadditto(cid:4)f − f (cid:5).ThevaluesaretabulatedinTable3.Note
atthemeanredshiftofthelenses.Wefindthattheexcessdensityratio 45
thatasimilarcorrectionisappliedinMandelbaumetal.(2006a).
alongthemajoraxisishigherthanalongtheminoraxis,mostnoticeably
forthe“red”sample.Pleasenotethedifferentscalesoftheverticalaxes. To compare the anisotropy of the distribution of satellites
to the literature, we now assume that at a narrow radial range
the excess galaxy density ratio can be described by 1+ f =
cg
tions. The excess source galaxy density ratio is a few percent Niso+N(cid:10)Δθcos(2Δθ).Wefitthistotheexcessdensityratiointhe
largeralongthemajoraxisthanalongtheminoraxiswithinthe majorand minoraxis quadrants,separately for each radialbin.
virialradiiofthelenssamples,mostnoticeablyforthe“red”lens We use these fits to compute (cid:4)θ(cid:5), the mean angle between the
sample. locationofthesatellitesandthemajoraxisofthecentralgalaxy,
The measured anisotropy is caused by two effects4: using
anisotropicmagnification,and the presence of physically asso- (cid:11)
π/2
creiadtsehdifstsouforcreosurthgaatlaaxreiesa,nwiseotcraonpnicoatldlyisednisttarnibguletetdh.eAtwsoweeffelacctsk. (cid:4)θ(cid:5)= (cid:11)0π/2dθθfcg(θ)· (18)
dθf (θ)
However, we estimate the impact of anisotropic magnification 0 cg
for the lens samples in Appendix E, and find that even in the
InFig.5,we show(cid:4)θ(cid:5) asa functionofprojectedseparationfor
case where the galaxy and the dark matter halo are perfectly
thethreelenssamples.
aligned, the effect is small. We conclude therefore that the ob-
We find that satellite galaxies preferentially reside near the
servedanisotropyistheresultoftheanisotropyofthe distribu-
majoraxisofthelenses,moststronglyforthe“red”lenses.We
tionofsatellitegalaxies. determinetheweightedmeanof(cid:4)θ(cid:5)withinthevirialradius,and
We correctthetangentialshearinthemajorandminoraxis find(cid:4)θ(cid:5) = 43.7◦±0.3◦ forthe“all”sample,(cid:4)θ(cid:5) = 41.7◦±0.5◦
quadrantforthecontaminationbysatellitesbymultiplyingwith for the “red” sample and (cid:4)θ(cid:5) = 42.0◦ ± 1.4◦ for “blue” sam-
theirrespectiveexcessgalaxydensity ratio,beforewe measure ple.Additionally,forthe“red”lenseswefindthat(cid:4)θ(cid:5) becomes
theshearratios.Tocalculatethecorrectionof(f − f ),weob-
45 moreisotropicatlargerprojectedseparations.Itisusefultocom-
servehow fΔΣ (r)changesinthepresenceofphysicallyasso-
iso pareourresultstopreviousstudies,thatarebasedonsimulations
ciatedgalaxiesinthesourcesamplethatareanisotropicallydis-
(e.g. Sales et al. 2007; Faltenbacher et al. 2008; Agustsson &
tributed.RatherthanEq.(7),thequantityweactuallymeasureis
Brainerd2010)andobservations(e.g.Brainerd2005;Agustsson
(cid:5)fΔ(cid:2)Σ (r) = A (cid:9)wiΔΣieαg,icos(2Δθi); &200B8r;aiNnieerrden2b0e0r6g,e2t0a1l0.;2F0a1l1te)n.bInacthheerseetwaol.rk2s0,0(cid:4)7θ;(cid:5)Bisaiflionunedt atlo.
