First detection of galaxy-galaxy-galaxy lensing in RCS. A new tool for studying the matter environment of galaxy pairs
P. Simon, P. Watts, P. Schneider, H. Hoekstra, M.D. Gladders, H.K.C. Yee, B.C. Hsieh, H. Lin
aa r X i v : . [ a s t r o - ph ] J a n Astronomy & Astrophysics manuscript no. 8197gggl c (cid:13)
ESO 2018October 24, 2018
First detection of galaxy-galaxy-galaxy lensing in RCS ⋆ A new tool for studying the matter environment of galaxy pairs
P. Simon , , P. Watts , P. Schneider , H. Hoekstra , M.D. Gladders , H.K.C. Yee , B.C. Hsieh ,and H. Lin The Scottish Universities Physics Alliance (SUPA), Institute for Astronomy, School of Physics, University of Edinburgh, RoyalObservatory, Blackford Hill, Edinburgh EH9 3HJ, UK Argelander-Institut f¨ur Astronomie ⋆⋆ , Universit¨at Bonn, Auf dem H¨ugel 71, 53121 Bonn, Germany Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada Department of Astronomy & Astrophysics, University of Chicago, 5640 S. Ellis Ave., Chicago, IL, 60637, US Department of Astronomy & Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada Institute of Astrophysics & Astronomy, Academia Sinica, P.O. Box 23-141, Taipei 106, Taiwan, R.O.C. Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510
ABSTRACT
Context.
The weak gravitational lensing effect, small coherent distortions of galaxy images by means of a gravitational tidalfield, can be used to study the relation between the matter and galaxy distribution.
Aims.
In this context, weak lensing has so far only been used for considering a second-order correlation function that relatesthe matter density and galaxy number density as a function of separation. We implement two new, third-order correlationfunctions that have recently been suggested in the literature, and apply them to the Red-Sequence Cluster Survey. As a steptowards exploiting these new correlators in the future, we demonstrate that it is possible, even with already existing data, tomake significant measurements of third-order lensing correlations.
Methods.
We develop an optimised computer code for the correlation functions. To test its reliability a set of tests involvingmock shear catalogues are performed. The correlation functions are transformed to aperture statistics, which allow easy testsfor remaining systematics in the data. In order to further verify the robustness of our measurement, the signal is shown tovanish when randomising the source ellipticities. Finally, the lensing signal is compared to crude predictions based on thehalo-model.
Results.
On angular scales between ∼ ′ and ∼ ′ a significant third-order correlation between two lens positions and onesource ellipticity is found. We discuss this correlation function as a novel tool to study the average matter environment of pairsof galaxies. Correlating two source ellipticities and one lens position yields a less significant but nevertheless detectable signalon a scale of ∼ ′ . Both signals lie roughly within the range expected by theory which supports their cosmological origin. Key words.
Galaxies: halos – Cosmology: large-scale structure of Universe – Cosmology: dark-matter – Cosmology: observations
1. Introduction
The continuous deflection of light rays propagatingthrough the Universe by inhomogeneities of the large-scaledistribution of matter generates a coherent, weak distor-tion pattern over the sky: the shear field. As the source ⋆ Based on observations from the Canada-France-HawaiiTelescope, which is operated by the National Research Councilof Canada, le Centre Nationale de la Recherche Scientifique andthe University of Hawaii. ⋆⋆ Founded by merging of the Institut f¨ur Astrophysikund Extraterrestrische Forschung, the Sternwarte, and theRadioastronomisches Institut der Universit¨at Bonn. of the shear field is gravity, the total matter distributioncontributes to this effect. It is, therefore, a probe of thefull matter content of the Universe. The shear field canbe investigated by correlating the shapes of distant faintgalaxies (e.g. Schneider 2006; Van Waerbeke & Mellier2003; Bartelmann & Schneider 2001). Since the first de-tection of this effect (Bacon et al. 2000; Kaiser et al. 2000;Van Waerbeke et al. 2000; Wittman et al. 2000) study-ing the weak gravitational lensing effect has become animportant tool for cosmology. Together with the next-generation weak lensing surveys such as the RCS2 (on-going), CFHTLS (ongoing), Pan-STARRS 1 (commenc-ing soon) or KIDS (planned to commence during the
Simon et al.: GGGL in RCS second half of 2008), the weak lensing effect will allowus to put tight constraints on cosmological parameters(Peacock et al. 2006).One important topic in contemporary cosmology is therelation between the dark matter and the galaxy popula-tion, the latter of which is thought to form under partic-ular conditions from the baryonic component within thedark matter density field. This relation can be studied bycross-correlating the shear signal and (angular) positionsof a selected galaxy population.As the shear is quite a noisy observable, higher or-der galaxy-shear correlation functions are increasingly dif-ficult to measure. For this reason, studies in the pasthave focused on 2 nd -order statistics (“galaxy-galaxy lens-ing”, GGL hereafter) which involve one galaxy of theselected population (foreground) and one source galaxy(background) whose ellipticity carries the lensing signal.The GGL-signal can be used to learn more about thetypical dark matter environment of single galaxies (mostrecently Kleinheinrich et al. 2006; Mandelbaum et al.2006b,a,c; Seljak et al. 2005; Hoekstra et al. 2005, 2004;Sheldon et al. 2004), or the so-called galaxy biasing(Simon et al. 2007; Pen et al. 2003; Hoekstra et al. 2002a,2001).Schneider & Watts (2005) introduced “galaxy-galaxy-galaxy lensing” (GGGL) correlation functions and esti-mators thereof which allow us to move to the next, 3 rd -order level (see also Watts & Schneider 2005). The corre-lation functions now involve either two foreground galaxiesand one background galaxy, or one foreground galaxy andtwo background galaxies. This idea was also discussed byJohnston (2006) who studied how to derive the galaxy-galaxy-mass correlation function, which is one of the fore-going two, from weak gravitational lensing. These func-tions, although more difficult to measure than the two-point GGL signal, offer the opportunity to study the typ-ical environment of pairs of galaxies, e.g., within galaxygroups (or more technically, the occupation statistics ofgalaxies in dark matter halos, see Cooray & Sheth (2002)for a recent review), or possibly even the shape of darkmatter haloes (Smith et al. 2006). More generally, theymeasure 3 rd -order moments between number densities ofgalaxies and the matter density of dark matter (cross-correlation bispectra). Hence, they “see” the lowest-order non-Gaussian features produced by cosmic structure for-mation.This paper applies for the first time the GGGL-correlation functions to existing data, the Red-SequenceCluster Survey (RCS; Gladders & Yee 2005), and demon-strates that with the current generation weak lensing sur-veys it is already possible to extract these particular 3 rd -order statistics.The outline of the paper is as follows. We will givea brief description of the survey in Sect. 2. In Sect. 3,we will define the correlation functions and their practicalimplementation as estimators for real data. In Sect. 4,our results will be presented, discussed and compared tohalo-model based predictions to verify if the signal has r e l a t i v e f r equen cy photometric redshift all galaxieslensessources Fig. 1.
