Source Selection for Cluster Weak Lensing Measurements in the Hyper Suprime-Cam Survey
Elinor Medezinski, Masamune Oguri, Atsushi J. Nishizawa, Joshua S. Speagle, Hironao Miyatake, Keiichi Umetsu, Alexie Leauthaud, Ryoma Murata, Rachel Mandelbaum, Cristóbal Sifón, Michael A. Strauss, Song Huang, Melanie Simet, Nobuhiro Okabe, Masayuki Tanaka, Yutaka Komiyama
PPubl. Astron. Soc. Japan (2014) 00(0), 1–19doi: 10.1093/pasj/xxx000 Source Selection for Cluster Weak LensingMeasurements in the Hyper Suprime-CamSurvey
Elinor Medezinski , Masamune Oguri , Atsushi J. Nishizawa ,Joshua S. Speagle , Hironao Miyatake , Keiichi Umetsu , AlexieLeauthaud , Ryoma Murata , Rachel Mandelbaum , Crist ´obal Sif ´on ,Michael A. Strauss , Song Huang , Melanie Simet , NobuhiroOkabe , Masayuki Tanaka and Yutaka Komiyama Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ08544, USA Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI),TokyoInstitutes for Advanced Study, The University of Tokyo, Chiba 277-8582, Japan Research Center for the Early Universe, University of Tokyo, Tokyo 113-0033, Japan Department of Physics, University of Tokyo, Tokyo 113-0033, Japan Institute for Advanced Research, Nagoya University, Nagoya 464-8602, Aichi, Japan Harvard University, 60 Garden St, Cambridge, MA 02138 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Institute of Astronomy and Astrophysics, Academia Sinica, P. O. Box 23-141, Taipei 10617,Taiwan Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156HighStreet, Santa Cruz, CA 95064 USA McWilliams Center for Cosmology, Department of Physics, Carnegie MellonUniversity,Pittsburgh, PA 15213, USA University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA Department of Physical Science, Hiroshima University, 1-3-1Kagamiyama,Higashi-Hiroshima, Hiroshima 739-8526, Japan Hiroshima Astrophysical Science Center, Hiroshima University,Higashi-Hiroshima,Kagamiyama 1-3-1, 739-8526, Japan National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Department of Astronomy, School of Science, SOKENDAI (The Graduate University forAdvanced Studies), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan ∗ E-mail: [email protected]
Received ; Accepted
Abstract
We present optimized source galaxy selection schemes for measuring cluster weak lensing(WL) mass profiles unaffected by cluster member dilution from the Subaru Hyper Suprime-CamStrategic Survey Program (HSC-SSP). The ongoing HSC-SSP survey will uncover thousandsof galaxy clusters to z (cid:46) . . In deriving cluster masses via WL, a critical source of systemat-ics is contamination and dilution of the lensing signal by cluster members, and by foregroundgalaxies whose photometric redshifts are biased. Using the first-year CAMIRA catalog of ∼ c (cid:13) a r X i v : . [ a s t r o - ph . C O ] J a n Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 clusters with richness larger than 20 found in ∼
140 deg of HSC-SSP data, we devise and com-pare several source selection methods, including selection in color-color space (CC-cut), andselection of robust photometric redshifts by applying constraints on their cumulative probabilitydistribution function (PDF; P-cut). We examine the dependence of the contamination on thechosen limits adopted for each method. Using the proper limits, these methods give massprofiles with minimal dilution in agreement with one another. We find that not adopting eitherthe CC-cut or P-cut methods results in an underestimation of the total cluster mass ( ± )and the concentration of the profile ( ± ). The level of cluster contamination can reachas high as ∼ % at R ≈ .
24 Mpc /h for low-z clusters without cuts, while employing eitherthe P-cut or CC-cut results in cluster contamination consistent with zero to within the 0.5% un-certainties. Our robust methods yield a ∼ σ detection of the stacked CAMIRA surface massdensity profile, with a mean mass of M = (1 . ± . × M (cid:12) /h . Key words: gravitational lensing: weak — dark matter — galaxies:clusters: general
Tracing the exponential tail of the halo mass function, clustersof galaxies are a powerful probe of cosmology, and in particular,their abundance is sensitive to the late-time nonlinear growthof structure. Placing competitive cosmological constraints withcluster abundances requires precise and accurate masses forthese objects. Calibrations of indirect mass proxies for clus-ters detected by the Sunyaev-Zel’dovich (SZ) effect (Sunyaev& Zeldovich 1972), X-ray or optical surveys typically rely onscaling relations calibrated via alternative methods. Some ofthese relations make assumptions about the cluster dynamicalstate, e.g., hydrostatic equilibrium (HSE) in the case of X-rayobservations.The distribution of mass within clusters provides further in-sight into dark matter (DM) and structure formation scenarios.Simulations of cold DM (CDM) dominated halos consistentlypredict mass profiles that steepen with radius, providing a dis-tinctive, fundamental prediction for this form of DM (Navarroet al. 1997). Furthermore, the degree of mass concentration, c vir = r vir /r s , the ratio of the virial radius r vir to the inner char-acteristic radius r s , should decline with increasing cluster massas the more massive clusters collapse later when the cosmologi-cal background density is lower. A precise determination of theinner ( <
200 kpc) density slope of DM halos is of great impor-tance for DM annihilation experiments (Su & Finkbeiner 2012). The best direct probe of the total mass and its (projected)distribution in clusters is via weak gravitational lensing (WL),as it requires no assumption for the dynamical state of the clus-ter or the nature of DM. WL gives rise to the coherent distor-tion of galaxy shapes, measured statistically over thousands ofbackground galaxies. Since lensing is only sensitive to the pro-jected matter density, the triaxiality of cluster halos leads to anintrinsic scatter of 15–20% for the mass of each cluster whencompared to other methods (Oguri et al. 2005; Corless & King 2007; Meneghetti et al. 2010; Becker & Kravtsov 2011). Withcurrent samples of hundreds of clusters, cluster mass profilescan be stacked to enhance the signal-to-noise ratio (S/N) andreduce the scatter due to triaxiality (Umetsu et al. 2014; Okabe& Smith 2016; Simet et al. 2017b). This way, cluster mass cali-bration has reached ∼ – precision at intermediate redshiftsand masses (von der Linden et al. 2014; Hoekstra et al. 2015;Penna-Lima et al. 2017; Smith et al. 2016; Melchior et al. 2017;Simet et al. 2017b).Several systematic effects inherent to WL can bias the clus-ter mass calibration. Instrumental and observational distor-tions can cause systematic signals that are similar in size tothe gravitational distortion of galaxy shapes (Mandelbaum et al.2005b). Following recent extensive tests, current techniques cannow reach an accuracy of 1–2% in shape measurement, follow-ing proper calibration with image simulations (Heymans et al.2006; Massey et al. 2007; Bridle et al. 2010; Kitching et al.2012; Mandelbaum et al. 2015).However, a major source of systematics comes in correctlyestimating the redshift distribution of source galaxies lying be-hind the clusters, required to convert the lensing signal into aphysical mass. Contamination by unlensed cluster and fore-ground galaxies causes a systematic underestimation of the truelensing mass profile (Broadhurst et al. 2005). In particular, in-clusion of cluster galaxies significantly dilutes the signal closerto the cluster center and causes an underestimation of the con-centration of the density profile. In contrast, the inclusion offoreground galaxies in the background source sample due tophotometric redshift errors produces a dilution of the clusterlensing signal that does not depend on the cluster-centric ra-dius. In this paper, we investigate both types of contaminationof the source sample.Acquiring spectroscopic redshifts (spec-z’s) for each sourceis not feasible, particularly to the depths WL observations nowreach. Photometric redshifts (photo-z’s) are typically used in- ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 stead, but until recently, cluster lensing studies relied on atmost two to three observed bands (e.g., Medezinski et al. 2010;Okabe et al. 2010; Oguri et al. 2012), so that reliable photo-z’scould not be determined. In turn, well-calibrated field photo-z catalogs such as COSMOS (Ilbert et al. 2009) were usedto determine the redshift distribution (Medezinski et al. 2010;Umetsu et al. 2012; Okabe et al. 2010). Such field surveys areoften limited to deep, small areas, and are subject to signifi-cant cosmic variance. Furthermore, this approach does not cor-rect for contamination of the source sample by cluster galaxies.The enhancement of lens-source pair counts relative to random(known as “boost” factor) are used to correct for cluster con-tamination (Sheldon et al. 2004; Hoekstra 2007), but those canbe unreliable if the cluster sample is small (Simet et al. 2017a)or if there is not enough spatial coverage to make use of fieldsadjacent to the cluster (Applegate et al. 2014). Additionally,magnification bias (Umetsu et al. 2014; Chiu et al. 2016; Ziparoet al. 2016), masking by cluster galaxies (Simet & Mandelbaum2015), and galaxy selection effects need to be carefully ac-counted for when determining the boost (Mandelbaum et al.2006).In the coming decade, many wide optical surveys are aimedat constraining cosmology via WL, e.g., the ongoing DarkEnergy Survey (DES; The Dark Energy Survey Collaboration2005), the Kilo Degree Survey (KiDS; de Jong et al. 2013), andthe Hyper Suprime-Cam (HSC; Aihara et al. 2017a, 2017b) sur-vey, and the planned Large Synoptic Survey Telescope (LSST;Ivezic et al. 2008). These will observe