iso N(cid:9)i 1+ fcg(r,Δθ) be in the range between 41◦ and 43◦ for red central galaxies,
whilst no anisotropy is observed for blue central galaxies. We
N = 2 we2αcos2(2Δθ), (14)
i g,i i canonlymakeausefulcomparisonforthe“red”lenssample,as
i thissampleiscomparabletopreviouslystudiedredgalaxysam-
4 AnothereffectismentionedinMandelbaumetal.(2006a)thatcould ples (i.e. predominantlycontaining red early-typegalaxies, the
majorityofthemexpectedtobecentralsbasedontheirluminos-
cause an anisotropic source density ratio: additional lensing by fore-
groundgalaxies.Weestimatethatthishasanegligibleimpactbecause itydistribution).Wefindthattheconstraintsagreewell.Forthe
thenumber of additional massive foreground galaxies issmall due to “blue”and“all” sample,we cannotmakea comparisonto pre-
ourlenssampleselection. viousworkasthesesamplescontainamixtureofearly-typeand
A71,page7of25
A&A545,A71(2012)
Table2.Shearratiosforthelenssamples.
Sample (cid:4)1/f (cid:5) (cid:4)1/fcorr(cid:5)
mm mm
All 1.15+0.10 0.87±0.09
−0.09
Red 0.93+0.10 0.81+0.11
−0.09 −0.10
Blue 1.16+0.19 1.04+0.21
−0.16 −0.17
lenslight,andatlargerseparationsneighbouringstructuresbias
thelensingsignalhigh.ThebestfitM ,r andr aregivenin
200 200 s
Table1.Notethatingeneral,thebestfitmassesarelowerthan
themeanhalomassbecausetheshearofNFWprofilesdoesnot
scalelinearlywithmass,andthedistributionofthehalomasses
isnotuniform(Tasitsiomietal.2004;Mandelbaumetal.2005b;
Cacciato et al. 2009; Leauthaud et al. 2012; van Uitert et al.
2011).Theresultinguncertaintyintheactualmassisnotimpor-
tanthereaswearemainlyinterestedintheextentofthehaloes,
which is affected less (an increase of 30% in mass leads to an
increaseofonly10%insize).
Fig.5.Meananglebetweenthelocationofthesatellitesandthemajor
3. Shearratio
axisofthelensgalaxyasafunctionofprojectedseparation.Theblack
triangles,purplediamondsandbluesquaresindicatetheresultsforthe
In this section we present the measurements of the ensemble-
“all”,“red”and“blue”lenssample.Thearrowsonthehorizontalaxis
averagedratioofthetangentialshearalongthemajorandminor
indicatethelocationofthevirialradiiatthemeanredshiftofthelenses,
axis of the lenses. This is a basic indicator of the presence of
andcorrespondto150kpc,280kpcand192kpcforthe“all”,“red”and
anisotropiesinthelensingsignal.Wenotethattheshearratiois
“blue” lens samples, respectively. The satellitegalaxies preferentially
residenearthemajoraxisofthelenses. notanoptimalestimatorastheweightissimplyastepfunction,
anddoesnotdependonthe ellipticityofthe galaxy.Itenables,
however,acomparisontoParkeretal.(2007).Furthermore,we
late-typegalaxies,andafairfractionofthemisexpectedtobea willusetheshearratiotoexaminehowPSFresidualsystematics
satelliteofalargersystem.Theconstraintsweobtainedarestill intheshapecataloguesaffecttheanisotropy(Sect.3.1).
interesting,however,assimilarselectioncriteriacanbeapplied For all elliptical non-power law profiles, the shear ra-
tosimulations,andtheresultscompared. tio varies as a function of distance to the lens. This radial
dependence differs for different dark matter density profiles
(Mandelbaumet al. 2006a).Hence to obtain constraintson the
2.5.Virialmassesandradii
haloellipticityofthedarkmatter,wehavetoadoptaparticular
To determine to which projected separations the dark matter densityprofile.TocompareourresultstothosefromParkeretal.
haloesofthe galaxiesdominatethelensingsignal, we estimate (2007), we first assume that the density profile follows an SIE
the averagehalo size of each lens sample. For this purposewe profile on small scales. In thatcase, the shear ratio is constant,
modeltheazimuthallyaveragedtangentialshear(afterapplying andwedeterminetheaverageandthe68%confidencelimitsas
thecontaminationcorrections)withanNFWprofile,andfitfor detailedinAppendixC.
