CFHTLenS: Higher-order galaxy-mass correlations probed by galaxy-galaxy-galaxy lensing

0.8 MB·English

by  P. Simon

Mon.Not.R.Astron.Soc.000,1–25(2008) Printed10January2013 (MNLaTEXstylefilev2.2)
CFHTLenS: Higher-order galaxy-mass correlations probed
by galaxy-galaxy-galaxy lensing
3 P. Simon1⋆, T. Erben1, P. Schneider1, C. Heymans2, H. Hildebrandt3,1,
1
0 H. Hoekstra4,5, T.D. Kitching2, Y. Mellier6, L. Miller7, L. Van Waerbeke3,
2
C. Bonnett8, J. Coupon9, L. Fu10, M.J. Hudson11,12, K. Kuijken4, B.T.P. Rowe13,14,

n
a T. Schrabback1,4,15, E. Semboloni4, and M. Velander4,7
J 1Argelander Institute for Astronomy, Universityof Bonn, Auf dem Hu¨gel71, 53121 Bonn, Germany.

9 2Scottish UniversitiesPhysicsAlliance, Institute for Astronomy,
Universityof Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK.

] 3Department of Physics and Astronomy, University of BritishColumbia, 6224 Agricultural Road, Vancouver, V6T 1Z1, BC, Canada.
O 4Leiden Observatory,Leiden University,Niels Bohrweg 2, 2333 CA Leiden, The Netherlands.
C 5Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada.
6Institut d’Astrophysique de Paris, Universit Pierre et Marie Curie - Paris 6, 98 bisBoulevard Arago, F-75014 Paris, France.
.
h 7Department of Physics, Oxford University,Keble Road, Oxford OX1 3RH, UK.
p 8Institut de Cienciesde lEspai, CSIC/IEEC, F. de Ciencies,Torre C5 par-2, Barcelona 08193, Spain.
- 9Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan.
o
10Key Lab for Astrophysics, Shanghai Normal University,100 Guilin Road, 200234, Shanghai, China.
r
t 11Department of Physics and Astronomy, Universityof Waterloo, Waterloo, ON, N2L 3G1, Canada.
s 12Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, ON, N2L 1Y5, Canada.
a
13Department of Physics and Astronomy, UniversityCollege London, Gower Street, London WC1E 6BT, UK.
[
14California Institute of Technology, 1200 ECalifornia Boulevard, Pasadena CA 91125, USA.
1 15Kavli Institute for Particle Astrophysics and Cosmology, Stanford University,382 ViaPueblo Mall, Stanford, CA 94305-4060, USA.
v
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6 Versionof10January2013
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1 ABSTRACT
0
3 We present the first direct measurement of the galaxy-matter bispectrum as a
1 function ofgalaxyluminosity, stellarmass and SEDtype. Our analysisuses a galaxy-
: galaxy-galaxy lensing technique (G3L), on angular scales between 9 arcsec to 50 ar-
v
i cmin,toquantify(i)theexcesssurfacemassdensityaroundgalaxypairs(excessmass
X
hereafter) and (ii) the excess shear-shear correlations around single galaxies, both of
r which yield a measure of two types of galaxy-matterbispectra. We apply our method
a
tothestate-of-the-artCanada-France-HawaiiTelescopeLensingSurvey(CFHTLenS),
spanning154squaredegrees.Thissurveyallowsustodetectasignificantchangeofthe
bispectrawithlensproperties(stellarmass,luminosityandSEDtype).Measurements
forlenspopulationswithdistinctredshiftdistributionsbecomecomparablebyanewly
devisednormalisationtechnique.Thatwillalsoaidfuturecomparisonstoothersurveys
or simulations.A significantdependence of the normalisedG3L statistics on luminos-
ity within −236Mr 6−18 and stellar mass within 5×109M⊙ 6M∗ 62×1011M⊙
is found (h=0.73).Bothbispectra exhibit a strongersignalfor more luminous lenses
orthosewithhigherstellarmass(uptoafactor2-3).Thisisaccompaniedbyasteeper
equilateral bispectrum for more luminous or higher stellar mass lenses for the excess
mass. Importantly, we find the excess mass to be very sensitive to galaxy type as
recently predicted with semi-analytic galaxy models: luminous (Mr <−21) late-type
galaxiesshownodetectablesignal,whileallexcessmassdetectedforluminousgalaxies
seemstobeassociatedwithearly-typegalaxies.Wealsopresentthefirstobservational
constraints on third-order stochastic galaxy biasing parameters.
Key words: dark matter - large-scale structure of Universe - gravitationallensing
⋆ Email:[email protected]
2 Simon et al.
1 INTRODUCTION Hoekstra et al. 2002; Pen et al. 2003; Sheldon et al. 2004;
Seljaketal.2005;Simonetal.2007;Julloetal.2012).More
Over the course of the last two decades, the gravitational recently,GGLincombinationwithgalaxyclusteringinred-
lensingeffecthasallowedustoestablishanewbranchofsci- shift surveys has been employed to test general relativity
encethatexploitsthedistortionoflightbundlesfromdistant (Reyesetal.2010),ortosuccessfullyconstraincosmological
galaxies (“sources”) in order to probe the large-scale gravi- parameters (Mandelbaum et al. 2012).
tational field produced by intervening matter. Strong tidal [SW05] Schneider & Watts (2005, SW05 hereafter) in-
gravitational fields cause an obvious distortion of individ- troduced two new GGL correlation functions that involve
ualgalaxyimages(“stronglensing”;cf.Meylanetal.2006), three instead of two galaxies, either two lenses and one
whereas weak deflections can only be inferred by statisti- source(“lens-lens-shear”)ortwosourcesandonelens(“lens-
cal methods utilising many galaxy images (“weak lensing”; shear-shear”).Therefore,thisnewclassofcorrelatorsrepre-
cf. Schneider2006). For the latter, usually shear image dis- sents the third-order level of GGL or simply “G3L”. Both
tortions are harnessed, although the study of higher-order correlators express new aspects of the average matter dis-
flexion distortions may also be feasible in the near future tributionaroundlenses, which can betranslated intothird-
(cf. Goldberg & Natarajan 2002; Goldberg & Bacon 2005; ordergalaxybiasingparameters(SW05),especiallyifrepre-
Velanderetal.2011).Recently,thelensingmagnificationef- sentedintermsofaperturestatistics(Schneider1998).This
fect hasalsomovedintothefocusofresearch asnewsource paper chooses the aperture statistics to represent the G3L
of information on cosmological large-scale structure (Hilde- signal. Thereby we essentially express the angular bispec-
brandt et al. 2009). As the gravitational field is solely de- trum of the (projected) matter-galaxy three-point correla-
termined by the mass density of the objects under exami- tion. A rigorous mathematical description of the aperture
nation, no further assumptions on their properties need to statistics is given in the following section.