in four to six broadbands, so that the photo-z’s are better inferred, to a mean levelof (cid:46) . However, these photo-z’s will still be plagued by alarge fraction of outliers ( ∼ − ) due to inherent color-redshift degeneracies. These degeneracies stem from having afinite number of broad optical bands that do not span a wideenough wavelength range, particularly ultraviolet or infrared(Ben´ıtez et al. 2009; Rafelski et al. 2009). Another complica-tion in the case of template-fitting codes (e.g., Ben´ıtez 2000) isthat the template libraries often may not include the full rangeof galaxy spectral energy distribution (SED) features, e.g., ac-counting for emission lines or dust obscuration. The photo-z un-certainties are folded in by incorporating the full probability dis-tribution function (PDF) of the individual photo-z’s (Applegateet al. 2014). However, the PDFs suffer from large dependencyon the assumed priors, and the representability of the referencespec-z sample used for training. Other approaches rely on morestringent color cuts to reject outlier photo-z’s (Medezinski et al.2010; Okabe et al. 2010), but then suffer from lower statisticalpower, as they result in lower source densities.In this paper, we aim to explore the systematic effects clus-ter and foreground contaminations have on cluster WL studiesby using the exquisite deep data from the Hyper Suprime-CamStrategic Survey Program (HSC-SSP; Miyazaki et al. 2017, Aihara et al. 2017a,b). HSC-SSP, an ongoing survey in five op-tical bands ( grizy ), will reach unprecedented depths ( i ∼ ) fora large area ( when finished). About a thousand clus-ters have already been identified to z (cid:46) . in its currently ob-served
240 deg HSC-Wide fields (Oguri et al. 2017) using thered-sequence-based cluster finding algorithm, CAMIRA (Oguri2014). The stacked CAMIRA cluster lensing signal will pro-vide a 4% (7%) statistical constraint on the mean cluster massat low (high) redshifts. Here we make use of this large statisti-cal cluster sample and test several source selection methods thatoptimize the use of robust photo-z’s and minimize the contami-nation by cluster galaxies in order to reduce the systematic levelbelow that required from statistics.This paper is organized as follows. In Section 2 we presentthe basic WL methodology. In Section 3 we present the HSCsurvey, the dataset and the CAMIRA cluster catalog derivedfrom HSC. In Section 4 we present the source selection methodsconsidered in this paper. In Section 5 we present the resultingmass profiles derived using the different selection methods andtheir biases as inferred from modeling. In Section 6 we presentvalidation tests on the level of contamination in each method,and we summarize and conclude in Section 7. Throughout thispaper we adopt a
WMAP9 (Hinshaw et al. 2013) Λ CDM cos-mology, where Ω M = 0 . , Ω Λ = 0 . , and h = H / km s − Mpc − . Weak lensing distorts source galaxy shapes. The amplitude ofthis distortion is proportional to all matter contained in the lens-ing cluster and along the line of sight to the lens. The tangentialdistortion profile is related to the projected surface-mass densityprofile of the average mass distribution around the cluster, γ T ( R ) = ∆Σ( R )Σ cr = ¯Σ( < R ) − Σ( R )Σ cr , (1)where R is the comoving transverse separation between thesource and the lens, Σ( R ) is the projected surface mass den-sity, ¯Σ( < R ) is the mean density within R , and Σ cr = c πG D A ( z s ) D A ( z l ) D A ( z l , z s )(1 + z l ) , (2)is the critical surface mass density, where G is the gravitationalconstant, c is the speed of light, z l and z s are the lens and sourceredshifts, respectively, and D A ( z l ) , D A ( z s ) , and D A ( z l , z s ) are the angular diameter distances to the lens, the source, andthe lens-to-source, respectively. The extra factor of (1 + z l ) comes from our use of comoving coordinates (Bartelmann &Schneider 2001).We estimate the mean projected mass density excess profile ∆Σ( R ) from Equation 1 by stacking the shear over a populationof source galaxies s over multiple clusters l that lie within agiven cluster-centric radial annulus R , Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 ∆Σ( R ) = 12 R ( R ) (cid:80) l,s w ls e t,ls [ (cid:104) Σ cr − (cid:105) ls ] − (1 + K ( R )) (cid:80) l,s w ls , (3)where the double summation is over all clusters and over allsources associated with each cluster (i.e., lens-source pairs). Inthe above expression, the measured tangential shape distortionof a source galaxy is e t = − e cos 2 φ − e sin 2 φ, (4)where φ is the angle measured in sky coordinates from the rightascension direction to a line connecting the lens and sourcegalaxy, and e , e are the shear components in sky coordinatesobtained from the pipeline (see below). The ◦ -rotated com-ponent, e × , is also similarly computed, and is expected to bezero. The mean critical density (cid:104) Σ − (cid:105) − ls is averaged with thesource photo-z PDF, P ( z ) , for each lens-source pair, such that (cid:104) Σ cr − (cid:105) ls = (cid:82) ∞ z l Σ cr − ( z l , z ) P ( z )d z (cid:82) ∞ P ( z )d z . (5)As long as the mean P ( z ) correctly describes the sample red-shift distribution, the above equation corrects for dilution bycluster or foreground source galaxies. However, with limitedwide optical bands, this is not always achievable, as we showbelow. The weight in Equation 3, w ls , is given by w ls = ( (cid:104) Σ cr − (cid:105) ls ) σ e,s + e ,s , (6)where σ e is the per-component shape measurement uncertainty,and e rms ≈ . is the RMS ellipticity estimate per component.The factor ( (cid:104) Σ cr − (cid:105) ls ) downweights pairs that are close in red-shift to the lens. The factor (1 + K ( R )) in Equation 3 cor-rects for a multiplicative shear bias m as determined from theGREAT3-like simulations (Mandelbaum et al. 2014, 2015) andis described in Mandelbaum et al (in prep.). It is calculated as K ( R ) = (cid:80) l,s m s w ls (cid:80) l,s w ls . (7)The ‘shear responsivity’ factor in Equation 3, R ( R ) = 1 − (cid:80) l,s e , s w ls (cid:80) l,s w ls ≈ . , (8)represents the response of the ellipticity, e , to a small shear(Kaiser 1995; Bernstein & Jarvis 2002). A full description andclarification of this procedure is given in Mandelbaum et al.(2017). Finally, the covariance matrix includes the statisticaluncertainty due to shape noise, C stat ( R ) = 14 R ( R ) (cid:80) l,s w ls ( e ,s + σ e,s ) (cid:10) Σ − (cid:11) − ls [1 + K ( R )] (cid:104)(cid:80) l,s w ls (cid:105) . (9)We only include in the covariance the statistical uncertaintydue to shape noise. While other sources of uncertainty shouldbe considered, e.g. due to uncorrelated large-scale structures(Hoekstra 2003), photo-z’s and the shear multiplicative bias cor-rection, in this paper we are only interested in comparing thesystematic error due to source selection with the uncertainty in-duced by statistics, rather that present a full mass calibration ofthe CAMIRA clusters. The HSC-SSP (Aihara et al. 2017b) is conducting an opticalimaging survey with the new 1.77 deg HSC camera (Miyazakiet al. 2017) installed on the Subaru 8.2m Telescope. The surveyis designed to have three layers: Wide, Deep and UltraDeep.The Wide survey, when completed, will span ∼ . Forthis study, we use the first 140 deg of full-depth full-color(FDFC) data of the S16A internal data release. It is incremen-tal to the first public data release, S15B, presented in Aiharaet al. (2017a). The Wide layer consists of five broad bands, grizy , reaching a typical limiting magnitude of i ∼ , and ex-ceptional mean seeing of FWHM = 0 . (cid:48)(cid:48) in the i band. The HSCdata are reduced with the HSC Pipeline, hscPipe (Bosch et al.2017), which is based on the LSST pipeline (Ivezic et al. 2008;Axelrod et al. 2010; Juri´c et al. 2015). Seven different photo-zcodes have been implemented by the team (Tanaka et al. 2017).The WL catalogs derived from the HSC observations aredescribed in detail in Mandelbaum et al. (2017). In short,galaxy shapes are measured from the coadded i -band imagesusing the re-Gaussianization method (Hirata & Seljak 2003)that was extensively used and tested in the Sloan Digital SkySurvey (SDSS; Mandelbaum et al. 2005a; Reyes et al. 2012;Mandelbaum et al. 2013) and its performance for HSC has beenfurther characterized in Mandelbaum et al. (2017). The descrip-tion of the shape catalog, its properties and the cuts applied arefurther described in Section 4.1. For this cluster WL analysis, we make use of the HSC CAMIRAcluster catalog (Oguri et al. 2017), based on HSC S16A data.The detailed methodology of the CAMIRA algorithm was pre-sented in Oguri (2014). In short, it fits all photometric galaxieswith the stellar population synthesis (SPS) models of Bruzual& Charlot (2003) to compute likelihoods of being red-sequencegalaxies as a function of redshift. From the likelihoods, a three-dimensional richness map is constructed to locate cluster can-didates from peaks. For each cluster candidate, the brightestcluster galaxy (BCG) is identified, and in an iterative processthe richness, cluster photo-z, and the BCG identifications arethen refined.The CAMIRA-HSC catalog contains 1921 clusters at esti-mated redshifts . < z < . and richness N mem > , whererichness is defined as the effective number of members abovestellar mass greater than . M (cid:12) , which roughly correspondsto . L ∗ . To be conservative, we only make use of clusters withrichness N mem > whose centers are within the FDFC area,totaling 921 clusters. The cluster redshifts are based on photo-z’s (from the SPS model fitting described above) of the clustermember galaxies and show overall good performance, with bias ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 Table 1.