Histogram of photometric redshifts of lenses, z ∈ [0 , . z ∈ [0 . , . . × lenses (¯ z ≈ .
30) and3 . × sources (¯ z ≈ . m = 0 .
3, for the matter density parameter, andΩ Λ = 0 .
7, for the dark energy density parameter, are as-sumed. Dark Energy is assumed to behave like a cosmo-logical constant. For the dark matter power spectrum nor-malisation we adopt σ = 0 .
2. Data: The Red-Sequence Cluster Survey
The data used in this paper were taken as part ofthe Red-Sequence Cluster Survey (RCS; Gladders & Yee2005), and comprise of approximately 34 square degrees of
B, V, R C and z ′ imaging data observed with the Canada-France-Hawaii Telescope (CFHT). The B and V bandswere taken after completion of the original RCS, to allowfor a better selection of clusters at low redshifts. Thesefollow-up observations also enable the determination ofphotometric redshifts for a large sample of galaxies. Thisphotometric redshift information is key for the work pre-sented here. A detailed discussion of these multicolourdata, the reduction, and the photometric redshift deter-mination can be found in Hsieh et al. (2005). In the red-shift range out to z ∼ . δz , and redshift is δz ∼ . z ). Thisrelation over-estimates the uncertainty for z < . imon et al.: GGGL in RCS 3 under-estimates it for z > . × galaxies with 18 < R c <
24 that areused in the analysis presented here. Hoekstra et al. (2005)also present a number of lensing specific tests, demonstrat-ing the usefulness of the RCS photometric redshift cata-logue for galaxy-galaxy lensing studies. The frequency dis-tribution of photometric redshifts in our galaxy samplesis shown in Fig. 1.The galaxy shapes were determined from the R C im-ages. The raw galaxy shapes are corrected for the ef-fects of the point-spread-function (PSF) as described inHoekstra et al. (1998, 2002c). We refer the reader to thesepapers for a detailed discussion of the weak lensing anal-ysis. We note that the resulting object catalogues havebeen used for a range of weak lensing studies. Of these,the measurements of the lensing signal caused by largescale structure presented in Hoekstra et al. (2002b,c) areparticularly sensitive to residual systematics. The varioustests described in these papers suggest that the systemat-ics are well under control. It is therefore safe to concludethat residual systematics in the galaxy shape measure-ments are not a significant source of error in the analysispresented here.
3. Method
Here we briefly summarise definitions of the three-pointcorrelation functions, their estimators and the relation be-tween aperture statistics and correlation functions. A de-tailed derivation and explanation of those can be lookedup in Schneider & Watts (2005).
For our analysis we consider two different classes of cor-relation functions. Both classes require triplets of galax-ies which are located at the positions θ , θ and θ onthe sky (see Fig. 2). In a cosmological context, randomfields – such as the projected number density of galaxies, N ( θ ), or the shear field, γ ( θ ) – are statistically homoge-neous and isotropic. For that reason, all conceivable cor-relations between the values of those fields depend merelyon the separation, | θ i − θ j | , and never on absolute posi-tions θ i . Therefore, our correlators are solely functions ofthe dimensions of the triangle formed by the galaxies. Weparameterise the dimension of a triangle in terms of thelengths of two triangle edges, ϑ and ϑ , and one angle, φ , that is subtended by the edges. Note that the sign of φ , i.e. the handedness of the triangle, is important. The galaxy-galaxy-shear correlator , G ( ϑ , ϑ , φ ) = (cid:28) κ g ( θ ) κ g ( θ ) γ (cid:18) θ ; ϕ + ϕ (cid:19)(cid:29) , (1)is the expectation value of the shear at θ rotated in thedirection of the line bisecting the angle φ multiplied bythe number density contrast of lens (foreground) galaxiesat θ , : κ g ( θ ) ≡ N ( θ ) N − . (2)A rotation of shear is defined as γ ( θ ; ϕ ) ≡ − e − ϕ γ c ( θ ) , (3)where γ c is the shear relative to a Cartesian coordinateframe. It should be noted that G and the following corre-lators are complex numbers.A second class of correlators are the galaxy-shear-shearcorrelators , G + ( ϑ , ϑ , φ ) = h γ ( θ ; ϕ ) γ ∗ ( θ ; ϕ ) κ g ( θ ) i ,G − ( ϑ , ϑ , φ ) = h γ ( θ ; ϕ ) γ ( θ ; ϕ ) κ g ( θ ) i , (4)which correlate the shear at two points with the lensgalaxy number density contrast at another point. Again,the shears are rotated, this time in the direction of thelines connecting the source (background) galaxies, at θ , ,and the lens galaxy at θ . With practical estimators for (1) and (4) in mind,Schneider & Watts (2005) introduced modified correlationfunctions. They differ from G and G ± in that they are de-fined in terms of the number density of the lens galaxies, N ( θ ), instead of the number density contrast, κ g :˜ G ( ϑ , ϑ , φ ) ≡ (cid:10) N ( θ ) N ( θ ) γ (cid:0) θ ; ϕ + ϕ (cid:1)(cid:11) N (5)= G ( ϑ , ϑ , φ ) + h γ t i ( ϑ )e − i φ + h γ t i ( ϑ )e +i φ , and˜ G + ( ϑ , ϑ , φ ) ≡ N h γ ( θ ; ϕ ) γ ∗ ( θ ; ϕ ) N ( θ ) i (6)= G + ( ϑ , ϑ , φ ) + h γ ( θ ; ϕ ) γ ∗ ( θ ; ϕ ) i , ˜ G − ( ϑ , ϑ , φ ) ≡ N h γ ( θ ; ϕ ) γ ( θ ; ϕ ) N ( θ ) i (7)= G − ( ϑ , ϑ , φ ) + h γ ( θ ; ϕ ) γ ( θ ; ϕ ) i . These correlators also contain, apart from the origi-nal purely 3 rd -order contributions, contributions from2 nd -order correlations: h γ t i ( θ ) is the mean tangentialshear about a single lens galaxy at separation θ (GGL), h γ ( θ ; ϕ ) γ ∗ ( θ ; ϕ ) i and h γ ( θ ; ϕ ) γ ( θ ; ϕ ) i are shear-shear correlations which are functions of the cosmic-shearcorrelators ξ ± ( θ ) (e.g. Bartelmann & Schneider 2001). Torecover pure 3 rd -order statistics, the 2 nd -order terms can Simon et al.: GGGL in RCS J f j j θ θ θ J J J j j θ θ θ f ( ) j j+ Fig. 2.