themass.TheNFWdensityprofileisgivenby In Fig. 6, we show the average tangential shear along the
major and minor axis, the averagecross shear in the quadrants
δ ρ
ρ(r)= c c , (19) that are rotated by 45 degrees, and the inverse of the shear ra-
(r/rs)(1+r/rs)2 tios fmm and fmcomrr.Thetangentialshearandthecrossshearhave
been multiplied with the projectedseparation in arcmin, to en-
with δ the characteristic overdensity of the halo, ρ the criti-
cal dencsityfor closure of the Universe,and r = r c/c the hance the visibility of the measurementson large scales where
s 200 NFW thesignalisclosetozeroandtheerrorbarsaresmall.Weshow
scale radius, with c the concentration parameter. We adopt
themass-concentratNioFnWrelationfromDuffyetal.(2008) the inverse of the ratios followingthe definitionused in Parker
et al. (2007).We do not observea clear signature for an align-
(cid:12) (cid:13)
c =5.71 M200 −0.084 (1+z)−0.47, (20) mentoranti-alignmentbetweenthelensesandtheirdarkmatter
NFW 2×1012h−1M(cid:8) haloes. Furthermore,we find that on small scales (<1 arcmin),
f and fcorr are consistent, which suggests that the systemat-
mm mm
which is based on numerical simulations using the best fit pa- icspresentonthesescalesaresmallerthanthemeasurementer-
rametersoftheWMAP5cosmology.M isdefinedasthemass rors.Onlargerscales,thedifferenceislarger,whichunderlines
200
inside a sphere with radius r , the radius inside of which the theimportanceofapplyingthecorrectionstoremovesystematic
200
densityis200timesthecriticaldensityρ .Wecalculatethetan- contributions.Thecorrectionislargestforthe“all”lenssample,
c
gential shear profile using the analytical expressions provided becauseitslensingsignalissmallestandthereforemostsuscep-
byBartelmann(1996)andWright&Brainerd(2000).Wefitthe tibletosystematiccontributions.Wedeterminetheaverageshear
NFWprofilebetween50and500kpcatthemeanlensredshift; ratiowithinthevirialradiusatthemeanlensredshift,andshow
closer to the lens the lensing signal might be contaminated by theresultsinTable2.
A71,page8of25
E.vanUitertetal.:Constraintsontheshapesofdarkmatterhaloes
Fig.6.Lensingsignalmultipliedwiththeprojectedseparationinarcminasafunctionofangulardistancefromthelens,forthe“all”lenssample
(left-handpanels),the“red”lenssample(middlepanels)andthe“blue”lenssample(right-handpanels).Inthetoppanels,thegreensquares(blue
triangles)showtheaveragerΔΣalongthemajor(minor)axis(quadrantsB(A)inFig.3).ThedashedlinesindicatethebestfitNFWprofiletimes
theprojectedseparation,fittedtotheazimuthallyaveragedlensingsignalonscalesbetween50and500kpcusingthemeanlensredshift.Inthe
middlepanel,thegreensquares(bluetriangles)showthecrossshearsignalaveragedinquadrantD(C)ofFig.3.Inthebottompanels,1/f and
mm
1/fcorr areshownbytheredsquaresandblacktriangles,respectively.Thedottedlinesindicatethevirialradiusfromthebest-fitNFWprofiles.
mm
Theshearratiodoesnotprovideclearsignsforthealignmentbetweengalaxiesandtheirdarkmatterhaloes.