Source sample properties for different selectionmethods
Sample n ag N bs (cid:104) z s , spec (cid:105) c (cid:104) z s , photo (cid:105) d WL 24.72 − − − photo-z (‘All’) 21.66 12 534 922 1.35 1.31CC-cut, z l < . z l < . z l ≥ . z l ≥ . b Number density (unweighted) in [arcmin − ]. a Total number of source galaxies within 5
Mpc /h of the CAMIRA clustercenters. c Mean source redshift, estimated from the lensing-weighted re-weighted spec-z’s(see Section 6.1, Eq. 13). d Mean source redshift, estimated from the lensing-weighted stacked P ( z ) (seeSection 6.1, Eq. 14). and scatter in ∆ z/ (1 + z ) being better than 0.005 and 0.01 overmost of the redshift range, respectively. The WL shape catalogs used here contain galaxies that arewithin the footprint of the Wide survey and have well-measuredshapes and photometry. To accomplish this, basic WL cuts havebeen explored and applied to the galaxy catalogs and are de-scribed in full detail in Mandelbaum et al. (2017), and so weonly briefly summarize them here. These include using galax-ies within the FDFC area, excluding regions in which the PSFis poorly measured, and applying bright star masks (Couponet al. 2017). Photometry flags have been applied to remove ob-jects with large deblendedness parameter, saturation, bad pixels,interpolated pixels, cosmic-rays, bad centroids, and those thathave duplicate entries. Shape flags have been applied to keepgalaxies with ellipticities in the range | e | < , shape uncertaintyin the range < σ e < . and resolution factor R ≥ . (asdefined by the pipeline; see Mandelbaum et al. 2017). Finally,galaxies are limited to be brighter than i < . , have S/N > in the i -band, and to have detections in at least two other bands.Since photo-z’s are used in the WL analysis, we further ap-ply photo-z quality cuts on the WL catalog to remove undefinedor inadequate photo-z’s. In this paper we compare results fromtwo photo-z codes: MLZ and
FRANKEN - Z , though there are sev-eral others (Tanaka et al. 2017). In the case of MLZ , we re-quire that the PDF standard deviation to be small, σ ( P ( z )) < ,and have high photo-z “confidence” factor, zConf > . (seeTanaka et al. 2017). In the case of FRANKEN - Z , we require thephoto-z χ fit to be small, < . In Section 5 we compare WLprofiles from these two photo-z methods, whereas elsewhere weadopt MLZ as the fiducial photo-z code.We combine the photometry, shape, photo-z and P ( z ) in-formation, applying the above quality cuts, to comprise a basic r-z [mag] g - i [ m a g ] R e d s h i f t Fig. 1. g − i versus r − z CC diagram for galaxies in the HSC footprint. Blackdots represent the color distribution of galaxies close (
R <
100 kpc /h ) tocluster centers, in order to highlight the red-sequence population. Coloredtracks show the color evolution with redshift of synthetic galaxy templates ofE, Sa and Sd-type galaxies (top right to bottom left). source catalog we refer to as ‘all’. The typical mean unweightedgalaxy number density in the catalog is . − , de-pending on seeing (see also Figure 9 in Mandelbaum et al.2017). We summarize the mean number density in the first col-umn of Table 1 after applying the WL/photo-z cuts (first tworows). We also give the mean source redshift of the ’all’ sam-ple based on the redshift distribution that is estimated belowin Section 6.1. The redshift distribution is calculated using twomethods – reweighted spec-z’s (third column) and stacked P ( z ) information (fourth column). As noted in Section 2, we makeuse of the full photo-z P ( z ) when calculating the WL quantities, ∆Σ( R ) and its weight, in a way that corrects for the dilution byobjects lying in front of the lens, assuming that the P ( z ) givesthe true description of the galaxy redshift distribution. The fur-ther restrictive selection methods explored in subsequent sub-sections will explore the validity of this assumption. The color-color (CC) selection method has been extensivelypresented and explored in Medezinski et al. (2010) and used inthe literature (Medezinski et al. 2011, 2013, 2016; Oguri et al.2012; Umetsu et al. 2012, 2014, 2015; Formicola et al. 2016;Wegner et al. 2017; Monteiro-Oliveira et al. 2017). It is basedon conservatively rejecting galaxies in areas of color space sus-pected as having a large fraction of cluster or foreground galax-ies. The region of cluster members is easily identified in CCspace by having an overdensity of red-sequence galaxies, partic-ularly when plotting the number density in CC space of galax-ies lying close to cluster centers; or alternatively, plotting themean cluster centric distance in CC space (see figures 1 & 2in Medezinski et al. 2010). Using the HSC CAMIRA cluster
Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 red red blue blue Fig. 2.
Color-Color diagrams for galaxies in the HSC footprint. Contours and gray-scale colors show the distribution of galaxies close (
R <
50 kpc /h ) tocluster centers, in order to highlight the red-sequence population. Red and blue points denote the red and blue source populations, as selected by our fiducialCC cuts (denoted on the right panel as red z < . clusters and the right panelshows galaxies around z ≥ . clusters. catalog (Oguri et al. 2017), we further demonstrate it here. ForHSC, we opt to use the g − i vs r − z CC space, which bestspans the optical range given the bands observed in HSC andtherefore maximizes the separation of different populations ofgalaxies in this space.To demonstrate this, similar to Medezinski et al. (2010), weoverlay on Figure 1 synthetic evolutionary tracks derived using
GALEV (Kotulla et al. 2009) for different types of galaxies (E,Sa and Sd, top-right to lower-left colored curves) on the galaxydistribution (within
100 kpc /h of cluster centers; gray points)in g − i and r − z space. This depicts where different populationsare expected to lie. At lower redshifts the tracks lie close to g − i ∼ , r − z ∼ . where an overdensity of galaxies is seen.At these colors, galaxies at z ∼ drop out of the g band (reddercolor of the curves at r − z ∼ that extend redder in g − i from 0–3.5), indicating a region of expected degeneracies between low-z ( ∼ . ) ˚ A -break galaxies and high-z ( ∼ ) star-formingLyman-break galaxies, causing large outlier fractions in photo-z’s based on limited wide optical bands (especially due to thelack of u -band or deep IR to distinguish between the two). Thuswe expect this region to contain a large fraction of foregroundgalaxies with possibly biased photo-z’s.We further select galaxies within
50 kpc /h from clustercenters in two cluster redshift bins, and plot their mean num-ber density in the g − i versus r − z space in Figure 2 (gray-scale and contours; left for z l < . clusters, right for z l ≥ . clusters). An overdensity of red-sequence cluster members isevident at g − i ∼ . , r − z ∼ for z l < . (left), and at g − i ∼ . , r − z ∼ . for z l ≥ . (right). To avoid dilution ofthe lensing signal by cluster members, it is therefore importantto exclude galaxies in this region from our source sample.To further explore the exact limits that best isolate back- ground galaxies from the foreground and cluster regions, wehereby make use of the CAMIRA cluster catalog, and selectbackground galaxies in two regions in CC space, marked by redand blue points in Figure 2. Using this large statistical lens-source sample, we can demonstrate the effect of dilution byvarying the color limits as we approach the suspected contam-inating regions. We define several limits, denoted red- (cid:104) ∆Σ (cid:105) (normalized bythe leftmost 3 color bins) for each color limit, as we extend thelimit further into the contaminating region – left to right panelsare for limits red- z l < . (upper panels) and z l ≥ . (lowerpanels). In general, as we extend the color cut (to the right ofeach plot), we are approaching the contaminating population,and therefore expect the signal to drop due to dilution; however,we also increase the sample size by including more galaxiesand therefore expect the shot noise to decrease, as indicated bythe smaller uncertainties (shaded regions for each curve), scal-ing by the square-root of source number density. We note thatcluster dilution will manifest as a decreasing signal as a func-tion of color limit for the inner radial bins only (brown curves),whereas dilution by foreground galaxies (of biased photo-z’s ofbackground galaxies) will manifest as decreasing signal at allradii, albeit inner radial bins will be noisier (due to lower sta- ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 ∆ Σ ( z l < . ) ( no r m a li ze d ) red ∆ color ∆ Σ ( z l ≥ . ) ( no r m a li ze d ) red ∆ color blue ∆ color blue ∆ color
200 h − kpc2200 h − kpc Fig. 3.