Geometry of the galaxy-shear-shear correlation, G ± ( ϑ , ϑ , φ ) ( left panel ), and the galaxy-galaxy-shear cor-relation, G ( ϑ , ϑ , φ ) ( right panel ). The figure is copied from Schneider & Watts (2005).either be subtracted, or even neglected, if we work in termsof the aperture statistics, as we will see in the next section.With respect to practical estimators, number densi-ties are more useful quantities because every single galaxyposition is an unbiased estimator of N ( θ ) /N . For thatreason, every triangle of galaxies that can be found in asurvey can be made an unbiased estimator of either ˜ G (two lenses and one source) or ˜ G ± (two sources and onelens). Since, generally, a weighted average of (unbiased)estimates is still an (unbiased) estimate , we can combinethe estimates of all triangles of the same dimension usingarbitrary weights, w j/k , for the sources. Note that for thefollowing sums only triangles of the same ϑ , ϑ and φ have to be taken into account inside the sums. We adopta binning such that ϑ , ϑ and φ need to be within somebinning interval to be included inside the sums, i.e. trian-gles of similar dimensions are used for the averaging:˜ G est+ = N l ,N s P i,j,k =1 w j w k ǫ j ǫ ∗ k e − ϕ j e +2i ϕ k N l ,N s P i,j,k =1 w j w k , (8)˜ G est − = N l ,N s P i,j,k =1 w j w k ǫ j ǫ k e − ϕ j e − ϕ k N l ,N s P i,j,k =1 w j w k , (9)where j, k ∈ { . . . N s } are indices for sources and i ∈{ . . . N l } is the index of the lenses; N l and N s are thenumber of lenses and sources, respectively. By ϕ j and ϕ k The weighting scheme only influences the statistical uncer-tainty of the average, i.e. the variance of the combined esti-mate. Note that the whole statement requires that the weightsare uncorrelated with the estimates that the average is takenof. we denote the phase angles of the two sources relative tothe foreground galaxy i .The statistical weights are chosen to down-weight tri-angles that contain sources whose complex ellipticities, ǫ i (Bartelmann & Schneider 2001), are only poorly deter-mined. Lenses, however, always have the same weight inour analysis.Similarly, we can define an estimator for ˜ G . However,one has to take into account that ǫ e − i( ϕ + ϕ ) of one singletriangle – consisting of two lenses and one source withellipticity ǫ – is an estimator of (cid:10) N ( θ ) N ( θ ) γ (cid:0) θ ; ϕ + ϕ (cid:1)(cid:11) h N ( θ ) N ( θ ) i = ˜ G ( ϑ , ϑ , φ )1 + ω ( | θ − θ | ) (10)and not ˜ G alone as has falsely been assumed inSchneider & Watts (2005). The function ω ( | ∆ θ | ) ≡ h κ g ( θ ) κ g ( θ + ∆ θ ) i (11)is the angular clustering of the lenses (Peebles 1980).Based on this notion, we can write down an estimatorfor ˜ G :˜ G est = N l ,N s P i,j,k =1 w k ǫ k e − i( ϕ i + ϕ j ) [1 + ω ( | θ i − θ j | )]( − N l ,N s P i,j,k =1 w k (12)that includes explicitly the clustering of lenses. Here, w k ( k ∈ { . . . N s } ) are the statistical weights of the sources.By ϕ i and ϕ j ( i, j ∈ { . . . N l } ) we denote the phase anglesof the two lenses relative to the source k . Again, onlytriangles of the same or similar dimensions (parameters insame bins) are to be included inside the sums. This becomes apparent if one sets γ ` θ i ; ϕ + ϕ ´ = γ ` θ ; ϕ + ϕ ´ = const in the Eqs. (34)and (32), respectively, of Schneider & Watts (2005).imon et al.: GGGL in RCS 5 For obtaining an estimate of ω ( θ ) in practice we em-ployed the estimator of Landy & Szalay (1993), which,compared to other estimators, minimises the variance tonearly Poissonian: ω ( θ ) = DDRR − DRRR + 1 . (13)It requires one to count the number of (lens) galaxy pairswith a separation between θ and θ + δθ , namely the num-ber of pairs in the data, denoted by DD , the number ofpairs in a random mock catalogue, RR , and the numberof pairs that can be formed with one data galaxy and onemock data galaxy, DR . The random mock catalogue iscomputed by randomly placing the galaxies, taking intoaccount the geometry of the data field, i.e. by avoidingmasked-out regions, see Fig. 12. We generate 25 randomgalaxy catalogues and average the pair counts obtainedfor DR and RR .When computing the ˜ G and ˜ G ± estimators, we suggestthe use of complex numbers for the angular positions ofgalaxies: ϑ = ϑ + i ϑ with ϑ , being the x / y -coordinatesrelative to some Cartesian reference frame (flat-sky ap-proximation). The phase factors turning up inside thesums (8), (9) and (12) are then simply (notation of Fig.2):e − ϕ = ϑ ∗ ϑ ; e − ϕ = ϑ ∗ ϑ ; e − i( ϕ + ϕ ) = ϑ ϑ | ϑ || ϑ | , (14)where ϑ ij ≡ ϑ i − ϑ j . In weak lensing, cosmological large-scale structure is oftenstudied in terms of the aperture statistics (Simon et al.2007; Kilbinger & Schneider 2005; Jarvis et al. 2004;Hoekstra et al. 2002a; Schneider 1998; Van Waerbeke1998) that measure the convergence (projected matterdistribution), κ , and projected number density fields ofgalaxies, κ g , smoothed with a compensated filter u ( x ), i.e. R ∞ d x xu ( x ) = 0: M ap ( θ ) = 1 θ Z ∞ d ϑ u (cid:18) | ϑ | θ (cid:19) κ ( | ϑ | ) , (15) N ( θ ) = 1 θ Z ∞ d ϑ u (cid:18) | ϑ | θ (cid:19) κ g ( | ϑ | ) , (16)where θ is the smoothing radius. M ap is called the aper-ture mass, while N is the aperture number count ofgalaxies. With an appropriate filter these aperture mea-sures are only sensitive