bemadewhenstudyinglensing.Thismakesitauniquetool A more intuitive interpretation (Simon et al. 2012) of
for cosmologists to examine the large-scale structure of the G3L is given by the definition of the real-space correlation
Universe, in particular the relation between luminous com- functions: the lens-lens-shear correlation function measures
ponents,such as galaxies, and thedark component. Within theaverageexcessshear(orexcessmass,Simonetal.2008)
the current ΛCDM standard model of cosmology (Peacock around clustered lens pairs, i.e., in excess of the average
1999; Dodelson 2003), the major fraction of matter is so- shear pattern around pairs formed from a hypothetical set
calleddarkmatter,whereasordinarybaryonicmatterissub- of lenses that is uniformly randomly distributed on the sky
dominant (Komatsu et al. 2011). Therefore, lensing plays a (unclustered)butexhibitthesameGGLsignalasthelenses
key role in scrutinising the dominant matter component or inthedata.Itisaprobeforthejointmatterenvironmentof
in testing thestandard model. galaxypairs,notsinglegalaxies.Thiscorrelatorpromisesto
Statistical methods have been developed that quan- putadditional constraints on galaxy models (Saghiha et al.
tify theaverage massdistribution aroundgalaxies bycross- 2012) as it appears to be very sensitive to galaxy types.
correlating tangential shear, as observed from background Ontheotherhand,thelens-shear-shearcorrelationfunction
sources, with foreground lens galaxy positions. Galaxy- measures the “excess shear-shear correlation”: it quantifies
galaxylensing(GGL),asthefirsthighlysuccessfulapplica- the shear-shear correlation function in the neighbourhood
tion, in effect measures the stacked projected surface mass of a lens in excess of shear-shear correlations as expected
density profiles around galaxies (Brainerd et al. 1996; Hud- from randomly scattered lenses. Thereby it picks up the
sonetal.1998;Fischeretal.1999;McKayetal.2001;Hoek- (projected) matter density two-point correlation function
stra et al. 2003; Sheldon et al. 2004; Seljak & Warren 2004; of matter physically associated with lenses. In a way this
Hoekstraetal.2004;Kleinheinrichetal.2006;Mandelbaum makes the lens-shear-shear correlator similar to the tradi-
et al. 2006; Parker et al. 2007; van Uitert et al. 2011; Man- tionalGGL,butnowalsoprobingthevarianceinthesurface
delbaumetal.2012;Leauthaudetal.2012).TheGGLsignal matterdensityaround lensesinstead ofmerely theaverage.
is thus a function of lens-source separation (and their red- Theangularmatter-galaxybispectraareFourier-transforms
shifts)only,i.e.,atwo-pointstatisticthatisbasedonalens of these correlators.
andtheimageellipticityofasourcegalaxy.Forareviewsee Simon et al. (2008) have demonstrated with the Red-
Schneider (2006) or Hoekstra & Jain (2008). GGL studies SequenceClusterSurvey(RCS1;Gladders&Yee2005)that
revealed, e.g., a mass distribution far exceeding the exten- bothG3Lcorrelationfunctionscanreadilybemeasuredwith
sion of visible light: lenses are embedded in a dark matter existing lensing surveys. The RCS1 study aimed to obtain
halo of a size with at least 100h−1kpc (Hoekstra et al. a high signal-to-noise ratio of the lensing signal, for which
∼
2004)andameandensityprofileconsistentwiththosefound all available lenses were combined into one lens catalogue.
in ΛCDM simulations (Navarro et al. 1996; Springel et al. Therefore,apartfromthisfeasibility studyinexistingdata,
2005). As extension of GGL, the light distribution within littlemoreisknownonthedependenceoftheG3Lsignalon
the lens can be utilised to align the stacked mass fields, galaxy properties. This paper is a first step to fill this gap
whichallows themeasurementofthemeanellipticityofthe by systematically measuring G3L for a series of lens sam-
halo mass distribution in a coordinate frame aligned with ples with varying properties. The amount of data available
thestellar light distribution of a lens (Hoekstra et al. 2004; through the CFHTLenS analysis allows this to be done for
Mandelbaumetal.2006; vanUitertetal.2012;Schrabback the first time. An accompanying paper by Velander et al.
&CFHTLenSteam 2012).Moregenerally,onlargerspatial (2012) explores the GGL signal of CFHTLenS in the light
scales the technique has been exploited to infer the spatial of thehalo model (Cooray & Sheth 2002).
distribution of lenses with respect to the matter distribu- Thepaperislaidoutasfollows.Sect.2summarisesthe
tion, thesecond-order galaxy biasing (Hoekstra et al. 2001; aperturestatisticsthatisdevisedtoexpresstheG3Lsignal,
Galaxy-galaxy-galaxy lensing in CFHTLenS 3
gives their practical estimators and lists possible sources of By θ∗ we denote the complex conjugate of θ. For this pur-
systematics. In Sect. 3, we outline the selection criteria of pose, the complex ellipticity of thegalaxy image
oursourceandlenssamples.Lensesareselectedbyluminos-
ǫ(θ) γ (θ)+ǫ ; ǫ =0 (4)
ity, stellar mass, redshift and two galaxy spectral types, all ≈ c s h si
to be analysed separately. Sect. 4 presents our G3L results. serves as a noisy estimator of γ ; the noise term originates
c
Foralargerangeofangularscalescoveredinthisstudy,the fromtheunknownintrinsicshapeǫ .Inaddition,duetothe
s
G3Lsignalischaracterisedbyasimplepowerlawwhosepa- finitenumberofsources,onealsoexperiencessamplingnoise
rametersaregiven.Sect.5offersaphysicalinterpretationof of the shear field. Note that we adopt the commonly used
the G3L statistics in terms of 3D galaxy-matter bispectra. complex notation of 2D vectors and spinors (in the case of
In this context, we also introduce a normalisation scheme shearsandellipticities), whererealandimaginary partsare
toremove,tolowest order,theimpactoftheexactshapeof the components along two Cartesian axes in a tangential
the lens redshift distribution and the source redshift distri- plane on thesky.
bution from the signal. Finally, the Sects. 6 and 7 present Galaxy-galaxy lensing techniques correlate the total
our discussion and conclusions. matter distribution κ(θ) with the relative number density
Throughout the paper we adopt a WMAP7 (Komatsu distribution κ (θ) of lens galaxies (“lenses”) on the sky by
g
et al.2011) fiducialcosmology for thematterdensityΩm = means of cross-correlating the lensing signal with positions
0.27,thecosmologicalconstantΩΛ =1 Ωm =0.73(bothin of foreground galaxies,
TcuonhnietsssteroapfinattrhsaemocberttiaetiircnsaeladrdfereoncmosintCsyi)sFtaHennTdtLwHeni0tSh=i−tgs1rea0lv0fih(tKaktmiilobsni−nag1lMelrepnects−ian1lg.. κg(θ)= ng(θn)g−ng =Z0χhdχpf(χ)δg(cid:16)fK(χ)θ,χ(cid:17), (5)
2012; Benjamin et al. 2012; Heymans et al. 2012). If not where pf(χ)dχ is the p.d.f. of the lens (foreground) comov-
stated otherwise, we explicitly use h = 0.73, in particular ing distances along the line-of-sight; ng(θ) is the projected
fortheabsolutegalaxymagnitudesandtheirstellarmasses. numberdensityoflensesandn¯g itsstatisticalmean.Forthe
scopeofthispaper,p (χ)isestimated from aredshiftp.d.f.
f
p (z)dz=p (χ)dχof a selected lens sample.
z f
2 FORMALISM
2.2 G3L aperture statistics
Thissection summarises thetheoryand notation ofG3L as
detailed in SW05, and lists possible G3L specific systemat- For practical purposes, the aperture statistics are a conve-
ics. nient measure for a lensing analysis (Schneider et al. 1998;
Schneider1998;vanWaerbeke1998;Crittendenetal.2002).