Mean lensing signal, ∆Σ (normalized by the signal in the first color bin), as a function of each color limit of the CC-cut method, measured at two radialbins (colored by distance, indicated in the legends). Upper panels are for lenses at z < . , bottom panels are for lenses at z > . . From left to right, panelsare for the red sample color limit g,r,i,z (see Appendix 1). In all panels, any contaminating population (e.g., red-sequencein red ∆ color). As can be seen in some of thepanels (e.g., red- tistical power). In all panels, the dashed vertical line marks thechosen cut below which sources are selected to have minimalcontamination ( (cid:46) relative signal dilution). We note, how-ever, that the exact location of the cut to within about ± . dexdoes not significantly affect our results.In Figure 3, the panels that explore the mean lensing signalas a function of color limit red- (cid:38) signal dilution in theinner radial bin (brown curve), but (cid:46) in the outer radialbins (blue curve), indicating mostly cluster contamination. Thepanel exploring the color limit blue- z l < . clusters(upper right) shows a ∼ decrease in the inner radial bin,and a noisy and smaller ( (cid:46) ) decreasing trend in the outerradial bin, which may indicate foreground contamination. In thelower panels, for sources behind z l ≥ . lenses, the trends arenot as significant, except perhaps for red- < level.This is probably because the effective source density is lower,leading to noisier trends. It may be that the cluster contamina-tion is less severe at higher lens redshifts simply because fewerfaint members are detected. As is also evident from Figure 2 (right), high-z clusters occupy a redder region in CC space thatis better separated from the background red and blue samples.To be conservative, we set the limits as indicated by the verticaldashed lines. We display the final CC selection of red and blue(red and blue points) background galaxies in Figure 2 behind z l < . and z l ≥ . clusters (left and right, respectively). Wesummarize the unweighted galaxy number density and meansource redshift after applying these sets of cuts in Table 1 (thirdand fifth rows) for a simple comparison. As can be seen, for thelow- z l cuts, this selection removes about 50% of the galaxies,whereas at high- z l , it removes about 67% of the objects.Finally, in Figure 4, we carry out a similar exercise, explor-ing the mean lensing signal as a function of color cut, now sep-arating into two cluster richness regimes, low ( N mem < , top)and high ( N mem ≥ , bottom). The dilution by cluster mem-bers is now stronger for high richness clusters, as expected.Within the noise, it appears that our chosen CC for the low- z l regime still succeed in removing most of the contaminationfor both richness bins. Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 ∆ Σ ( N m e m < ) ( no r m a li ze d ) red ∆ color ∆ Σ ( N m e m ≥ ) ( no r m a li ze d ) red ∆ color blue ∆ color blue ∆ color
200 h − kpc2200 h − kpc Fig. 4.
Same as Figure 3, but divided into low richness ( N mem < , top) and high richness ( N mem ≥ , bottom) clusters, instead of by cluster redshift. Thevertical dashed lines denote the same fiducial cuts adopted for low-z clusters in Figure 3. ∆ z ∆ Σ ( no r m a li ze d )
277 h − kpc494 h − kpc878 h − kpc1562 h − kpc Fig. 5.
Mean lensing signal, ∆Σ (normalized by the signal in the largest ∆ z bin), as function of lens redshift threshold, ∆ z , defined in Equation 10 forthe P-cut method, measured at several radial bins (colored by distance).The vertical dashed line indicates the optimal redshift threshold, ∆ z = 0 . ,that minimizes contamination. P ( z ) cuts A second secure source selection method examined in this pa-per has been presented in Oguri (2014). It relies on photo-zselection, but rather than using photo-z point estimates, this ap-proach utilizes the full P ( z ) information for each galaxy (here-after P-cut). With this method, we define a sample of galaxiesthat satisfy: ∞ (cid:90) z l +∆ z P ( z ) d z > p cut and z p < z max (10)where P ( z ) is the photo-z PDF and z p is the redshift pointestimate. Here we choose to use the Monte-Carlo derivedpoint estimate, photoz mc . In Oguri (2014), the minimum red-shift cut used was z l + 0 . , i.e. a threshold of ∆ z = 0 . above the cluster redshift. The sample was further defined suchthat p cut = 0 . , i.e., 98% of the P ( z ) lies beyond this lensredshift threshold. Finally, the maximum redshift was set to z max = 1 . , since for the CFHTLenS data used in that WL anal-ysis, the photo-z’s above that limit are thought to be less secure(Kilbinger et al. 2013).In this section we further attempt to establish the requiredcuts more robustly, by exploring them in a similar fashion asintroduced in the previous section, using the sources behindCAMIRA clusters in HSC data. First, since HSC data are muchdeeper than the CFHTLenS sample used in Oguri (2014), andextend to the y -band, we set z max = 2 . . We also adopt as be-fore, p cut = 0 . . We vary both of these limits, in the range z max = 1 . – and p cut = 0 . – , and measure the mean lens-ing signal, but find this has little to no effect on the recoveredlensing signal. On the other hand, we find that varying ∆ z hasa significant dilution effect. In Figure 5, we present the meanlensing signal, (cid:104) ∆Σ (cid:105) (normalized), as a function of ∆ z for alllens-source pairs, and measured in different cluster-centric an-nuli (color-coded in the legend). As can be seen, up to ∆ z = 0 . , ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 ∆ z ∆ Σ ( N m e m < ) ( no r m a i ze d ) ∆ z ∆ Σ ( N m e m ≥ ) ( no r m a i ze d )
277 h − kpc494 h − kpc878 h − kpc1562 h − kpc Fig. 6.