to a very narrow range of spa-tial Fourier modes so that they are extremely suitablefor studying the scale-dependence of structure, or eventhe scale-dependence of remaining systematics in the data(Hetterscheidt et al. 2007). Moreover, they provide a verylocalised measurement of power spectra (band power), inthe case of (cid:10) N n M m ap (cid:11) for n + m = 2, and bispectra, inthe case of n + m = 3, without relying on complicated transformations between correlation functions and powerspectra. The aperture filter we employ for this paper is: u ( x ) = 12 π (cid:18) − x (cid:19) e − x / (17)as introduced by Crittenden et al. (2002). For an aper-ture radius of θ the filter peaks at a spatial wavelengthof ℓ = √ θ which corresponds to a typical angular scale of πℓ = π √ θ .As shear and convergence are both linear combinationsof second derivatives of the deflection potential, the aper-ture mass can be computed from the shear in the followingmanner (Schneider et al. 1998): M ap ( θ ) + i M ⊥ ( θ ) = 1 θ Z ∞ d θ ′ q (cid:18) | θ ′ | θ (cid:19) γ ( θ ′ ; ∠ θ ′ ) , (18) q ( x ) ≡ x Z x d s s u ( s ) − u ( x ) , (19)where we denote by ∠ θ ′ the polar angle of the vector θ ′ .Note that in Eq. (18) we place, for convenience, the originof the coordinate system at the centre of the aperture.In expression (18), M ap is the E-mode, whereas M ⊥ isthe B-mode of the aperture mass. Of central importancefor our work is that we can extract E- and B-modes of theaperture statistics from the correlation functions. Since B-modes cannot be generated by weak gravitational lensing,a zero or small B-mode is an important check for a success-ful PSF-correction of real data (e.g. Hetterscheidt et al.2007), or the violation of parity-invariance in the data(Schneider 2003), which is also a signature of systematics.Another argument in favour of using aperture statis-tics at this stage of our analysis is that 2 nd -order termsin ˜ G and ˜ G ± do not contribute to the 3 rd -order aperturestatistics (Schneider & Watts 2005). Therefore, a signifi-cant signal in the aperture statistics means a true detec-tion of 3 rd -order correlations.The 3 rd -order aperture statistics can be computedfrom ˜ G via: (cid:10) N M ap (cid:11) ( θ , θ , θ ) = (20) ℜ (cid:16) I h ˜ G ( ϑ , ϑ , φ ) A N N M ( ϑ , ϑ , φ | θ , θ , θ ) i(cid:17) , (cid:10) N M ⊥ (cid:11) ( θ , θ , θ ) = (21) ℑ (cid:16) I h ˜ G ( ϑ , ϑ , φ ) A N N M ( ϑ , ϑ , φ | θ , θ , θ ) i(cid:17) , where we have introduced for the sake of brevity an ab-breviation for the following integral: I [ f ] ≡ ∞ Z d ϑ ϑ ∞ Z d ϑ ϑ π Z d φ f . (22)By ℜ ( x ) and ℑ ( x ) we denote the real and imaginarypart, respectively, of a complex number x . Eq. (20) is theE-mode of the aperture moment hN ( θ ) N ( θ ) M ap ( θ ) i ,whereas Eq. (21) is the corresponding parity mode that Simon et al.: GGGL in RCS is non-zero in the case of violation of parity-invariance;the latter has to be zero even if
B-modes are present inthe shear pattern that may be produced to some degreeby intrinsic source alignment (e.g. Heymans et al. 2004)or intrinsic ellipticity/shear correlations (Hirata & Seljak2004) – that is, if we assume that the macroscopicworld is parity-invariant. The integral kernel A N N M forour aperture filter can be found in the Appendix ofSchneider & Watts (2005).The aperture statistics associated with the GGGL-correlator ˜ G ± are the following: (cid:10) M N (cid:11) ( θ , θ , θ ) = (23) ℜ [ h M M
N i ( θ , θ , θ ) + h M M ∗ N i ( θ , θ , θ )] / , (cid:10) M ⊥ N (cid:11) ( θ , θ , θ ) = (24) ℜ [ h M M ∗ N i ( θ , θ , θ ) − h M M
N i ( θ , θ , θ )] / , h M ⊥ M ap N i ( θ , θ , θ ) = (25) ℑ [ h M M
N i ( θ , θ , θ ) + h M M ∗ N i ( θ , θ , θ )] / , where we used the following definitions h M M
N i ( θ , θ , θ ) ≡ (26) I h ˜ G − ( ϑ , ϑ , φ ) A MM N ( ϑ , ϑ , φ | θ , θ , θ ) i , h M M ∗ N i ( θ , θ , θ ) ≡ (27) I h ˜ G + ( ϑ , ϑ , φ ) A MM ∗ N ( ϑ , ϑ , φ | θ , θ , θ ) i . Eq. (23) is the E-mode of h M ap ( θ ) M ap ( θ ) N ( θ ) i , Eq.(24) is the B-mode which should vanish if the shearpattern is purely gravitational, and Eq. (25) is again aparity-mode which is a unique indicator for systemat-ics. As before, the integral kernels A MM N and A MM ∗ N for our aperture filter may be found in the Appendix ofSchneider & Watts (2005). In the last section, we outlined the steps which have tobe undertaken in order to estimate the 3 rd -order aperturemoments from a given catalogue of lenses and sources.The three steps are: 1) estimating the angular clusteringof lenses yielding ω ( θ ), 2) estimating ˜ G and ˜ G ± for somerange of ϑ , and for φ ∈ [0 , π [, and finally 3) transform-ing the correlation function to (cid:10) N M ap (cid:11) and (cid:10) N M (cid:11) in-cluding all E-, B- and parity-modes.There are several practical issues involved here. Oneissue is that, in theory, for the transformation we require˜ G , ˜ G ± for all ϑ ∈ [0 , ∞ ], see Eq. (22). In reality, we willhave both a lower limit (seeing, galaxy-galaxy overlap-ping), ϑ low , and an upper limit (finite fields), ϑ upper . Onthe other hand, the GGGL-correlators drop off quickly forlarge ϑ and the integral kernels A N N M , A MM N , A MM ∗ N have exponential cut-offs for ϑ , ϑ ≫ θ , , . Therefore,we can assume that there will be some range where we Fig. 3.