Parkeretal.(2007)used22squaredegreesoftheCFHTLS shearonlytendstoincrease1/f ;ifsystematicswerepresent,
mm
tomeasuretheshapesof∼2×105lenses,selectedwithabright- the discrepancy would be even larger. Secondly, it is not clear
nesscutof19 < i(cid:3) < 22.Theirlenssampleconsistedofamix- whetherParkeretal. (2007)accountedforthenon-Gaussianity
tureofearly-typeandlate-typegalaxieswith amedianredshift oftheratiooftwoGaussiandistributedvariablesindetermining
of0.4.Theshearratiowasdeterminedusingmeasurementsout theshearratio;thisisparticularlyimportantwhenthesignal-to-
to 70 arcsec (correspondingto 250 h−1kpc at z = 0.4), with a noise of the lensing measurementsis not very high. Generally,
best-fit value of (cid:4)1/f (cid:5) = 0.76± 0.10. Excluding the round accountingfor the non-Gaussianityincreases the positive error
mm
lenseswithe < 0.15,thebest-fitratiois(cid:4)1/f (cid:5) = 0.56±0.13. baroftheshearratio,anddecreasesthenegativeone.Thiscould
mm
ThelenssamplefromParkeretal.(2007)canbebestcompared bringtheirresultclosertoours.Finally,itisnotdescribedhow
to our “all” sample; comparing the relative number of early-/ the average ratio was determined. These differences could ex-
late-types in both samples using the CFHTLS W1 photomet- plainthediscrepancybetweentheresults.
ric redshift catalogue (Hildebrandt et al. 2012), we find they
are similar. Also, the average mass of the lenses are compara-
3.1.ImperfectPSFcorrection
ble. Fitting the shear ratio on the same physical scale, we find
(cid:4)1/fmcomrr(cid:5)=0.98±0.08forthe“all”sample,whichis∼2σlarger Tomeasuretheellipticitiesofgalaxies,wehavetocorrecttheir
thanParkeretal.(2007).Excludinglenseswithe<0.15,wefind observedshapesfor smearingby the PSF. The precision of the
(cid:4)1/fmcomrr(cid:5) = 0.95−+00..1101, which is evenalmost 3σ apart.Since the PSFcorrectionislimited,whichismainlyduetotheinaccuracy
lenssamplesarecomparable,thisismostlikelytheresultofdif- ofthePSFmodel(Hoekstra2004).Hence,residualPSFpatterns
ferencesintheanalysis.Firstly,Parkeretal.(2007)donotapply may still be present in the shape catalogues. These residuals
a correction for systematic contributions. However, systematic affect both the ellipticity estimates of the lens and the source
A71,page9of25
A&A545,A71(2012)
Table3.Best-fitvaluesfortheanisotropyofthegalaxy-masscross-correlationsfunction,(cid:4)f − f (cid:5),andtheratioofthedarkmatterhaloellipticity
45
andthegalaxyellipticity, f ,foranSIEandanellipticalNFWprofile.
h
Sample α (cid:4)feff(cid:5) (cid:4)f − f45(cid:5) fh(SIE) fh(NFW)
All 0.0 1.3±0.6×10−3 0.19±0.10 0.47±0.37 0.96+0.83
−0.80
All 0.5 1.1±0.7×10−3 0.21+0.11 0.57±0.40 1.19+0.89
−0.10 −0.85
All 1.0 0.8±0.8×10−3 0.23±0.12 0.70±0.46 1.50+1.03
−1.01
All 1.5 0.6±1.0×10−3 0.26±0.15 0.83±0.55 1.80+1.23
−1.19
All 2.0 0.4±1.2×10−3 0.29±0.17 0.97±0.65 2.12+1.45
−1.42
Red 0.0 11.9±1.8×10−3 0.13±0.15 0.00±0.58 −0.19+1.09
−1.08
Red 0.5 11.3±2.1×10−3 0.19±0.16 0.05±0.60 −0.14+1.12
−1.10
Red 1.0 9.3±2.5×10−3 0.28±0.18 0.25±0.70 0.20+1.34
−1.31
Red 1.5 7.2±3.1×10−3 0.40±0.22 0.61±0.86 0.87+1.67
−1.63
Red 2.0 5.2±4.0×10−3 0.54±0.27 1.09±1.07 1.82+2.12
−2.08
Blue 0.0 1.5±1.4×10−3 −0.16+0.18 −0.56±0.68 −1.24+1.62
−0.19 −1.65
Blue 0.5 2.0±1.6×10−3 −0.25±0.19 −0.75±0.70 −1.62+1.69
−1.72
Blue 1.0 2.3±1.9×10−3 −0.35+0.21 −1.01±0.81 −2.17+1.97
−0.22 −2.03
Blue 1.5 2.5±2.3×10−3 −0.45±0.26 −1.24±0.96 −2.67+2.36
−2.44
Blue 2.0 2.5±2.7×10−3 −0.53+0.31 −1.44±1.17 −3.06+2.85
−0.32 −2.95
galaxies,albeitwithadifferentamount.Lensgalaxiesaretypi-
callylargeandbright,whilesourcegalaxiesaresmallandfaint,
andhencehardertocorrectfor.Regardlessofthat,PSFresiduals
tendtoalignthelensandsourcegalaxies.Ifnotaccountedfor,
itcouldaddafalseanti-alignmentsignaltotheshearanisotropy
measurement(seeHoekstraetal.2004).