Same as Figure 5, further divided into low richness ( N mem < , left) and high richness ( N mem > , right) clusters. For the high richness bin, clusterdilution is present even for the highest threshold, ∆ z = 0 . . there is a significant dilution of the lensing signal. It is mostsignificant in the innermost radial bin (brown curve), suggestingthat it is due to contamination by cluster members. However, wenote that the outermost radial bin (blue curve) also shows somedecrease, indicating that some of the contamination is due toforeground galaxies or faint cluster members. In this selectionscheme, it is not possible to separate cluster from foregroundcontamination as in the CC-cut scheme. This plot overall indi-cates that contamination is present up to a higher threshold thanpreviously adopted ( ∆ z = 0 . ). We therefore set the limit to ∆ z = 0 . in subsequent analyses.Here we also repeat the test of dividing the sample intosources behind low- and high-richness clusters. Figure 6 showsthe mean lensing signal as a function of redshift threshold fortwo richness bins. The selected cut, ∆ z = 0 . still applies forlow-richness clusters, but as can be seen from the right panel,for high-richness clusters the dilution in the innermost bin ispresent even at ∆ z = 0 . , albeit the uncertainty on these curvesis much larger due to the small number of clusters available forthis test. This method therefore seems less successful in remov-ing the cluster dilution for very massive clusters.The unweighted galaxy number density and mean sourceredshift after applying the chosen cuts are summarized inTable 1 (fourth and sixth rows). We set (cid:104) z l (cid:105) = 0 . for a se-lection relative to low-z clusters, and (cid:104) z l (cid:105) = 0 . for a selectionrelative to high-z clusters (since a lens redshift must be assumedfor this procedure), to compare with the typical CC selection for z l < . and z l ≥ . . Similar to the CC selection, at low lensredshift about 35% of galaxies are removed, and at high lens redshifts about 75% are removed. In this section, we present and compare the cluster WL pro-files, as derived using the different source selection methods de-scribed in the previous section. In the top panel of each subplotin Figure 7, we calculate the surface mass over-density, ∆Σ( R ) ,in 10 logarithmically-spaced bins spanning . – /h , forfour different lens redshift bins spanning 0.1–0.9 (top left to bot-tom right). The errors represent the statistical uncertainties dueto shape noise. The middle panel of each subplot shows the ◦ -rotated shear. As expected, this cross-shear is consistent withzero within the errors. Finally, the bottom panel of each subplotshows the effective source density profile, after accounting forthe lensing weight (see Equation 6; so this depicts the densityof only those sources used in the lensing calculation as deter-mined by the photo-z PDF). The different curves show differentselection schemes – using ‘all’ galaxies (i.e., only WL+photo-zcuts applied and incorporating the photo-z PDF; black circles),after applying P-cut (cyan squares), and after applying CC-cuts(blue triangles). For all lens redshift slices, the CC-cuts (blue)provide profiles that are consistently higher than without thecuts (black), especially for the case of low-z lenses (upper leftpanel). When comparing CC-cuts and P-cut profiles, the con-servative cut made for P-cut, ∆ z = 0 . , results in very consistentprofiles within the errors up to z l < . . At the highest lens bin(bottom right), P-cut gives a somewhat lower signal than theother methods, but consistent within the large errors. In conclu- Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 ∆ Σ + [ h M ⊙ p c − ] cl < 0.3 all, S/N=40.7P-cut, S/N=40.2CC-cut, S/N=36.0 −505 R ∆ Σ × n g /10 [ h M p c − ] cl < 0.5 all, S/N=36.3P-cut, S/N=34.2CC-cut, S/N=30.3 −2001.01.52.0020406080100120140160180 ∆ Σ + [ h M ⊙ p c − ] cl < 0.7 all, S/N=25.7P-cut, S/N=23.5CC-cut, S/N=20.0 −200 R ∆ Σ × R [Mpc/h] n g /10 [ h M p c − ] cl < 0.9 all, S/N=16.2P-cut, S/N=12.2CC-cut, S/N=13.8 −2500.2 0.5 1 2 5 R [Mpc/h]
Fig. 7.
Stacked WL profiles around CAMIRA clusters, comparing between different source selection methods, for four different lens redshift bins, as indicated.Top panels show the surface mass density contrast profile, ∆Σ , middle panels show the ◦ -rotated shear (B-mode), expected to agree with zero; bottompanels show the effective source number density. All quantities in this plot were calculated using the MLZ photo- z PDFs. Different lines in each panel showdifferent source selection schemes: using all galaxies behind the lens and simply incorporating P ( z ) (black), using P-cut selected galaxies (for which 98% of (cid:80) P ( z ) lies behind z l + 0 . ; cyan), or CC-cut selected galaxies as depicted in Table 2 (blue). S/N values for each selection are given in the legend of eachpanel. sion, for nearly all lens redshifts, not applying any cut but ratherrelying on the photo-z PDF to correct for dilution will result insignificantly diluted profiles (black).To test how much these biases are due to photo-z codes, wereproduce the same plots using the FRANKEN -z photo-z code(Speagle et al., in prep) in Figure 8. Overall the same trend isobserved, where for all lens redshift bins, CC-cuts and P-cutprofiles agree, and slight differences are seen only for the high-est lens bin (lower right panel). A more comprehensive com-parison between the performance of different photo-z codes,of which seven different variants are run for HSC, will be dis-cussed in More et al. (in prep), and so we defer this discussionthere. We will note that More et al. also find that the differencesin the redshift distributions derived from different photo-z codesare most apparent at higher redshifts, z > . In this case, ourhigh- z l lensing profile (lower right panel of Figure 7) will alsobe most affected by photo-z code differences, since most of thesources in that bin lie beyond z > . .The source density profiles in the bottom of each subplotshow that the P-cut method provides more source galaxies at most lens redshift slices, up to z l ∼ . , where both P-cut andCC-cuts remove the same fraction of galaxies (for FRANKEN - Z , P-cut removes even more). The same is also indicated bythe S/N level in the legend of each subplot and the raw numberdensities in Table 1. In conclusion, the P-cut method appears toperform slightly better for low-z clusters, and slightly worse forhigh-z clusters, than the CC selection. We also note that using‘all’ galaxies results in number density profiles that show some-what of an excess at small scales compared to the conservativeselection methods, especially noticeable at low redshifts (topleft panel). This is, as discussed, due to cluster contamination,although the effect is harder to see in this plot since the pro-files are not normalized. We will address and compare numberdensity profiles more clearly below (see Section 6.2).Since dilution by cluster members is expected to be worsefor high richness clusters, where the fraction of cluster galax-ies is by definition higher, we also explore the performance ofthese methods as a function of cluster richness. In Figure 9we plot the lensing profiles for each method in two richnessbins, ≤ N mem ≤ (left) and ≤ N mem (right). At low ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 ∆ Σ + [ h M ⊙ p c − ] cl < 0.3 all, S/N=41.2P-cut, S/N=38.8CC-cut, S/N=36.0 −808 R ∆ Σ × n g /10 [ h M p c − ] cl < 0.5 all, S/N=36.5P-cut, S/N=31.6CC-cut, S/N=30.7 −2001.01.52.0020406080100120140160 ∆ Σ + [ h M ⊙ p c − ] cl < 0.7 all, S/N=27.0P-cut, S/N=20.4CC-cut, S/N=20.4 −200 R ∆ Σ × R [Mpc/h] n g /10 [ h M p c − ] −50050100150200 cl < 0.9 all, S/N=17.7P-cut, S/N=10.1CC-cut, S/N=14.3 −50−2500.2 0.5 1 2 5 R [Mpc/h]
Fig. 8.
Same as Figure 7, but using the
FRANKEN -z photo- z PDFs. ∆ Σ + [ h M ⊙ p c − ]
20 ≤ N mem < 50 all, S/N=62.2P-cut, S/N=58.7CC-cut, S/N=52.8 −505 R ∆ Σ × R [Mpc/h] n g /10 [ h M p c − ]
50 ≤ N mem < 200 all, S/N=29.5P-cut, S/N=29.3CC-cut, S/N=25.5 −200200.2 0.5 1 2 5
R [Mpc/h]
Fig. 9.
Same as Figure 7, but for different richness bins: left for ≤ N mem < and right for ≤ N mem . The CC and P-cut selections are consistent forlow richness clusters, but for high richness clusters the P-cut method shows a hint of dilution in the innermost bin. Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0
Table 2.
Best-fit NFW model parameters
Method M c M c [ M (cid:12) /h ] [ M (cid:12) /h ]All . ± .
05 1 . ± .
13 0 . ± .
03 1 . ± . P-cut . ± .
05 2 . ± .
16 1 . ± .
03 1 . ± . CC-cuts . ± .
06 2 . ± .
19 1 . ± .