They quantify moments of fluctuations in κ(θ) and κ (θ)
g
2.1 Galaxy-galaxy lensing preliminaries withinaperturesofavariableangularscaleθ.Themoments
are determined from thesmoothed fields κ(θ) and κ (θ),
The weak gravitational lensing effect (see Schneider 2006, g
and references therein) probes the three-dimensional rela- d2ϑ
tive matter density fluctuations δm(R⊥,χ) = ∆ρm/ρ¯m in Map(θ) = Z θ2 u |ϑ|θ−1 κ(ϑ) , (6)
(cid:0) (cid:1)
projectionalongtheline-of-sightintermsofthelensingcon- d2ϑ
vergence N(θ) = Z θ2 u |ϑ|θ−1 κg(ϑ), (7)
(cid:0) (cid:1)
κ(θ)= 3Ωm χhdχg(χ)fK(χ)δ f (χ)θ,χ . (1) whereu(ϑ/θ)θ−2 isthesmoothingkernel.Formathematical
2DH2 Z0 a(χ) m(cid:16) K (cid:17) convenience, we placed the aperture centre at θc = 0 in
Here R⊥ =fK(χ)θ is a 2D vector perpendicular to a refer- thepreviousdefinition.Third-ordermomentsaredefinedby
enceline-of-sightandθtheangularpositiononthesky.The considering the ensemble average of
comoving angular diameter distance f (χ) is written as a
K
function of comoving radial distance χ. By DH :=c/H0 we hN2Mapi(θ1;θ2;θ3):=(cid:28)N(θ1)N(θ2)Map(θ3)(cid:29), (8)
definetheHubblelength,anda(χ)isthecosmicscalefactor
atadistanceχ;weseta(0)=1bydefinition;cisthevacuum M2 (θ ;θ ;θ ):= (θ )M (θ )M (θ ) , (9)
hN api 1 2 3 (cid:28)N 1 ap 2 ap 3 (cid:29)
speedoflight.Byχ wedenotethecomovingHubbleradius
h
of today as the theoretical maximum distance at which we overallrandomrealisationsofthefieldsκ(θ)andκ (θ).Due
g
canobserveobjects.Thelensingefficiencyaveragedoverthe totheassumed statistical homogeneityof thefields,theav-
probability densitydistribution function (p.d.f.) pb(χ)dχof eragesdonotdependontheaperturecentreposition.There-
background galaxies (“sources”) is expressed by fore,inpractice,whereonlyonerealisationorsurveyisavail-
g(χ)=Zχχhdχ′pb(χ′)fKf(Kχ(′χ−′)χ) . (2) fauebcrtleesn,Nttha(peθs1ee)rNtquu(rθae2nc)teMintiateprse(sθa3rc)eoavenesrtdiinmNga(ttθeh1de)Mbsuyarpva(veθye2r)aaMrgeianap.g(θSt3eh)eefoFprirgdo.idf1--
Although the convergence in principle is observable for an illustration.
∞
through magnification of galaxy images, past weak lensing For a compensated filter u, i.e., dθθu(θ) = 0, the
0
analyses and this paper focus on the related gravitational aperturemasscaninprinciplebeobtaRineddirectlyfromthe
shear (Kaiser & Squires1993) observable shear through (Schneideret al. 1998)
1 1 ∞ 2π dϕdϑϑ
γ (θ)= d2ϑ (ϑ θ)κ(ϑ) ; (θ):= . (3) M (θ)= q ϑθ−1 (γ(ϑ;ϕ)), (10)
c π Z D − D −(θ∗)2 ap Z Z θ2 ℜ
0 0 (cid:0) (cid:1)
4 Simon et al.
j 1(j +j )
θ
2 2 1 2
N(θ) θ
2 j
1
θ
J
hN(θ)M2(θ)i 1
ap 2
hN2(θ)Map(θ)i f J
3
1
θ
galaxies
3
j
θ
2 j
1
Map(θ) θ
θ
2
1
J J
f
2 1
3
θ
3
convergence
Figure 1. Illustration of the aperture statistics. Fluctuations Figure 2. Illustration of the parametrisation of the lens-
(θ) in the projected galaxy number density (top panel),
sNmoothed to the characteristic filter scale θ, are statistically lens-shear three-point correlator Ge(ϑ1,ϑ2,φ3) (top panel), and
compared to the filtered projected matter fluctuations Map(θ) pthaenelle)n.s-Tshheeaser-sshtaetairstciocsrrealraetieomnpfulonycetdiontoGee±st(iϑm1a,tϑe2,tφhe3)a(pbeortttuorme
(lensing convergence; bottom panel). We take N2(θ)Map(θ) or statisticsinFig.1.ThefigureiscopiedfromSW05.
(θ)M2 (θ), and average these for different aperture centres
N ap
(dashed circles) to estimate third-order moments of the joint
probabilitydistributionofN(θ)andMap(θ). Wegenerally denote a Fourier transform of f(θ) by f˜(ℓ) in
the following. The exponential filter u˜(ℓ) peaks in Fourier
space at an angular wave number of ℓ = √2, which deter-
where γ(ϑ;ϕ) := e−2iϕγ (ϑ) denotes the Cartesian shear
c minesacharacteristic angular scale selected byan aperture
γ at angular posi−tion ϑ rotated by the polar angle ϕ. The
c radius of θ.
real part of γ(ϑ;ϕ) is the tangential shear, the imaginary
part the cross shear. The relation between the filters u(x)
and q(x) is given by
2.3 Aperture statistics estimators
2 x
q(x)= dssu(s) u(x). (11) To obtain the third-order moments of the galaxy-matter
(cid:18)x2 Z0 (cid:19)− aperturestatistics, we utilise thelens-lens-shear correlation
function in thecase of 2M and thelens-shear-shear
Thispaperusestheexponentialaperturefilterfromvan ap
G hN i
Waerbeke(1998), exponential filter hereafter, correlation function G± for hNMa2pi. This section provides
only a brief description of this approach. For a more de-
u(x)= 1 1 x2 e−x2/2 , (12) tails, its computationally optimised implementation as well
2π (cid:18) − 2 (cid:19) asverification,wereferthereadertoSect.3ofSimonetal.
(2008).
whicheffectivelyhasafinitesupportbecauseoftheGaussian
Inpractice,theaperturemoments 2M or M2
factor that suppresses the filter strongly to zero for ϑ&3θ hN api hN api
are not computed from the aperture mass M or aperture
(SW05). The Fourier transform of theaperture filter is ap
number counts directly. The information contained in
N
u˜(ℓ)= d2θu(θ)e+iℓ·θ = ℓ2e−ℓ2/2. (13) the aperture statistics is also contained inside two classes
Z 2 of three-point correlation functions (SW05), which are rel-
Galaxy-galaxy-galaxy lensing in CFHTLenS 5
atively straightforward to estimate. Once the correlation functions.AsshowninSW05(Sect.7.2.therein),thetrans-
functionshavebeendetermined,theycanbetransformedto formation from to 2M yields the same result when
ap
G hN i
thecorrespondingaperturestatisticsbyanintegraltransfor- is taken instead of . Therefore, the integral transforma-
G G
mation. The estimation process thus proceeds in two basic tion automatically ignores unconnected second-order terms
steps.Inthefirststep,for 2Map oneestimatesthesource ien the triple correlator, resulting in an aperture statistics
hN i
tangentialellipticityrelativetothemidpointconnectingtwo that are only determined by pure (connected) third-order
lenses, correlationterms.ThesameholdstrueforG± andhNMa2pi.