Test run of our computer code with mock databased on some arbitrary convergence field. The mockdata has been prepared such that (cid:10) N ( θ ) M ap ( θ ) (cid:11) = (cid:10) N ( θ ) M ( θ ) (cid:11) = (cid:10) N ( θ ) (cid:11) ; h N N N i ≡ (cid:10) N ( θ ) (cid:11) is thevalue that has to be found by the code (only equally sizedapertures are correlated for test: θ = θ = θ = θ ). Thebinning range is between ϑ ∈ [0 . , lenses and the same number of sources.For radii greater than ∼ θ &
10 pixel be-cause the aperture size becomes comparable to the fieldsize. The error bars denote the 1 σ sampling uncertaintydue to finite galaxy numbers. The B- and parity modes(P) of the statistics are two orders of magnitude smallerthan the E-modes and are oscillating about zero (plottedis modulus).can compute the aperture statistics with satisfactory ac-curacy. We perform the following test to verify that thisis true: by using theoretical 3D-bispectra of the galaxy-dark matter cross-correlations (Watts & Schneider 2005)we compute both the GGGL-correlation functions andthe corresponding aperture statistics (Eqs. 37, 38, 40, 51,52 of Schneider & Watts 2005). By binning the GGGL-correlators we perform the transformation including bin-ning and cut-offs in ϑ . We find that one can obtain anaccurate estimate of the aperture statistics within a fewpercent between roughly θ & ϑ low and θ . ϑ upper / ϑ , and 100 linear bins for φ ).Therefore, with RCS-fields of typical size 139 ′ we can ex-pect to get an accurate result between about 0 ′ . . θ . ′ .Another issue is with step two above, in which theGGGL-correlators themselves need to be estimated. Theestimators – Eqs. (8), (9) and (12) – in terms of galaxy po-sitions and source ellipticities are simple but the enormousnumber of triangles that need to be considered is compu-tationally challenging (roughly 10 per field for RCS). Tooptimise this process we employ a data structure based on imon et al.: GGGL in RCS 7 a binary tree, a so-called tree code (e.g. Jarvis et al. 2004;Zhang & Pen 2005). The tree-code represents groups ofgalaxies within some distance to a particular triangle ver-tex as “single galaxies” with appropriate weight (and aver-age ellipticity). This strategy effectively reduces the num-ber of triangles. Moreover, we optimise the code such thatonly distinct triangles are found. Then, the other triangleobtained be exchanging the indices of either the two lenses( ˜ G ) or the two sources ( ˜ G ± ) is automatically accountedfor; this reduces the computation time by a factor of two.In order to test the performance and reliability of thecode, we create a catalogue of mock data. In order to dothis we use a simulated convergence field ( κ -field) on agrid, 512 ×
512 pixel , which has been obtained by ray-tracing through an N-body simulated universe. Actually,the only requirement that has to be met by the test fieldis that it behaves like a density contrast δ , i.e. h δ i = 0and δ ≥ −
1, and that it has non-vanishing 3 rd -order mo-ments, (cid:10) δ (cid:11) = 0. Based on this field we simulate a shearand lens catalogue. The shear catalogue is generated byconverting the κ -field to a shear field and by randomlyselecting positions within the field to be used as sourcepositions. The positions and associated shear provide themock shear catalogue; for details see Simon et al. (2004).In a second step, we use the κ -field as density contrast, κ g , of the number density of lenses to make realisationsof lens catalogues. This means one randomly draws posi-tions, θ , within the grid area and one accepts that posi-tion if x ≤ κ ( θ )1+ κ max , where x is a random number between x ∈ [0 ,
1] and κ max the maximum value within the κ field.Following this procedure one gets mock data for which κ = κ g and therefore (cid:10) N n M m ap (cid:11) = hN n + m i . In particularwe must get, apart from the statistical noise due to finitegalaxy numbers, (cid:10) N ( θ ) M ap ( θ ) (cid:11) = (cid:10) N ( θ ) M ( θ ) (cid:11) when running our codes with the mock data.Parallelly, we smooth the test shear field within aper-tures according to the definitions (18) with our aperturefilter and estimate the test data aperture statistics directlyby cross-correlating the smoothed fields. This also has tobe comparable (apart from shot noise) to our code output.The result of this test can be found in Fig. 3.As a further test we take the same mock data butrotate the ellipticities of the sources by 45 degrees, i.e.we multiply the complex ellipticities by the phase factor − e − φ with φ = 45 ◦ . This generates a purely B-mode sig-nal that should only be picked up by the B-mode channelsof the aperture statistics, yielding a plot similar to Fig. 3.The parity mode in (cid:10) N M (cid:11) has to be unaffected. This isindeed the case (figure not shown).The test results make us confident that the computercode is working and that we achieve a good accuracy eventhough we are forced to bin the correlation functions andto use a tree-code that necessarily makes some additionalapproximations. Fig. 5.
Combined measurement of angular clusteringof our sample of lenses (no correction for the integralconstraint). Error bars were obtained by looking at thefield-to-field variance. The solid line is a power-law fit, ω ( θ ) = A ω θ − β , to the regime θ ∈ [0 ′ . , ′ ].
4. Results and discussion
We applied the previously outlined method to the RCSshear and lens catalogues.Lenses were selected between photometric redshifts0 < z < .
4, whereas sources were from the range0 . < z < .
4. Compared to Hoekstra et al. (2005), inwhich photometric redshifts smaller than 0 . z ∼ . As a first result we would like to draw the reader’s at-tention to the angular clustering of lenses which is plot-ted in Fig. 5. This measurement was required for the es-timator ˜ G in Eq. (12). As widely accepted, the angularcorrelation function ω ( θ ) is, for the separations we areconsidering here, well approximated by a simple power-law, depending on galaxy type, colour and luminosity (e.g.Madgwick et al. 2003). As can be seen in Fig. 5, the power-law behaviour is also found for our lens galaxy sample. Theangular clustering plotted is still affected by the so-called Simon et al.: GGGL in RCS
Fig. 4.