03 1 . ± . richness, as before, the CC- and P-cut methods show consistentprofiles, whereas applying no cuts (‘all’) results in a lower sig-nal profile. For high-richness clusters, on the other hand, theP-cut method appears slightly diluted at the innermost bin rel-ative to the CC-cut profile, although they are consistent withinthe large statistical errors. For this measurement, only 38 high-richness clusters are available. When the full HSC-Wide surveyis complete, this type of test can be done with smaller statisti-cal errors, as we expect to have ∼ N mem > clusters inthe full CAMIRA catalog. We conclude from this test that extracare has to be taken in removing cluster contamination whenstudying very rich (or massive) clusters, in which case the CCselection method is then preferred. The interesting quantity for deriving cosmological clustercounts is the total cluster mass. Furthermore, the shape of theprofile, as quantified by its concentration, provides insight intothe formation history of a cluster (e.g., Umetsu et al. 2014), and Λ CDM simulations give predictions for this mass-concentrationrelation (Bhattacharya et al. 2013; Dutton & Macci`o 2014).To derive the total mass and concentration, and demonstratethe effect of source dilution on these quantities, we fit ourstacked ∆Σ( R ) profiles for each selection method with a spher-ically symmetric central Navarro, Frenk, & White (1996, NFW)model. The free parameters in this model are the mass, M c ,and concentration, c c , both in overdensity of 200 times thecritical density of the universe. We fix the lensing-weightedmean cluster redshift and fit for the mass and concentration us-ing the Markov Chain Monte Carlo (MCMC) algorithm EM - CEE (Foreman-Mackey et al. 2014). We exclude the innermostradial bin, where masking and deblending of BCGs may af-fect our photo-z’s or shape measurements (see discussion inSection 6.2; also Murata et al. 2017). We also exclude thelast two radial bins from the NFW analysis since we are onlyinterested in the 1-halo term (the cluster) in this fit, and fit inthe range . – /h . For the sake of computational effi-ciency, we set flat priors on the mass and concentration in therange ≤ M / M (cid:12) /h ≤ , ≤ c ≤ . We do notinclude miscentering in our model since it will equally affectall our source selection methods. Here we are only interestedin the effect of dilution on our selections. We note, however,that neglecting the effect of miscentering will lead to overall lower concentrations, as compared with Λ CDM predictions,and slightly lower masses. As mentioned in Section 2, the co-variance only includes the statistical uncertainty due to shapenoise, since we are only interested in comparing the systematicerror due to source selection with the uncertainty induced bystatistics. In what follows we have tested the effect covariancedue to uncorrelated large-scale structure, and find it is negligi-ble, since we are stacking over about a thousand clusters in awide enough area.Here we fit the WL profile stacked over all CAMIRA clus-ters in the redshift range . < z l < . without subdividinginto lens redshift or richness slices as in the previous section.The resulting profile (points) and its best-fit NFW profile (solidcurves with shaded error interval) are shown in the left panelof Figure 10 for each of the selection methods (‘all’ in black,P-cut in cyan, CC in blue). The corresponding posterior dis-tributions of the mass and concentration fitted parameters fromthe MCMC chains are shown in the right panel of Figure 10,with contours representing 1,2- σ confidence bounds. The fit-ted values and their statistical uncertainties for each method aresummarized in Table 2. Since quantities derived by other massproxies (e.g., X-ray, SZ) are often quoted in overdensities of ∆ = 500 , we also convert and quote M , c in Table 2.To be complete, we furthermore fit in the same way the stackedlensing signal derived from the CC-cut method for clusters atlow ( z l < . ) and high ( z l < . ) redshifts. We find meanmasses of M = (1 . ± . × M (cid:12) /h for low-z clus-ters, and M = (2 . ± . × M (cid:12) /h for high-z clus-ters. These statistical mass constraints (4% for low-z and 7%for high-z) set the tolerance for the required systematic level.We now estimate how much bias is caused to the M , c derived values by dilution. In order to accountfor the statistical correlation between the ‘all’ and CC-cut sam-ples due to the latter being strictly a ∼
50% subset of the for-mer, we bootstrap each of the source samples 100 times, andfollow the same stacking and fitting procedure. We find thatusing ‘all’ galaxies results in a mass that is underestimated by − M , all /M , CC = (13 ± %. The level of bias on theconcentration parameter is higher, and can cause an underesti-mation by as much as − c , all /c , CC = (24 ± whencomparing between ‘all’ and CC-selected sources. Althoughnot highly significant here, this level of bias, if true, may emergein future surveys such as LSST detecting thousands of clusterswith percent level statistical errors on the mean mass. So far, we have relied on comparing the lensing profiles of thedifferent source selection methods as an indication that theyminimize cluster and foreground contamination. Since the sig-nal is a combination of both the shear and the redshift of the ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 R [Mpc/ h ] ∆ Σ + [ h M fl M p c − ] all, M = (1 . ± . × M fl /h P-cut, M = (1 . ± . × M fl /h CC, M = (1 . ± . × M fl /h . . . . M [ M fl /h ] . . . . c . . . . c Fig. 10.
NFW model fits to the WL profiles obtained from different selection selection schemes. Left: datapoints show the stacked surface mass densityprofiles, ∆Σ( R ) , for all galaxies (black), CC-cut galaxies (blue) and P-cut galaxies (cyan), and solid lines and shaded regions show the equivalent NFWfit. The median total mass from each fit is given in the legend. Right: 1,2- σ confidence levels on the posterior 1- and 2-D distributions of the fitted NFWparameters, M and c , for each selection method (same color scheme as left panel). galaxies, all methods may still suffer from the same redshiftbias that would not be apparent in such a relative comparison.Furthermore, some residual cluster contamination may still bepresent in our CC- or P-cut samples. To more directly test thelevel of contamination, we present in this section independentvalidation tests. We first estimate the level of photo-z bias usingspec-z’s in section 6.1, and in section 6.2 we examine the use ofboost factors to estimate the residual cluster member contami-nation. We now attempt to estimate the reliability of the underlying red-shift distribution from photo-z by comparing it with that derivedfrom spec-z samples compiled in the HSC footprint (for detailsof the spec-z surveys used see Tanaka et al. 2017). However,spectroscopic samples are much shallower than those of photo-metric samples, and their color distribution may be very differ-ent since spectroscopic follow-up typically misses certain areasof color space (Masters et al. 2015). For these reasons, they maynot be representative of the photometric sample and its redshiftbehavior. We can account for these differences by re-weightingthe spec-z’s according to their distribution in color and magni-tude space (Lima et al. 2008).The Lima et al. re-weighting method assigns weights togalaxies in a spectroscopic subsample such that the weighteddistributions of photometric observables, e.g., multiband mag-nitudes, colors and sizes, match those of the corresponding dis-tributions of the photometric sample. The weight is calculated as u i = ρ p i ( k ) ρ s i ( k ) , (11)where ρ s i ( k ) ≡ k/V s i ( k ) is a local density of galaxies in color,magnitude and size space. The density is defined by the spher-ical volume centered on the i -th spec-z galaxy in which the k nearest neighbor galaxies are included. ρ p i is the correspondingdensity defined in the same manner using the photometric sam-ple. We define the photometric sample as all the galaxies usedfor our analysis above, and located within 3 Mpc /h from theCAMIRA cluster centers. We further calculate this for the P-cutand CC-cut constrained samples. We separate the clusters intotwo redshift bins, above and below z l = 0 . , as was done for ourCC-selection analysis. This weight ensures that the distributionof the spec-z sample in magnitude and color space is identicalto that of the target photometric source sample. The effects ofincompleteness or large-scale structure are all absorbed in thisweight. The redshift distribution can then be estimated as, N S ( z j ) = N spec (cid:88) i u i ( z j ) , (12)For each spec-z, we derive the weights of Equation 11. Wefurther apply the lensing weight of each spec-z source, w li , fromEquation 6 to the redshift distribution, N S w ( z j ) = N spec (cid:88) i w li u i ( z j ) . (13)Since the spec-z samples are not necessarily associated withclusters, lens redshifts are randomly drawn from the redshiftdistribution of the clusters. With this prescription, all the spec-z samples can be used to evaluate the contamination rate due Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 N S w ( z ) allP-cutCC-cut z −0.40.0 N S w ( z ) − N P w ( z ) N S w ( z ) allP-cutCC-cut z −0.60.0 N S w ( z ) − N P w ( z ) Fig. 11.
Source redshift distributions, based on re-weighted spec-z’s. The bottom panels show the difference between the re-weighted spec-z distribution andthe stacked P ( z ) distribution (see text). Source samples are selected based on CC-cuts (blue) and P-cuts (cyan) and with no cuts (black). Lensing weightshave been applied to each galaxy. The left panel shows the redshift distribution of sources behind low-z ( z l < . ) clusters, and the right for high-z ( z l ≥ . )clusters. For low-z clusters, the foreground contamination is negligible, − . For high-z clusters, the contamination is much higher, − (see Table3). to foregrounds. In practice, we only a portion of the cross-validation spec-z sample (ID = 5 ; see Tanaka et al. 2017), sincemost of the cross-validation samples (ID = 1 − ) are used totrain and calibrate the photo-z and should therefore not be usedto evaluate the performance.Figure 11 shows the re-weighted spec-z distributions, af-ter applying the lensing weight as described above for low-z(left) and high-z (right) clusters. For comparison, in the bottompanels of Figure 11 we also show the difference between thereweighted spec-z distribution and the lensing-weighted stacked P ( z ) distributions derived from MLZ . The weighted stacked P ( z ) distribution is defined as N P w ( z j ) = N p (cid:88) i w li P i ( z j ) , (14)where we use the lensing weight of Equation 6, and the sumis taken over all N p galaxies in our photometric source sam-ple within 3 h − Mpc from the center of each cluster, applyingcorresponding cuts of either CC- or P-cut.We define the fraction of foreground contamination in asource sample for a specific lens at redshift z l as, f FG ( z l ) = (cid:90) z l dzN S w ( z ) (cid:90) ∞ dzN S w ( z ) . (15)For a lens sample with redshift distribution p ( z l ) , the mean fore-ground fraction is computed as a weighted average of the red-shift distribution, (cid:104) f FG (cid:105) = (cid:82) d z l p ( z l ) w ( z l ) f FG ( z l ) (cid:82) d z l p ( z l ) w ( z l ) , (16)where the lens weight w ( z l ) is the sum of weights w li over allsources with respect to the lens redshift. We estimate errors from bootstrapping over the spec-z sample. We summarize theresults for each selection method in Table 3, separately consid-ering lenses above and below z l = 0 . . We find that although theforeground contributions exist, they are small for the low-z case,typically (cid:46) even without any cuts. For high-z lenses, thecontamination is much higher, reaching without any cutsapplied, but much improved for the P-cut and CC-cut methods,reaching only ∼ .We furthermore estimate the photo-z calibration bias. Thisis defined as the ratio between the measured ∆Σ and the true ∆Σ T estimated from the reweighed spec-z’s (see Mandelbaumet al. 2008; Leauthaud et al. 2017), b z ( z l ) + 1 = ∆Σ∆Σ T = (cid:80) i w li u i (cid:104) Σ − , P (cid:105) − i / Σ cr , T , i (cid:80) i w li u i (17)and b z gives the photo-z calibration bias. Here, Σ cr , P , Σ cr , T arethe photo-z and spec-z estimated critical densities, respectively.When inserting the expression for the weight from Equation 6, w li ∝ (cid:104) Σ − , P (cid:105) , the sum in the numerator is a finite and wellbehaved quantity, proportional to ∝ Σ − , T (cid:104) Σ − , P (cid:105) . Similar tothe mean foreground fraction estimate (Equation 16), we esti-mate the mean photo-z calibration bias by integrating over theredshift distribution of our cluster sample, for the low-z case . < z l < . , and the high-z case . ≤ z l < . . We presentthe results in Table 4. For the low-z case, the bias is small, . for the full sample, . for the P-cut sample and only . for the CC-cut sample. For high-z clusters, the bias issignificantly larger, estimated at for the full sample, whiledropping to for the P-cut sample and for the CC-cutsample. The overall agreement between the foreground frac-tion estimated above and the level of photo-z calibration biasindicates that indeed the foreground contamination dominatesthe ∆Σ bias due to photo-z error (with only small contributions ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 Table 3.