(ϑ ,ϑ ,φ )= 1 n (θ )n (θ )γ θ ;ϕ1+ϕ2 . (14) e
G 1 2 3 n2g(cid:28) g 1 g 2 (cid:16) 3 2 (cid:17)(cid:29)
2.4 Relation to 3D galaxy-matter bispectra
e
The meaning of the notation is illustrated in the left panel
of Fig. 2. For M2 one estimates the correlation of the Theaperturestatisticsaredirectlyconnectedtotheangular
ellipticities ofhtNwo aspoiurces relative to the line connecting cross-bispectraoftheprojectedmatterandlensdistribution:
the sources as a function of separation from one lens (right 2M (θ ;θ ;θ )= (19)
ap 1 2 3
panel), hN i
d2ℓ d2ℓ
G±(ϑ1,ϑ2,φ3)= n1g(cid:28)γ(θ1;ϕ1)γ±(θ2;ϕ2)ng(θ3)(cid:29). (15) ZM(22π()1θ2 Z;θ(;2θπ))22=u˜(ℓ1θ1)u˜(ℓ2θ2)u˜(|ℓ1+ℓ2|θ3)bggκ(ℓ1,ℓ(22)0,)
e hN api 1 2 3
iHnerγe±anmdeainnsthγefoforlloγw−in(gineqcausaetioonfsGa−)suapnedrsctrhipetco“m±”pleaxs Z (d22πℓ)12 Z (d22πℓ)22u˜(ℓ1θ1)u˜(ℓ2θ2)u˜(|ℓ1+ℓ2|θ3)bκκg(ℓ1,ℓ2),
conjugate γ∗ for γ+ (in case of G ).
+ e
Both correlation functions are estimated inside bins of where theangular galaxy-galaxy-matter bispectrum is
e
stiiomniloafrctormianpgalreasb,lie.es.i,dleenlesn-sgotuhrsceϑtripalnesdwopitehnininagcaonngfilegsuφra-, hκ˜g(ℓ1)κ˜g(ℓ2)κ˜(ℓ3)i=(2π)2δD(2)(ℓ1+ℓ2+ℓ3)bggκ(ℓ1,ℓ2)(21)
1,2 3
by summing over all relevant galaxy triplets. Any triple of and theangular matter-matter-galaxy bispectrum is
threegalaxy positions θ ,θ ,θ that meetsthecriteria of a
relevanttriangleisflaggeidbjy∆kϑ1ϑ2φ3 =1and∆ϑ1ϑ2φ3 =0 hκ˜(ℓ1)κ˜(ℓ2)κ˜g(ℓ3)i=(2π)2δD(2)(ℓ1+ℓ2+ℓ3)bκκg(ℓ1,ℓ2).(22)
ijk ijk
otherwise.Forthisstudy,weutilise100logarithmicbinsfor For statistically homogeneous random fields, the triple cor-
both ϑ1 and ϑ2, and 100 linear bins for the opening angle relators on the left-hand side of the previous two equations
φ3. For estimating G we utilise can only be non-vanishing when ℓ1 +ℓ2 +ℓ3 = 0, which
est(ϑ1,ϑ2,φ3)= e (16) is reflected by the 2D Dirac delta functions δD(2)(x) on the
G right-hand sides. Owing to homogeneity, thebispectra thus
e −iNP=d1jNP=d1kPN=s1wkǫke−i(ϕi+ϕj)h1+ω(|θi−θj|)i∆ϑij1kϑ2φ3 , wdeepaernbditroanrliylyocnhotowsoe ℓin1daenpdenℓd2e.nTthaisrgauumtoemntastiℓca,llfyorimwphliicehs
Nd Nd Ns wk∆ϑij1kϑ2φ3 tℓh3a=t t−h(eℓb1i+spℓe2c)t.raInaraeddsoitlieolynftuhnecsttioantisstoicfatlhiesomtroodpuyliimofpℓli1e,2s
iP=1jP=1kP=1 and theangle enclosed by both wave vectors.
and for G± theestimator AscanbeseenfromEqs.(19),(20),theaperturestatis-
tics are a locally filtered version of the bispectrum because
Gest(ϑ ,eϑ ,φ )= (17)
± 1 2 3 theexponentialu-filterisrelativelylocalisedinℓ-spacewith
e Nd Ns Ns wjwkǫjǫ±ke−2iϕje±2iϕk∆ϑij1kϑ2φ3 athfieltaepremrtauxriemsutamtiastticℓsmabxas=ic√all2y/θb.eBcoymmeseaansbaonfdthpeofiwlteerribnigs-,
iP=1jP=1kP=1 ,
Nd Ns Ns wjwk∆ϑij1kϑ2φ3 Epeqcst.ru(8m),v(e9rs)iomneoafsbugrgeκtowrobκdκigff.eHreenntceatnhgeualaprergtaulraexsyt-amtiasttticesr
iP=1jP=1kP=1 band power cross-bispectra.
where Nd and Ns are the number of lenses and sources, wi By virtue of the Limber approximation (Kaiser 1992;
are statistical weights of sources, ϕi are polar angles of the Bartelmann & Schneider 2001) the angular bispectra and
position vectors of galaxies with respect to the coordinate thereby the aperture statistics Eqs. (19), (20) can directly
origin, ǫi are the source ellipticities, and berelatedtothe3Dcross-bispectrumofthematterandlens
distribution(SW05)asprimaryphysicalquantitiesthatare
ω(∆θ)= κg(θ)κg(θ+∆θ) (18) assessed by thestatistics:
| | (cid:28) (cid:29)
b (ℓ ,ℓ )= (23)
is the angular two-point clustering of the lenses (e.g. Pee- ggκ 1 2
bles 1980). In this paper, the angular clustering of lenses 3Ωm χhdχ g(χ)p2f(χ)B ℓ1 , ℓ2 ,χ ,
is estimated by means of the estimator in Landy & Szalay 2DH2 Z0 fK3(χ)a(χ) ggm(cid:16)fK(χ) fK(χ) (cid:17)
(1993) prior to the estimation of and then interpolated. b (ℓ ,ℓ )= (24)
G κκg 1 2
Sources are weighed by the inverse-variance uncertainty in
thelensfitellipticity measurement e(Miller et al. 2012). 9Ω2m χhdχ g2(χ)pf(χ)B ℓ1 , ℓ2 ,χ ,
In a second step, we transform the estimates of and 4DH4 Z0 fK2(χ)a2(χ) mmg(cid:16)fK(χ) fK(χ) (cid:17)
G
G± to the aperture statistics by devising the transforma- wherethe3DbispectraaredeterminedbytheFouriertrans-
tion integrals Eqs. (63), (57), and (59) in SW05. Theere is forms of the matter density contrast, δ˜ (k,χ), and galaxy
neoneedtoremovetheunconnectedtermsinthecorrelation number density contrast, δ˜ (k,χ), atmradial distance χ,
g
6 Simon et al.
namely which is a parity violation indicator, a P-mode channel. In
hδ˜g(k1,χ)δ˜g(k2,χ)δ˜m(k3,χ)i= (25) AthsesfhoollwowniinngSwWe0w5,iltlhdeeBno-taentdhPes-emsotdateisstoifcsthaesshtNat2isMtic⊥si(cθa)n.