WecorrectforPSFresidualsystematicsinthecataloguesby
subtractingthecrossshearsignalinthequadrantsthatarerotated
by45degreeswithrespecttothemajorandminoraxes(γx,C−D
and f Δ (r) in fcorr and (f − f ), respectively). To quantify
45 iso mm 45
howmuchPSFresidualsactuallycontributetothesecorrection
terms,andtestwhethertheyareproperlyremoved,weintroduce
onpurposeanadditionalbiasinthePSFcorrection,andrecalcu-
latetheshapesofthegalaxies.Usually,theellipticitiesofgalax-
iesintheKSBmethodarecomputedasfollows:
(cid:14) (cid:15)
1 Psm
e = (cid:10)−(1+b)× (cid:10)(cid:11) , (21)
g P Psm(cid:11)
γ
with P the shear polarisability, Psm the smear susceptibility
γ
tensor,and (cid:10) the polarizations(Kaiser et al. 1995).The starred
quantitiesaredeterminedusingthePSFstars.Thebiasbisnor-
mally equal to zero, but to mimic an imperfect PSF correction Fig.7.DifferencebetweentheoriginalandthePSFbiasedshearratios
we set it to −0.05, and recalculate the shapes of all galaxies. (1/fcorr)−(1/fcorr) (blacktriangles)and(1/f )−(1/f ) (red
mm mm bias mm mm bias
Wecreatenewrandomshearcatalogues,andrepeattheanalysis squares)asafunctionofprojectedseparationfromthelensforthethree
usingthese biasedshapes.We showthe differencebetweenthe lenssamples.WefindthatthePSFresidualsareproperlyremovedfrom
originalandthebiasedshearratiosofthelenssamplesinFig.7. thecorrectedratio1/fmcomrr,asthedifferenceisconsistent withzeroon
We find that the difference of the shear ratios that are all scales. For the uncorrected ratio 1/fmm, the difference is negative
anddecreaseswithprojectedseparation.Thisresultshowsthatthecross
determinedusing the originaland the PSF biased cataloguesis
termeffectivelyremovesPSFresidualsintheshearratioestimators.
consistentwithzeroonallscalesfor1/fcorr,theshearratioesti-
mm
matorthatiscorrectedwiththecrossshearterms.Fortheuncor-
rectedshearratioestimator,1/f ,wefindthatthedifferenceis themselves.Inthissection,weestimatetheimpactofthesemul-
mm
consistentwithzeroonsmallscales,butturnsnegativeforpro- tiplelensingeventsonthehaloellipticitymeasurements.Wealso
jected separations larger than a few arcmin. This shows that if studytheimpactoftheclusteringofthelenses,andthecorrela-
PSFresidualsarestillpresentintheshapecatalogues,itaffects tionbetweentheirshapes,ontheshearanisotropy.
1/f ,butnot1/fcorr.HenceweconcludethatPSFresidualsare
mm mm
properlyaccountedforusingthecrossshearsignal. 4.1.Multipledeflections
Someforegroundgalaxiesinourdatalensboththelensesfrom
4. Impactofmultiplelenses
thelenssamplesandthesourcegalaxies.Wedenotethesefore-
More than one lens may contribute to the shearing of a sin- groundgalaxieswith L2, and our selected lenses with L1. The
gle source galaxy. Furthermore, some of the lenses are lensed impact of these “multiple deflections” on the halo ellipticity
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