ForegroundContamination Level, f FG [%] Method z < . z ≥ . All . ± . ± P-cut . ± . ± CC-cut . ± . ± . Table 4.
Photo-z Calibration Bias, b z [%] Method z < . z ≥ . All − . ± . − ± P-cut − . ± . − ± CC-cut − . ± . − ± due to the scatter and bias in background galaxy photo-z).Overall, the P-cut and CC-cut selection methods succeed inmitigating the bias introduced by removing foregrounds, but thelevel is still high for high-z clusters. This bias should be cor-rected for when deriving masses of high-z clusters, after fur-ther improving the sampling of spec-z’s in color and magnitudespace.There are a few important caveats regarding the validityof the re-weighting method worth mentioning: for one, thismethod is applicable only as far as the spec-z sample covers andsamples all of the color/magnitude space of the source sample.If some area of that space has no (or too few) spec-z galax-ies, then this method fails. When more complete spectroscopiccampaigns are completed (e.g., Masters et al. 2017 are targetingundersampled regions of color space to study the color-redshiftrelation) these estimates can be made more robustly. Anothercaveat is that we assume the spectroscopic completeness is in-dependent of redshift, and even at fixed color and magnitudethis may not be the case. This assumption is a known limitationof the re-weighting method, and has been so far adopted in otherWL studies (e.g., Hildebrandt et al. 2017), albeit at much shal-lower survey depths. There are currently no studies in the liter-ature that rigorously test the impact of this assumption for sam-ples as deep as HSC. These assumptions may well contributeto a systematic error that exceeds the remaining uncertainty inthe photo-z bias estimate, however, we are unable to reliablyquantify it with the datasets currently at hand. With these careful selection methods (P-cut and CC-cut), wehave attempted to create restrictive source samples with mini-mal cluster contamination, so that no further correction for di-lution is required. Figure 3 and Figure 5 should demonstratethat our selection is, by design, set to minimize the contamina-tion – we selected the cuts where the signal dilution is negli-gible. However, it is useful to validate the residual contamina-tion level independently. Boost factor corrections are typically used to account for any residual cluster contamination, by virtueof the radial correlation of source galaxies compared to a fieldsample which is not expected to be related to the clusters. Toaccomplish this, one can either stack the source sample aroundrandom points as the reference (Sheldon et al. 2004), or alterna-tively, stack “fake” sources around the lens sample (Murata etal., in prep, Melchior et al. 2017).For the former, a larger (thousands deg ) survey area is ideal,in order for the correlation function to correctly yield a large-separation baseline (Melchior et al. 2017). We find that theboost factor derived this way displays some non-typical be-havior, where the number counts gradually declines toward thecenter starting at (cid:46) /h , instead of rising as expected forcluster contamination. To determine if this decline is caused byblending with cluster galaxies, Murata et al. (2017, in prep) ex-amine the blending effect using fake objects around CAMIRAclusters in HSC. They find this effect should only be significantbelow scales of ∼ . /h , and cannot explain the behaviorwe see. We therefore cannot utilize the boost factor derived thisway at this stage.For the latter, one would require a fake source catalog, overthe entire observed area, on which the same photometry, WLand photo-z cuts need to be applied, as well as the same color(in case of CC-cuts) or P ( z ) cuts (in case of P-cut) for eachselection method to be assessed. This is due to the non-trivialeffect the color/photo-z selection may have on the number den-sity of selected galaxies and which varies from field to field.Although Huang et al. (2017) and Murata et al. (2017, in prep)have implemented a fake object pipeline, SynPipe , it currentlydoes not carry the color and photo-z information needed for ourtest here. We conclude an absolute boost factor cannot be de-rived in this way using the currently available products from
SynPipe .Instead, we attempt to estimate the boost factor by decom-posing the redshift distribution of our source sample into a fieldgalaxy component and a cluster member component. A variantof this method was first presented in Gruen et al. (2014), andlater adapted for the DES cluster lensing analysis by Melchioret al. (2017). For each method, in each radial bin, we mea-sure the lensing-weighted mean photo-z p ( z ) of sources aroundcluster positions, p ( z ) = (cid:80) l,s w ls p s ( z ) (cid:80) l,s w ls . (18)We furthermore measure the lensing-weighted mean photo-zP(z) of sources around random positions, p b ( z ) = (cid:80) r,s w rs p s ( z ) (cid:80) r,s w rs . (19)which constitutes the ”field” p ( z ) . We then decompose the ob-served photo-z distribution p ( z ) as a sum of the field distribu-tion p b ( z ) and a cluster member contribution, p m ( z ) , p ( z ) = (1 − f cl ) p b ( z ) + f cl p m ( z ) . (20) Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 p ( z ) all, f cl = 9.46 % p(z)model (1 ¡ f c ) p b ( z ) f c p m ( z ) z p ( z ) CC, f cl = -0.10 % p(z)model (1 ¡ f c ) p b ( z ) f c p m ( z ) R [Mpc/ h ] −0.020.000.020.040.060.080.10 boo s t f c l ( R ) allP-cutCC-cut Fig. 12.
Left: p ( z ) decomposition in the first radial bin, R ≈ .
24 Mpc /h , for sources behind clusters in the lens redshift range . < z cl < . . The blacksolid curves shows the mean p ( z ) of the sources behind clusters and the blue dashed curve shows the mean p ( z ) of sources around random points, i.e. thefield p b ( z ) . The red dashed-dotted line shows the cluster member contribution, modeled as a Gaussian. The sum of the modeled cluster contribution and thefield is given by the magenta dotted line and should overlap with the observed p ( z ) . The upper panel presents the decomposition of ‘all’ sources (i.e., no cutsapplied but the WL cuts), and the bottom panel presents the decomposition of CC-cut sources, depicting no cluster contamination even at the inner radial bin.Right: Boost factor profile, f cl ( R ) , based on the p ( z ) decomposition. Different curves show factors for different selection methods: all (black), P-cut (cyan)and CC-cuts (blue). We model the cluster member distribution as a Gaussian dis-tribution, jointly fitting its mean and width to all radial bins atonce, while we fit the amplitude of the Gaussian, f cl ( R ) , in eachradial bin independently. We show an example for this proce-dure in the upper left panel of Figure 12 using the full sourcesample (‘all’), done for sources behind lenses in the redshiftrange . < z < . , for all richnesses, in the first radial bin, R = 0 .
24 Mpc /h , where contamination is expected to be max-imal. The contribution in this bin is estimated to be as highas ∼ . . We repeat this test for the P-cut and CC-cut sam-ples, but fixing the Gaussian model mean and width values tothose determined from the full sample and only fitting for theamplitude f cl ( R ) , since for those methods the cluster membercontribution is expected to be negligible. This is also evident inthe bottom left panel of Figure 12, showing the photo-z distri-butions in the first radial bin using CC-cut sources. There, theestimated cluster contamination is consistent with zero. The re-sulting boost factor profile, f cl ( R ) , carried over all radial bins,is shown in the right panel of Figure 12, for ‘all’ source (black),P-cut sources (cyan) and CC-cut sources (blue). As can beseen, the CC- and P-cut methods have curves consistent withzero down to scales of . /h , suggesting there is no clus-ter contamination to within the 0.5% uncertainties. The fullsample (‘all’), on the other hand, shows significant cluster con-tamination that is not corrected for by the photo-z information,starting at (cid:46) /h and reaching as high as . ± . percent contamination at the inner radial bin, R ≈ .