(2π)3δD(3)(k1+k2+k3)Bggm(k1,k2,χ), be computed from G and G± directly by utilising an alter-
δ˜ (k ,χ)δ˜ (k ,χ)δ˜ (k ,χ) = (26) nativeintegralkernelinthetransformationfromcorrelation
h m 1 m 2 g 3 i functions to apertureestatisetics; see their Sect. 7.1 and 7.2.
(2π)3δ(3)(k +k +k )B (k ,k ,χ).
D 1 2 3 mmg 1 2
Thevectorkisthecomovingwavenumberofmodesentering
2.6 Reduction of II- and GI-contributions
the triple correlator. As before with the angular bispectra,
the spatial bispectra are also isotropic, i.e., they are only Onepossible source of systematics are correlations with in-
functions of |k1|, |k2| and theangle spanned by k1 and k2. trinsic ellipticities ǫs of sources. A correlation between ǫs
To refine the previous RCS1 measurement in Simon of different sources (II-correlations) or between ǫ and a
s
et al. (2008) for different galaxy populations, we focus fluctuation in the mass density field generating shear (GI-
on equally-sized apertures with θ1 = θ2 = θ3 only. correlations) isknowntocontributetotheshearcorrelation
This leads us to the short hand notations 2Map (θ) := functions (e.g Hirata & Seljak 2004; Heymans et al. 2006;
hN2Mapi(θ;θ;θ), likewise for hNMa2pi. DuehNto theiaction Joachimi et al. 2011). For a discussion of intrinsic align-
of the u-filter in the Eqs. (19) and (20) this picks up ments in CFHTLenS see also Heymans et al. (2012). We
mainly bispectrum contributions from equilateral triangles argue here that selecting lenses and sources from well sep-
ℓ1 = ℓ2 = ℓ1+ℓ2,albeitalsomixinginsignalfromother arated distances ideally removes contaminations by II- or
| | | | | |
trianglesbecauseofthefinitewidthoftheu-filterinℓ-space. GI-correlations in theG3L statistics.
Consider thegalaxy numberdensitycontrasts κ and
g,1
κ in two arbitrary line-of-sight directions θ and θ , re-
2.5 Systematics indicators g,2 1 2
spectively,andasourceellipticityǫ +γ inathirddirection
s
Thegravitationalshearofdistantgalaxyimagesisproduced θ3. The shear γ and ǫs are rotated in direction of the mid-
bysmallfluctuationsδφintheinterveninggravitationalpo- point between the two lenses according to the definition of
tential. To lowest order in δφ/c2 this is expected to only .If lenses and sources are well separated in distance, then
G
producecurl-freeshearfields(B-modesvanish).Currentsur- theirpropertiesarestatistically independent.Thelens-lens-
veysdonothavethepowertomeasurehigher-ordereffects, shear correlator measures
such that we expect these to be undetectable in our data.
= κ κ (γ+ǫ ) (27)
Shear-relatedcorrelationfunctions,oraperturemomentsin- G h g,1 g,2 s i
volvingtheaperturemass,hencevanishafterrotationofall = κg,1κg,2γ + κg,1κg,2 ǫs
h i h ih i
sdoautracaensablyys4is5,◦a,4i.5e◦.,raofttaetrioγnc(oθf)t7→he−soiuγrcc(eθe)l.liTprtaicnitsileastesdhoinutldo = hκg,1κg,2γi,
free ofany systematic contribution from theintrinsic shape
result in ameasurement thatis statistically consistent with
ǫ , if ǫ is statistically independent of the lens numberden-
the experimental noise (e.g. Hetterscheidt et al. 2007). We s s
sity fluctuation κ , i.e.,
use this as a necessary (but not sufficient) indicator for the g
absence of systematics in thedata. κ κ ǫ = κ κ ǫ , (28)
g,1 g,2 s g,1 g,2 s
TheestimatorGest inEq.(17)incorporatestwosources h i h ih i
±
vanishingdueto ǫ =0.
with two uniquely different possibilities to probe systemat- h si
e Now,consideralensnumberdensitycontrastκ inone
ics:rotatingtheellipticitiesǫ andǫ ofbothsourcesresults g
j k
intheso-calledB-modechannelof M2 (θ),denotedhere direction and the ellipticities ǫs,i+γi of two source images
by hNM⊥2i(θ), and the P-mode chhaNnnela,phiNM⊥Mapi(θ), if iin=di1r,e2ctiinontwofoliontehecrondnireeccttiniogntsh.eTshoeureclleisptinicaitciceosradraenrcoetwatiethd
onlyeitherǫ orǫ arerotated.AspointedoutbySchneider
j k
(2003),aP-modeisasignatureofaparity-invarianceviola- thedefinition of G±. The triple correlator measures
tciaonnoinnlythbeesgheenaerradtaetda,bwyhsiycshteimnaatipcasriintyt-hinevParSiaFnctourrneicvteirosne G± = hκg(γ1±+ǫ±s,1)(γ2+ǫs,2)i (29)
= κ γ±γ + κ ǫ± γ + κ γ±ǫ + κ ǫ± ǫ
pipeline,orinthealgorithmforthestatisticalanalysisofthe h g 1 2i h g s,1 2i h g 1 s,2i h g s,1 s,2i
data. Non-vanishing B-modes, on the other hand, can have = κ γ±γ + κ ǫ± γ + κ γ±ǫ + κ ǫ± ǫ
h g 1 2i h g s,1 2i h g 1 s,2i h gih s,1 s,2i
a physical cause. For example, they can be associated with = κ γ±γ + κ ǫ± γ + κ γ±ǫ .
theintrinsicclusteringofsources(Schneideretal.2002),in- h g 1 2i h g s,1 2i h g 1 s,2i
trinsic alignment correlations of physically close sources or The last term in the third line vanishes because κ in
g
intrinsicshape-shearcorrelations(Heymansetal.2006,and the foreground is independent of the intrinsic shape of the
references therein). Especially the latter two are a concern sources in the background and because of κ = 0. The
g
h i
for this analysis, as these effects are known to affect the E- latter follows from thedefinition of density fluctuationsκ .