Left : Aperture statistics (cid:10) N M ap (cid:11) ( θ, θ, θ ) for different aperture radii θ as measured in RCS. The upper panelis the E-mode, the lower panel is the parity mode which is consistent with zero. Error bars denote the field-to-fieldvariance between the ten RCS fields. Statistical errors are strongly correlated. The lines are tentative halo model-basedpredictions with arbitrary HODs for a ΛCDM cosmological model (see text). Right : Aperture statistics (cid:10) N M (cid:11) ( θ, θ, θ )for different aperture radii θ as measured in RCS. The upper panel contains the E-mode measurement, while B-mode(stars) and parity mode (squares) are plotted inside the lower panel. Error bars that extend to the bottom of theupper panel denote data points that are consistent with zero.integral constraint (Groth & Peebles 1977), which shiftsthe estimate of ω downwards by a constant value depend-ing on the geometry and size of the fields. For small θ . ′ this bias is negligible so that we used only the regime θ ∈ [0 ′ . , ′ ] to find the maximum likelihood parameters ofthe power-law.For ˜ G this power-law fit was used. Possible deviationsof the true clustering from a power-law for θ ≥ ′ werenegligible because for the estimator one actually needs 1+ ω instead of ω . Since ω is roughly smaller than ∼ .
05 anddecreasing for θ ≥ ′ , we gather that a certain remaininginaccuracy in ω has no big impact on 1+ ω . The power-lawindex is, with β = 0 .
58, fairly shallow, which is typical fora relatively blue sample of galaxies (e.g. Madgwick et al.2003).In a second step, the correlation functions ˜ G and ˜ G ± were computed separately for each of the ten RCS fields.The total combined signal was computed by taking theaverage of all fields, each bin weighted by the number oftriangles it contained. For the binning we used a rangeof 0 ′′ . ≤ ϑ ≤ ′ with 100 bins, thus overall 10 triangleconfigurations. By repeatedly drawing ten fields at ran-dom from the ten available, i.e. with replacement, andcombining their signal we obtained a bootstrap sample ofmeasurements. The variance among the bootstrapped sig-nals was used to estimate the sum of cosmic variance andshot noise, thus the remaining statistical uncertainty ofthe correlation functions.Finally, the correlation functions were transformed tothe aperture statistics considering only equally sized aper-tures, i.e. θ = θ = θ , see Fig. 4. For the scope of thiswork, equally sized apertures are absolutely sufficient. In future work, however, one would like to harvest the fullinformation that is contained in these statistics by explor-ing different θ i which then would cover the full (projected)bispectrum.For a start, we would like to focus on ˜ G . The left panelin Fig. 4 reveals a clean detection of (cid:10) N M ap (cid:11) for aper-ture radii between 0 ′ . . θ . ′ (with the adopted filterthis corresponds to typical angular scales between 1 ′ and11 ′ ) demonstrating the presence of pure 3 rd -order corre-lations between shear and lens distribution in RCS. Theparity mode of this statistic is consistent with the zero asexpected. Fig. 4 is one of the central results of this paper.We would like to further support that this is a real,i.e. cosmological, signal by comparing the measurement tocrude halo model-based predictions (see Cooray & Sheth2002, for a review). The halo model was used to pre-dict a spatial cross-correlation bispectrum, B gg δ (Eq. 12 inSchneider & Watts 2005), for a particular fiducial cosmo-logical model and halo occupation distribution (HOD) ofgalaxies (see Berlind & Weinberg 2002). By applying Eqs.(21), (52) in Schneider & Watts (2005), B gg δ was trans-formed, taking into account the correct redshift distribu-tion of lenses and sources (Fig. 1), to yield the aperturestatistics. A standard concordance ΛCDM model was em-ployed (Bardeen et al. 1986) with parameters Ω Λ = 0 . m = 0 .
30, for the (cold) darkmatter density, σ = 0 . .
21 for the shape parameter. This is inagreement with constraints based on the first WMAP re-lease (Spergel et al. 2003).The latest constraints favour a somewhat smaller valuefor σ (Benjamin et al. 2007; Hetterscheidt et al. 2007) imon et al.: GGGL in RCS 9 Fig. 6.
Residual signal (squares) of GGGL in RCS when the ellipticities of the sources are randomised ( left : (cid:10) N M ap (cid:11) , right : (cid:10) N M (cid:11) ). For comparison, the original signal before randomisation is also plotted (crosses). The line is a crudehalo-model prediction of a blue galaxy population as in Fig. 4. The error bars of the randomised signal quantify thebackground noise of a null-signal. This indicates that we have a significant detection of (cid:10) N M ap (cid:11) in the left panel butonly a weak detection of (cid:10) N M (cid:11) , most significant at about 2 ′ , in the right panel.which would shift the expected amplitude of GGGL to-wards smaller values. If we apply the scaling relation ofJain & Seljak (1997), given for the convergence bispec-trum, as a rough estimate of this shift, B κ ∝ σ . , we ob-tain a correction factor of about two for σ = 0 . N and M ap should have the same σ -dependence for unbiasedgalaxies).The halo-model predictions depend strongly on theadopted HOD. The basic set up for this model was thatoutlined in Takada & Jain (2003), which splits the occupa-tion function, N ( M ), into contributions from “red”, N R ,and “blue”, N B , galaxies: h N B i ( M ) = (cid:18) mm B (cid:19) γ B + A exp (cid:0) − A (log ( m ) − m B s ) (cid:1) h N R i ( M ) = (cid:18) mm R (cid:19) γ R exp (cid:18) − h m R m i , (cid:19) . (28)As parameters we used m B = 2 . × M ⊙ , A =0 .
65, A = 6 . m B s = 11 . m R = 1 . × M ⊙ and m R = 4 . × M ⊙ . Blue galaxies have a peak halooccupancy of around 10 M ⊙ and a shallow power law( γ B = 0 .
93) at high halo masses. In this simple prescrip-tion, red galaxies are relatively more numerous in highermass halos ( γ R = 1 .
1) and are excluded from low mass ha-los by an exponential cutoff around 5 × M ⊙ . Factorialmoments of the occupation distribution - the cross bis-prectra B ggδ and B δδg require the mean and variance- were as prescribed in the model of Scoccimarro et al.(2001). In this way, the moments are Poissonian forhigher mass halos, becoming sub-Poissonian for massesbelow 10 M ⊙ , i.e. (cid:10) N (cid:11) ( M ) = α [ h N i ( M )] , where α = 0 . ( m/ M ⊙ ).We stress at this point that we made no attemptto “fit” parameters to the data, we merely intended to bracket a range of possible results. To choose a range ofplausible scenarios, we constructed the theoretical aper-ture statistics for “red” galaxies, “blue” galaxies and for“all” galaxies (in which the occupation functions for redand blue galaxies are added together directly). We alsoshowed predictions for the unbiased case, in which the oc-cupation function N ( M ) ∝ M with Poisson moments for (cid:10) N ( M ) (cid:11) . Galaxies were assumed to follow the CDM halodensity profile (NFW) with no assumption of a centralgalaxy. Other parameters that define the halo model setup (e.g. concentration of the NFW profile) were as usedin Takada & Jain (2003).Our measurement of (cid:10) N M ap (cid:11) lies somewhat abovethe lower bound of the expected physical range of values,giving support as to the cosmological origin of the signal.Moreover, taken at face value, our result appears to fitthe picture that the lens population consists of rather bluegalaxies as has been concluded from the shallow slope ofthe angular correlation function ω .We randomised the ellipticities of the sources and re-peated the analysis. Since the coherent pattern, and itscorrelation to the lens distribution, is responsible for thesignal, destroying the coherence by randomising the ellip-ticity phase should diminish the signal. That this is thecase can be seen in Fig. 6 (left panel).Analogous to (cid:10) N M ap (cid:11) we computed and predicted (cid:10) N M (cid:11) , the result for which is shown in Fig. 4 (rightpanel). Here a signal significantly different from zero wasonly found for aperture radii 1 ′ ≤ θ ≤ ′ and at about θ ∼ ′ .