24 Mpc /h .This demonstrates the strength of our applied selection meth-ods in significantly removing the cluster contamination fromour source sample, and allowing us to derive the lensing sig- nal on smaller scales.We note, for sources behind N mem > clusters, the mean p ( z ) appears biased by the cluster contamination excess, asabove, but also shows a dearth around . < z < , compared tothe field. We speculate this is caused by the intra-cluster lightcontaminating the photometry of z ∼ blue galaxies, makingthem redder, and scattering them to lower redshifts. This mayexplain why for high-richness clusters the P-cut method appearsdiluted compared to the CC-cut (in Figure 9). It would thenmake this test inapplicable to high-richness clusters. However,our high-richness cluster sample is small, so we cannot explorethis robustly. We leave this analysis to a future paper when thefull HSC dataset is at hand. In this paper, we have investigated robust source selection tech-niques that separate the background galaxies used for clusterWL shape measurement and alleviate dilution of the lensingsignal. We have compared different source selection schemescommonly used: (1) relying on photo-z’s and their full PDFs tocorrect for dilution (all), (2) selecting background galaxies ac-cording to their colors (CC-cut), and (3) applying PDF cuts sothat the cumulative PDF lies beyond the cluster redshift (P-cut).We explored the exact boundaries to set within these methods toderive the least contaminated background galaxy samples. Wehave used 912 CAMIRA clusters detected in HSC-SSP first-year data that span the redshift range . < z < . , allowingus to further explore these cuts for different lens redshifts andrichnesses. ublications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0 We demonstrate that simply relying on the photo-z PDFs re-sults in a cluster surface mass density profile that suffers fromcluster dilution at small radii. On the other hand, both theCC- and P-cut selection methods perform comparatively wellin removing most of the cluster and foreground contamina-tion, and are consistent with each other. Differences are onlyseen for the most rich clusters, N mem > , where the P-cutmethod produces a marginally diluted signal compared to theCC-cuts. However, our cluster sample is currently not largeenough to demonstrate this dilution with high significance. Wehave shown by virtue of modeling the signal with an NFW pro-file that not applying these cuts will result in a ( ± )% under-estimation of the mass, M , and up to ( ± )% underesti-mation of the concentration of the mass density profile, c .We have attempted to verify these selection methods by ap-plying them to reweighted spectroscopic samples, and find themto be consistent and largely free from foreground contaminationto below the level of − for low redshift lenses, z l < . . Athigher redshifts, z l ≥ . , the methods succeed in reducing thelevel of bias from bias to (cid:46) . Although the source se-lection methods examined here greatly reduce the photo-z biasby removing foreground contaminants, the adopted reweightingmethodology may still suffer from systematic uncertainties dueto spectroscopic selection effects that may exceed the 1% level.By stacking the source photo-z distributions for each methodand comparing to the field, we have modeled the cluster contam-ination fraction to be quite significant for low-z clusters whenno cuts are applied, reaching . at R ≈ .
24 Mpc /h , yetbeing consistent with zero at all scales to within the 0.5% un-certainties if applying the P-cut or CC-cut methods.In summary, we conclude that applying either the P-cut orthe CC-cut selection method is crucial for removing cluster andforeground contamination and achieving an undiluted clustermass profile. We note, however, that for very massive or nearbycluster samples, the conservative CC selection yields a more se-cure source sample with minimal contamination and less diluteddensity profile. Acknowledgments
We thank the anonymous referee for insightful comments that im-proved the manuscript. EM acknowledges fruitful discussions with AndyGoulding, Peter Melchior and Nick Battaglia. The Hyper Suprime-Cam(HSC) collaboration includes the astronomical communities of Japanand Taiwan, and Princeton University. The HSC instrumentation andsoftware were developed by the National Astronomical Observatory ofJapan (NAOJ), the Kavli Institute for the Physics and Mathematics ofthe Universe (Kavli IPMU), the University of Tokyo, the High EnergyAccelerator Research Organization (KEK), the Academia Sinica Institutefor Astronomy and Astrophysics in Taiwan (ASIAA), and PrincetonUniversity. Funding was contributed by the FIRST program fromJapanese Cabinet Office, the Ministry of Education, Culture, Sports,Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the TorayScience Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and PrincetonUniversity. This paper makes use of software developed for the LargeSynoptic Survey Telescope. We thank the LSST Project for makingtheir code available as free software at http://dm.lsst.org. The Pan-STARRS1 Surveys (PS1) have been made possible through contribu-tions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating in-stitutes, the Max Planck Institute for Astronomy, Heidelberg and the MaxPlanck Institute for Extraterrestrial Physics, Garching, The Johns HopkinsUniversity, Durham University, the University of Edinburgh, Queen?sUniversity Belfast, the Harvard-Smithsonian Center for Astrophysics,the Las Cumbres Observatory Global Telescope Network Incorporated,the National Central University of Taiwan, the Space Telescope ScienceInstitute, the National Aeronautics and Space Administration under GrantNo. NNX08AR22G issued through the Planetary Science Division of theNASA Science Mission Directorate, the National Science Foundation un-der Grant No. AST-1238877, the University of Maryland, and EotvosLorand University (ELTE) and the Los Alamos National Laboratory.Based [in part] on data collected at the Subaru Telescope and retrievedfrom the HSC data archive system, which is operated by Subaru Telescopeand Astronomy Data Center at National Astronomical Observatory ofJapan. This paper makes use of packages available in Python’s openscientific ecosystem, including NumPy (Walt et al. 2011), SciPy (Joneset al. 2001–), matplotlib (Hunter 2007), IPython (P´erez & Granger2007), AstroPy (Astropy Collaboration et al. 2013), and cluster-lensing(Ford 2016). The work reported on in this paper was substantially per-formed at the TIGRESS high performance computer center at PrincetonUniversity which is jointly supported by the Princeton Institute forComputational Science and Engineering and the Princeton UniversityOffice of Information Technology’s Research Computing department.HM is supported by the Jet Propulsion Laboratory, California Institute ofTechnology, under a contract with the National Aeronautics and SpaceAdministration. This work was supported in part by World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan,and JSPS KAKENHI Grant Number 26800093 and 15H05892. KU ac-knowledges support from the Ministry of Science and Technology ofTaiwan through the grant MOST 103-2112-M-001-030-MY3. RM is sup-ported by the US Department of Energy Early Career Award Program.
Appendix 1 Color-Color Selection Recipe
Here we give a full description of the CC limits used in theCC selection method, along with the exact values chosen. Wemake use of the g − i and r − z colors, using cModel magni-tudes measured by the pipeline (see Bosch et al. 2017 for thedefinition of cModel). In this CC space, we define a line whichbroadly follows the red-sequence of galaxies at z ∼ . , seen asan overdensity in Figure 2, CCseq (cid:107) = 2 . × ( r − z ) − . , (A1)and a perpendicular line as CCseq ⊥ = 1 / . × ( r − z ) − . / . . (A2)We further define a line that follows the red-sequence in r − z as a function of z magnitude, as rz seq (cid:107) = − . × z + 1 . . (A3) Publications of the Astronomical Society of Japan , (2014), Vol. 00, No. 0
The red sample limits are then defined with respect to theabove lines as red∆color g − i ) − CCseq (cid:107) (A4) red∆color g − i ) − CCseq ⊥ ) / (1 + 1 / . ) (A5)The specific cuts determined for each source sample (red orblue) at each lens redshift bin are explored in Section 4.2 andare shown in Figure 3 (dashed vertical lines). For red sourcesassociated with clusters at low redshifts, z l < . , these defini-tions are red∆color < − . < r − z > . z > . (A6)For red sources associated with clusters at high redshifts, z l ≥ . , the cuts are red∆color < − . < . r − z > . z > . (A7)The blue source sample limits are defined as blue∆color r − z ) − rz seq (cid:107) (A8)and the blue ∆color is the same as that defined for the redsample in Equation A5. The cuts used for blue sources associ-ated with clusters at low redshifts, z l < . , are (cid:2) blue∆color < − . | (cid:0) blue∆color < . g − i < (cid:1) (cid:3) & r − z < . z > . (A9)The cuts used for blue sources associated with clusters at highredshifts, z l ≥ . , are (cid:2) blue∆color < − . | (cid:0) blue∆color < . g − i < (cid:1) (cid:3) & r − z < . z > . (A10) References
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