g
mode channel of the aperture statistics, which is the prime The last two terms in the last line are less clear. For
focus of this work. However, currently it is unclear by how examplein κ ǫ± γ ,γ could becorrelated with both ǫ
h g s,1 2i 2 s,1
much this really affects G3L. We discuss in the following (GI signal, if source 2 is behind source 1) and κ (GGL
g
Sect.2.6thattheinfluenceofthesesystematicscanbesup- signal). However, on the level of accuracy of the Born ap-
pressed by separating lenses and sources in redshift, which proximationthatisusedinEq.(1),theshearγ islinearin
2
is carried out in ouranalysis. thematter density contrast δ up to thedistance of source
m
Sincetheestimator est inEq.(16)involvesonesource, 2.Wecan,therefore,splitthecontributionstoγ intothree
2
there is only a single sysGtematics indicator of 2M (θ), parts γ =γ +γ +γ , namely (i) in contributions from
e hN api 2 κ ǫ rest
Galaxy-galaxy-galaxy lensing in CFHTLenS 7
10-8 2.5x105
lenses
<M 3>(q ) sources
ap
2.0x105
s
e
xi
ala 1.5x105
g
10-9 of
ber 1.0x105
m
u
n
5.0x104
0.0x100
10-10 0.2 0.4 0.6 0.8 1 1.2
1 q [arcmin] 10 photometric redshift
Figure 3.Predictionofthe third-ordermomentofthe aperture Figure 4. Total number of lenses (red) and sources (blue) in
masshMa3piforsourcesatredshiftzs=0.4. ltehnesecsataolrog1u7e.5b6etwi′e<en204..276fozrphsooutorc<es1..2Thanedfi1g7u.r5es6cio′m<p2r2is.e5faolrl
galaxies, complying with the selection cuts, contained within all
matter within correlation length to thelens, γκ, (ii) matter 172 pointings. For the G3L analysis, the lens sample is further
withincorrelation distancetosource1,γ ,and(iii)therest subdividedinluminosity,stellarmassbinsandphotometric red-
ǫ
γrest, which is neither correlated with κg nor with ǫs,1. In shift,whilesourcesarerejectedforaphoto-z ofzphoto<0.65.
this case we find
hκgǫ±s,1γ2i=hκgγκihǫ±s,1i+hκgihǫ±s,1γǫi+hκgihǫ±s,1ihγresti.(30) of θ∼1′ and sourcesat zs∼1.0. Asthisincludescontribu-
tionsfromtheentireintegratedmatteruptoz ,whereasthe
Allthree termsvanish owing to κ = ǫ± =0. A similar s
h gi h s,1i G3Lmagnificationeffectonlycontributionsfromthematter
rationalshowsthatalso κ γ±ǫ vanishestolowestorder,
h g 1 s,2i integrateduptothelensredshiftszd 0.4,weconsiderthis
such that we expect to findin theweak lensing regime ∼
anempiricalupperlimitforthemagnificationeffect.InFig.
G± =hκgγ1±γ2i. (31) f3o,rwaeWshMowAaP7p-rleikdeicctoiosnmooflohgMya3bpa(sθe)diwoniththseotuhrecoesryatdezssc=rib0e.d4
in Semboloni et al. (2011). This result implies that the im-
2.7 Magnification of lenses pact of lens magnification on the G3L aperture statistics is
smaller than .10−8.
Another conceivable systematic effect is through cosmic
magnification (Narayan 1989; Bartelmann & Schneider
2001) that is generated by matter density fluctuations in
front of lenses. To lowest order, foreground matter density 3 DATA
fluctuationswith lensing convergenceκ (Eq.1) integrated
< 3.1 Object selection and photometric redshifts
tothelensdistance modifytheobservedclusteringoflenses
on thesky abovea certain fluxlimit flim according to This work uses the full CFHTLenS data set, which origi-
κ′ =κ +λκ + (κ2), (32) nates from theCFHTLS-Wide Survey.The CFHTLS-Wide
g g < O < imaged171MegaCam(mountedontheCFHT)pointingsin
comparedtotheunmagnifiedlensnumberdensityκ .Here, the five broad-band filters u∗, g′, r′, i′, and z′. During the
g
wehaveλ:=2(ν 1)withn¯ (>f ) f−ν beingthemean observationcampaignofCFHTLS,thei′-bandfilterwasre-
− g lim ∝ lim
number density of lenses with flux greater than f . Nor- placed by a new filter with a slightly different transmission
lim
mally ν 1 is of the order of unity (van Waerbeke 2010) curve. For some of the pointings only the updated i′-band
−
or smaller. Likewise the shear distortion γ = γ + γ , filtermagnitudes areavailable, which aretreated astheold
< >
Eq. (3), into the same l.o.s. direction contains a contri- filtermagnitudesintheanalysis.Fordetails,seeErbenetal.
bution γ related to κ , and γ that is the shear orig- (2012).
< < >
inating from matter fluctuations beyond the foreground. CFHTLenS has an effective area (different pointings
This in combination produces as additional contribution to partly overlap) of about 154 square-degrees with high-
= κ′ κ′ (γ +γ ) the term λ2 κ κ γ and to quality photometric redshifts down to i′ 24.7. The data
GG± =hhκg,′g1(γg<,2,1+<γ>,1)(>γ<i,2+γ>,2)ithehte<rm,1 λ<h,κ2<<γ<i,1γ<,2i. setandtheextractionofourphotometricr≈edshiftcatalogue
These termsarebasically third-ordercosmic shearcor- are described in Hildebrandt et al. (2012). Our data pro-
relationsor,intermsoftheaperturestatistics,relatedtothe cessing techniquesand recipes are described in Erben et al.
M3 (θ) statistics(Schneideretal.2005).Third-ordershear (2009) and Erben et al. (2012). As primary selection cri-
hcorraeplatiions have been measured (Bernardeau et al. 2003; terion, we select sources brighter than i′ <24.7 and lenses
Pen et al. 2003; Jarvis et al. 2004; Semboloni et al. 2011), brighter than i′ <22.5. This will be further subdivided in
and M3 (θ) hasbeenfound(Jarvisetal.2004; Semboloni thefollowingbyusingphotometricredshifts(Fig.4)and,in
et alh. 20a1p1) tio be of the order of .10−7 for aperture scales thecaseoflenses,M restframemagnitudes,stellarmasses
r
8 Simon et al.
Table1.SelectioncriteriaoflenssamplesandsourcesamplefortheG3LanalysisappliedtothesamplesinFig.4,followingMandelbaum
etal.(2006) forthelenses.Theluminositybins(L), stellarmassbins(sm)andgalaxytypebins (ETG:early-typegalaxies; LTG:late-
typegalaxies)areagainsubdividedby0.26zphoto<0.44(“low-z”)and0.446zphoto<0.6(“high-z”).Sourcesattributednostatistical
weight w by lensfitare not used in the source sample. The galaxy numbers are for all pointings of which the final analysis discards
roughly 25%. Luminosities and stellar masses assume h = 0.73. (1) z¯: mean redshift, σz: r.m.s. variance of p(z); (2) and (3): best-fit
parameters of ω(θ)=Aω(θ/1′)−λ+IC within0′.26θ<10′; (4): samplecompleteness; (5): mean r-band luminosity;(6): mean stellar
massinunitsof1010M⊙.