5. Below θ ∼ ′ . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) 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cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) III IV originalparity index swapping
III (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) lens 2 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) lens 1
Fig. 7.
Sketch illustrating how ˜ G or G are plotted. Seetext for details. Fig. 8.
Signal in ˜ G originating from pure 2 nd -order statis-tics (GGL) that was subtracted from ˜ G to obtain Fig. 9(left). The gray-scale intensity is the modulus of ˜ G , thesticks indicate the average shear at the source positionin the lens-lens-source triangle configuration. The units ofthe axis are in h − Mpc which corresponds to the meanphysical scale at the lens distance of about z = 0 .
30. Thetwo lenses are located at the positions of the crosses, leftand right from the centre.the signal is on average smaller than the lowest theoreticalvalue from our crude models. However, as discussed above,a lower σ easily brings the model down towards smallervalues. The signal disappeared if the ellipticities of thesources were randomised (Fig. 6, right panel). Therefore,we found a tentative detection of (cid:10) N M (cid:11) in our data. The aperture statistics clearly have advantages: the B- andparity-modes allow a check for remaining systematics inthe data, and 2 nd -order statistics do not make any contri-butions so that we can be sure to pick up a signal solelyfrom connected 3 rd -order terms. This is what we did inthe forgoing subsection. The result suggests that we havea significant detection of G .The disadvantage of using aperture statistics is, how-ever, that they are hard to visualise in terms of a typi-cal (projected) matter distribution (lensing convergence)about two lenses, say. Therefore, we introduce here an al-ternative way of depicting G which is similar to the workthat has been proposed by Johnston (2006).A similar way of visualising G ± probably could bethought up as well. However, since we found only a weakdetection of GGGL with two sources and one lens we post-pone this task to a future paper and focus here on G alone.The following summarises what essentially is done if weestimate ˜ G from the data for fixed lens-lens separations.We pick out only lens-lens-source triangles from our dataset in which the lenses have a fixed separation or a sepa-ration from a small range. Each triangle is placed insidethe plot such that the line connecting the lenses is paral-lel to the x -axis and that the centre of this line coincideswith the centre of the plot, as seen for the triangles in Fig.7. The ellipticities of the sources of all triangles are thenmultiplied by 1 + ω ( | θ − θ | ) (rescaled according to Eq.10) and (weighted) averaged at the source positions. Forthis paper, we used 128 ×
128 grid cells for binning the el-lipticities. Following this procedure we effectively stackedall shear patterns about a lens-lens configuration – rotatedappropriately – to obtain an average shear field about twolenses. This is, in essence, the meaning of ˜ G . The full ˜ G is abundle of such plots with continuously changing lens-lensseparations.Note that the ellipticity at the source position, storedin G , is rotated by φ / G appropriately.The resulting plot has symmetries. Firstly, we do notdistinguish between ”lens 1” and “lens 2”. Both lenses aredrawn from the same galaxy sample . This means for everytriangle, we will find the same triangle but with the posi-tions of “lens 1” and “lens 2” exchanged. Therefore, thetwo lenses and the source of the triangle named “original”in Fig. 7 will make the same contribution but complex con-jugated at the source position of the triangle named “in-dex swapping”. Thus, quadrants I and III will be identicalapart from a complex conjugate and mirroring the posi-tions about the x - and y -axis. The same holds for quad-rants II and IV. This would no longer be true, of course, ifwe chose the two lenses from different catalogues in order imon et al.: GGGL in RCS 11 Fig. 9.
Plots of G after subtraction of the 2 nd -order signal from ˜ G . The units used are h − Mpc, which correspondsto the mean comoving physical distance at the lenses’ distance of, on average, z = 0 . Left:
Lenses were selected tohave a mutual angular separation between 40 ′′ and 80 ′′ corresponding a projected physical scale of about 250 h − kpc. Right:
Lenses were chosen to have a separation between 4 ′ and 8 ′ , or equivalently a projected comoving separationbetween 1 − h − Mpc.to, for instance, study the matter distribution around ablue and a red galaxy.A second symmetry can be observed if the Universe(or the PSF-corrected shear catalogue) is parity invariant.Mirroring the triangle “original” with respect to the lineconnecting the two lenses ( x -axis) results in another tri-angle coined “parity”. For parity invariance being true theellipticity at the source position of “parity” is on average identical to the ellipticity at the source position of tri-angle “original”. In this case, quadrant IV is statisticallyconsistent with quadrant I and quadrant II with quad-rant III (after mirroring about the x -axis). Taking paritysymmetry for granted could be used to increase the signal-to-noise in the plots by taking the mean of quadrants IVand I (or II and III).Since the way of binning in the plot is completely dif-ferent from the way used to get the aperture statisticsout of RCS, we made two reruns of the estimation of ˜ G with our data. For the first run we only considered lens-lens separations between 40 ′′ and 80 ′′ , the second run se-lected triangles in which the lenses had a separation be-tween 4 ′ and 8 ′ . For a mean lens redshift of z ∼ . h − kpc and 1 . h − Mpc, respectively. Asusual, the results from the ten individual fields were av-eraged by weighting with the number of triangles insideeach bin and the statistical weights of the sources.Since we effectively stacked the shear fields about allpairs of lenses, aligned along the lens-lens axis, we ob-tained the average shear about two lenses. The shear pat-tern still contained a contribution stemming from GGLalone. This contribution could, however, easily be sub- tracted according to Eq. (5) after estimating the meantangential shear, h γ t i ( ϑ ), about single lens galaxies (seee.g. Simon et al. 2007). A typical shear pattern due to2 nd -order GGL can be seen in Fig. 8. This is the shearpattern that is to be expected if the average shear abouttwo lenses is just the sum of two mean shear patternsabout individual lenses. They contain all contributionsthat are statistically independent of the presence of theother lens. Therefore, contributions (contaminations) tothe shear from lens pairs that are just accidentally closeto each other by projection effects, but actually too sepa-rated in space to be physically connected, are removed.Now, Fig. 9 shows the shear patterns after removingthis signal. Clearly, there is a residual coherent patternwhich is most pronounced for the smaller lens-lens sep-arations. This proves that one finds an additional shearsignal around two galaxies if they get close to each other.Hence, the average gravitational potential about two closelenses is not just the sum of two average potentials aboutindividual lenses.Unfortunately, all physically close galaxies with a fixedprojected angular separation contribute to the excessshear– independent of whether they are in galaxy groupsor clusters. Exploiting lens redshifts and rejecting lensesfrom regions of high number densities on the sky mighthelp to focus on galaxy groups, for example. This, how-ever, is beyond the scope of this paper.One can relate the residual shear pattern in Fig. 9 to anexcess in projected convergence (matter density) using thewell known relation between convergence and cosmic shear Fig. 10.