Sample Selection #Galaxies z¯±σz(1) Aω/0.1(2) λ(3) fc(4) hMri(5) hM∗i(6)
L1low-z 186Mr < 17 36,372 0.22 0.16 2.40 0.29 0.45 0.11 0.14 -17.75 0.04
− − ± ± ±
L1high-z ” – – – – – – –
L2low-z 196Mr < 18 157,306 0.28 0.15 1.91 0.23 0.35 0.05 0.45 -18.60 0.10
− − ± ± ±
L2high-z ” – – – – – – –
L3low-z 206Mr < 19 220,329 0.34 0.14 1.41 0.12 0.43 0.05 0.81 -19.52 0.26
− − ± ± ±
L3high-z ” 75,902 0.48 0.11 1.63 0.18 0.54 0.08 0.42 -19.72 0.29
± ± ±
L4low-z 216Mr < 20 149,190 0.34 0.12 1.63 0.07 0.53 0.03 0.95 -20.50 0.91
− − ± ± ±
L4high-z ” 185,286 0.51 0.10 1.62 0.08 0.69 0.04 0.82 -20.53 0.98
± ± ±
L5low-z 226Mr < 21 88,916 0.34 0.11 2.19 0.14 0.60 0.05 0.98 -21.48 3.09
− − ± ± ±
L5high-z ” 134,369 0.51 0.09 2.06 0.05 0.74 0.02 0.99 -21.49 3.06
± ± ±
L6low-z 236Mr < 22 31,373 0.35 0.10 3.02 0.24 0.65 0.07 0.99 -22.40 8.56
− − ± ± ±
L6high-z ” 55,315 0.52 0.08 2.50 0.10 0.92 0.04 1.00 -22.42 8.11
± ± ±
sm1low-z 0.56M∗/1010M⊙<1.0 78,181 0.34 0.12 2.41 0.34 0.43 0.09 0.94 -20.49 0.71
± ± ±
sm1high-z ” 69,784 0.50 0.10 1.72 0.33 0.58 0.15 0.77 -20.66 0.73
sm2low-z 1.06M∗/1010M⊙<2.0 61,650 0.34±0.11 3.75±0.82 0.36±0.11 0.98 -20.98 1.42
± ± ±
sm2high-z ” 82,411 0.51 0.09 2.39 0.07 0.60 0.07 0.90 -20.99 1.45
sm3low-z 2.06M∗/1010M⊙<4.0 48,632 0.34±0.10 3.47±0.31 0.51±0.07 0.99 -21.46 2.85
± ± ±
sm3high-z ” 81,305 0.51 0.08 2.44 0.13 0.72 0.05 0.98 -21.45 2.85
sm4low-z 4.06M∗/1010M⊙<8.0 33,218 0.35±0.09 4.05±0.39 0.59±0.08 0.99 -21.91 5.60
± ± ±
sm4high-z ” 57,049 0.51 0.08 2.72 0.11 0.77 0.04 0.99 -22.00 5.59
± ± ±
sm5low-z 8.06M∗/1010M⊙<16.0 15,527 0.36 0.08 5.00 0.41 0.70 0.07 1.00 -22.40 10.86
± ± ±
sm5high-z ” 27,598 0.51 0.08 3.56 0.24 0.81 0.07 1.00 -22.81 10.88
± ± ±
sm6low-z 16.06M∗/1010M⊙<32.0 4,605 0.36 0.07 6.58 0.50 1.51 0.07 1.00 -23.00 21.13
± ± ±
sm6high-z ” 7,121 0.52 0.07 4.18 0.78 1.58 0.16 1.00 -23.22 20.90
± ± ±
sm7low-z 32.06M∗/1010M⊙<64.0 526 0.38 0.06 8.89 1.37 1.64 0.15 1.00 -23.60 40.81
± ± ±
sm7high-z ” 775 0.52 0.07 5.61 1.30 1.28 0.21 1.00 -23.67 38.52
± ± ±
ETGlow-z 06TB<2 236Mr < 21 89,359 0.34 0.10 3.43 0.08 0.68 0.02 0.99 -21.88 5.91
| − − ± ± ±
ETGhigh-z ” 137,144 0.51 0.08 2.90 0.09 0.83 0.03 1.00 -21.91 5.74
± ± ±
LTGlow-z 26TB<6 236Mr < 21 30,926 0.35 0.13 0.70 0.13 0.87 0.18 0.96 -21.64 1.73
| − − ± ± ±
LTGhigh-z ” 52,527 0.51 0.10 1.33 0.16 0.78 0.11 0.99 -21.73 2.05
± ± ±
SOURCES 0.656zphoto<1.2|w>0 2,926,894 0.93±0.26 – – – – –
orSEDinformation (detailsbelow).43pointingsoutof171 endowed with photometric redshifts, three classes of lens
exhibit a significant PSF residual signal, according to the samples are selected (Table 1):
detailed tests in Sect.4.2 ofHeymanset al. (2012), and are
thereforediscardedfortheanalysis( 25%area);129point- A luminosity or L-sample class, which consists of six
∼ •
ingsareincludedintheanalysis.Thisleavesatotaleffective distinctrest-frameMr bins(SDSSr-filter;Yorketal.2000),
surveyareaof 120deg2thatiseventuallyusedintheanal- labelled L1toL6.Thesameformal luminosity binlimits as
∼
ysis. Of this area an additional 20% percent is lost due in Mandelbaum et al. (2006) or Velander et al. (2012) are
∼
to masking. The analysis is performed on individual fields applied,althoughwedonotautomaticallyexpectequivalent
which allows us to use field-to-field variances of the mea- completeness of thesamples. Toquantify thecompleteness,
surementstoestimatethecovarianceofmeasurementerrors we introducethe fc parameter below.
directly from thedata. Astellarmassorsm-sampleclass,whichisalsofurther
•
subdividedusingsevendistinctstellarmassbins.Again,we
are guided by Mandelbaum et al. (2006) for compiling this
sample class. The sm class has sub-classes with labels sm1-
sm7.
3.2 Lens samples
A galaxy type class using the T B parameter in BPZ
•
To guarantee a high reliability of the photo-z estimates for (Benitez 2000), which provides the most likely galaxy SED
thelenses,amagnitudecutofi′ 622.5isapplied.Adetailed foragivengalaxyanditsestimatedphoto-z;seeErbenetal.
account and tests of the CFHTLenS photo-z pipeline can (2012) for more details. T B=2 as division line, we separate
befoundinHildebrandtetal.(2012).Basedonthegalaxies early-type galaxies (“ETG”), which have T B<2, from late
Galaxy-galaxy-galaxy lensing in CFHTLenS 9
0.25 0.25
low-z sources high-z sources
L1 L3
L2 L4
0.2 0.2
L3 L5
L4 L6
L5
bility 0.15 L6 bility 0.15
a a
b b
o o
pr 0.1 pr 0.1
0.05 0.05
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
redshift redshift
0.35 0.3
low-z sources high-z sources
sm1 sm1
0.3 sm2 0.25 sm2
sm3 sm3
0.25 sm4 sm4
sm5 0.2 sm5
ability 0.2 ssmm67 ability 0.15 ssmm67
b b
o 0.15 o
pr pr
0.1
0.1
0.05 0.05
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
redshift redshift
0.25 0.3
low-z sources high-z sources
ETG ETG
LTG 0.25 LTG
0.2
0.2
ability 0.15 ability 0.15
b b
o o
pr 0.1 pr
0.1
0.05
0.05
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
redshift redshift
Figure 5.In contrast to Fig. 4, this figure shows the fullBPZ redshiftposterior of the various samples. Low-z lenses areselected from
zphoto ∈ [0.2,0.44], high-z lenses from zphoto ∈ [0.44,0.6], and sources from zphoto ∈ [0.65,1.2]. The dashed black line enclosing the
sourcep.d.f.isaparametrisedfit,seeSect.3.3forbest-fitparameters.
typegalaxies(“LTG”).1 Inordertodefineavolume-limited etal.(2012). Theestimators assumeaChabrier(2003) star
sample of ETG and LTG, we select only luminous galaxies initial mass function.
with restframe luminosities 23 6 M < 21. With this
r
− −
luminosity cut, ETG and LTG are actually subsamples of
L5 and L6 combined.
The stellar masses of the lenses are determined from the
galaxymulti-colourdataasdescribedinSect.2.1ofVelander Allthreeclassesarefurthersplitintotwophoto-z bins:
a “low-z” bin with 0.26z <0.44 and a “high-z” bin
photo
with0.446z <0.60.Asredshiftestimatorsweusethe
photo
1 Within BPZ values of T B denote best-fitting galaxy tem- maximumprobabilityredshiftsoftheredshiftposteriorpro-
plates:1=CWW-Ell,2=CWW-Sbc,3=CWW-Scd,4=CWW-Im, videdbyBPZ.Theredshiftboundariesgivecomparablenum-
5=KIN-SB3,and6=KIN-SB2.Notethatthetemplatesareinter- bers of lenses prior to attributing them to one of the three
polated,suchthatfractionalnumbersoccur. lensclasses(Fig.4).Notcountingthehigh-zL1andL2sam-
10 Simon et al.
ples,which havetoofaint limits tocontain lenses2,wehave posterior redshift p.d.f. of individual sources, shown in ev-
in total 28 lens subsamples. erypanelofFig.5incomparisontotheredshiftdistribution
The true redshift distribution of a lens sample is of the lens samples. The individual posteriors are weighted
not identical to the distribution of their photometric red- with thesourceweight that isalso used inthelensing anal-
shifts due to the errors in the photo-z estimators. For a ysis. The source redshift p.d.f. is well fitted by a broken
magnitude cut of i′ <22.5, the errors are approximately exponentialdistribution
σ .0.04(1+z)witha 3%outlierrate(Hildebrandtetal.
2b0uz1t2io)n. Wfunecctoiomnbsin(pe.dth.fe.)∼poofsatlelrlieonrsreesdgsihviefnt pbryoBbPaZb,ilsietyeFdiigst.r5i-, pb(z)∝(cid:26) eexxpp((−pp02((zz0−zz0))pp13)) iofthze6rwzis0e, (33)
− −
to quantify the redshift uncertainties of complete lens sam-
with fit parameters p = 91.14, p = 2.623, p = 4.093,
0 1 2
ples. The depicted redshift probability distributions will be
p = 1.378 and z = 0.794 (dashed black lines). With our
3 0
utilised when normalising theG3L aperturestatistics.
selections wefindabout 3 106 sources with mean redshift
To help the comparison of our G3L results to future ×
z¯ 0.93.AscanbeseeninFig.5,theoverlapofthevarious
studies, we also quote theangular clustering and complete- ≈
p (z)andthesourcep (z)issmallbutnotentirelyvanishing,
f b
ness of the lens samples. The results are listed in Table 1,
mainlyatz=0.5 0.7forthehigh-z andatz 0.6forthe
thedetails are described in AppendicesB1 (clustering) and − ∼
low-z samples. The typical overlapping area of the redshift
B2 (completeness). In short, for the angular clustering of
probabilitydistributionfunctions(visibleinFig.5)is 12%
lenses, we approximate the angular galaxy two-point cor- ∼
for thehigh-z samples and 4% for the low-z samples.
relation function by a power law over the angular range ∼
0.′26θ<10′. For each lens sample with the photo-z bin
[z ,z ], we quote the completeness factor f that expresses
1 2 c
theaverageV(z ,z )/V(z ,z )ofalllensesinthesample;
1 max 1 2
V(z ,z ) is the light cone volume between redshift z and
1 2 1
z , and z 6 z is the maximum redshift up to which a 4 RESULTS
2 max 2
lensisstillabovethefluxlimiti′ =22.5.Asmallf isasign
c
4.1 Measurements and their uncertainties
ofastrongincompletenessbecausemanygalaxiessimilarto
those observed near z1 are presumably missing at higher In order to obtain measurements for the lensing aperture
redshifts. Due to the magnitude limit, samples containing statistics, we use the method outlined in Sect. 2.3. As the
a substantial portion of faint galaxies are most affected by binninggridfor est andGest,100log-binsrangingbetween
±
G
incompleteness, most notably L1 and L2. As expected, the 9arcsecand50arcminaresetupforϑ ,100linearbinsare
e e 1/2
completenessdropsifonemovesfromthelow-z tothehigh- usedfortheopeningangleφ ,yieldingoverall106 binswith
3
z bin in almost all cases. The few minor exceptions, L5 for bin widths ∆φ =3.6deg and ∆lnϑ=0.058. All measure-
3
instance, are probably due to shot noise in the fc estima- ments are performed separately on every individual point-
tor. We conclude that L4-L6, sm3-sm7 and ETG/LTG are ing, out of 129 square pointings with roughly 1deg2 each.
the most complete, volume-limited samples for our study Adjacent pointings partly overlap, however, which reduces
(fc > 0.80 for both low-z and high-z). In Table 1 we also the area that is actually used. In our study, we crop the
quotetheaverageabsoluter-bandfluxofthesamples,listed pointings to remove the overlap. For the final result, indi-
asmagnitude Mr andthetheiraveragestellarmass M∗ . vidual estimates are combined by averaging the individual
h i h i
est and Gest weighted by the number of triangles within
±
G
each bin.
e e
3.3 Source sample Finally, the combined estimates are transformed to
the aperture statistics by the integral transformations dis-
All details concerning the galaxy shape measurement (em-
cussed in SW05. In his way,theaperturestatistics between
ploying the lensfitalgorithm; Kitching et al. 2008, Miller
0.′56θ 610′fortenaperturescaleradiiarecomputed.As
etal.2007,andMilleretal.2012), CFHTLenSsourcecata- ap
addressedinSimonetal.(2008),thetransformation from ˜
lporgeuseengteendeirnatHioeny,manandstheteadli.sc(u20ss1i2o)naonfdshMeailrlesrysettemal.at(i2c0s1a2r)e. or G˜± to aperture statistics becomes biased towards smaGll
and large aperture radii due to an insufficient sampling of
We account for the multiplicative shear bias by employing
the correlation functions. A similar transformation bias is
the Miller et al. (2012) normalisation scheme adjusted to
alsoknownfortheaperturemassstatistics (Kilbingeretal.
our estimators (see AppendixA).
2006). For the small separations, the bias depends in detail
In order to reduce the level of undesired II- and GI-
on themean numberdensity of the galaxies, most crucially
correlations in the measurements, we attempt to separate
the lenses, and the clustering of the lenses, which in com-
sources and lenses by redshift, utilising photometric red-
bination determines the sampling of the correlation func-
shiftsasestimators.Asacompromisebetweenaccuratered-
tions by small triangles. By comparison to simulated data,
shift estimates and a large numbers of sources, we apply a
magnitude limit i′ <24.7 to the lensfitshear catalogue and wweithminadtehesurraengtehaotf t1h′i.s bθi.as1i0s′ninegoliugribcleas(eb(esloeewF∼ig.110%in)
selectsourcesbetween0.656z <1.2.Asforthelenses,
photo
Saghiha et al. 2012 for an illustration of the transforma-
the true redshift distribution is derived from the combined
tion bias). The variance of themeasurements across all 129
pointingsisusedtoestimatethecovarianceofmeasurement
2 Actually, we find a few galaxies in the high-z L1/L2 samples. errors (Jackknifing; Appendix B1). The inverse covariance
Theseareprobablyextremeoutlierswithgreatlyinaccuratered- matrixisestimatedfromthepointing-to-pointingcovariance
shiftestimates. according to the method in Hartlap et al. (2007).