Convergence fields obtained by transforming the shear fields in Fig. 9. They are related to the (average)excess in matter density around two galaxies of fixed angular separation after subtraction of the matter density profilethat is observed about individual galaxies.
Left:
Residual convergence for two lenses with projected comoving distanceof roughly 250 h − kpc. The box-size is 1 . h − Mpc × . h − Mpc.
Right:
Residual convergence at about 1 . h − Mpcprojected lens-lens distance. The box-size is 8 . h − Mpc × . h − Mpc. Note that the convergence in this figure islower by roughly an order of magnitude compared to the left figure.in weak gravitational lensing (Bartelmann & Schneider2001; Kaiser & Squires 1993): γ ℓ = ℓ − ℓ + 2i ℓ ℓ ℓ + ℓ κ ℓ , (29)where γ ℓ and κ ℓ are the Fourier coefficients of theshear and convergence fields, respectively, on a grid and ℓ = ( ℓ , ℓ ) is a particular angular mode of the grid inCartesian coordinates. We obtained the γ ℓ ’s by employ-ing Fast-Fourier-Transforms (and zero-padding to reduceundesired edge effects) after binning the residual shearpatterns onto a 512 ×
512 grid. We assumed that the con-vergence is zero averaged over the box area which makes κ ℓ = 0 for ℓ = 0. Fig. 10 shows the thereby computedmaps. The plots were smoothed with a kernel of a sizeof a few pixels.As a cross-check we also transformed the shear pat-tern produced by the 2 nd -order terms in ˜ G (Fig. 8) andfound, as expected, that the corresponding convergencefields were just two identical radially symmetric “matterhaloes” placed at the lens positions in the plot.In the same way as in the previously discussed shearplots, parity invariance can also be checked in the conver-gence plots: quadrants I and IV (or II and III), mirroredabout the x -axis, have to be statistically consistent. If wewould like to enforce parity invariance, we could take theaverage of the two quadrants. Secondly, if one obtains the convergence field from the shear field via a Fourier trans-formation as described before, the convergence field willbe a field of complex numbers. In the absence of any B-modes, however, the imaginary part will be zero or purenoise. Thus, the imaginary part of the convergence can beused to either check for residual B-modes or to estimatethe noise level of the E-mode (real part).This was done for Fig. 11. We found that the residualconvergence for the small lens-lens separation is highlysignificant within the central region of Fig. 10, left panel,whereas the convergence in the right panel of Fig. 10 isnoise dominated. This means we did not find any excessconvergence beyond the noise level for the lens-lens pairsof large separation.To sum up, one can see that closer lens pairs areembedded inside a common halo of excess matter, whilethe lenses with larger separation appear relatively discon-nected; the convergence for the lenses of larger separationis lower by at least one order of magnitude and slips be-low noise level in our measurement. This result definitelydeserves further investigation which we will do in a forth-coming paper.
5. Conclusions
We found a significant signal of GGGL in RCS – at leastfor the case for which we considered two lenses and one imon et al.: GGGL in RCS 13
Fig. 11.
Plots similar to the plots in Fig. 10 except the the shear has been rotated by 45 ◦ (B-mode) before transformingto the convergence fields. The thereby obtained convergence quantifies the statistical noise in the plots of Fig. 10.source. The signal is of an order of magnitude which isexpected from a crude halo model-based prescription. Thissuggests a cosmological origin of the observed correlation.In particular, our finding demonstrates that wide-fieldsurveys of at least the size of RCS allow us to exploitGGGL. As can be seen in Fig. 4 (left), the remaining sta-tistical uncertainties of the measurement are much smallerthan the shift of the signal expected for different HODs ofthe adopted halo model. This means with GGGL we nowhave a new tool to strongly constrain galaxy HODs, andpossibly even spatial distributions of galaxies inside haloesin general, which is a parameter in the framework of thehalo model. As the wide-field shear surveys of the nextgeneration will be substantially larger than RCS thoseconstraints will become tighter. Further subdivisions oflens samples into different galaxy types and redshifts willtherefore still give a reasonable signal-to-noise ratio.Leaving the interpretation in the context of the halomodel aside, the measurement of GGGL can be trans-lated into a map of excess convergence around two galax-ies of a certain mutual (projected) distance. For RCS, wedemonstrated that there is a significant excess in conver-gence about two lenses if galaxies are as close as roughly250 h − kpc. Although the details need still to be workedout, this promises to be a novel way of studying the matterenvironment of groups of galaxies. Acknowledgements.
We would like to thank Jan Hartlapfor providing us with simulated shear catalogues used asmock data. We are also grateful to Emilio Pastor Mira,who kindly computed the aperture statistics in our mockdata using his aperture based code, Oliver Cordes who helped us with the Linux cluster and Lindsay King forcomments on the paper. This work was supported by theDeutsche Forschungsgemeinschaft (DFG) under the projectSCHN 342/6–1 and by the Priority Programme SPP 1177‘Galaxy evolution’ of the Deutsche Forschungsgemeinschaft un-der the project SCHN 342/7–1. Patrick Simon was also sup-ported by PPARC. Henk Hoekstra acknowledges support fromNSERC and CIAR.
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Fig. 12.
Determined geometry of the individual RCS fields (field masks). Each mask has a size of approximately139 ′ × ′ or, equivalently, 20 . × .
000 pixel2