PSZ2LenS. Weak lensing analysis of the Planck clusters in the CFHTLenS and in the RCSLenS
Mauro Sereno, Giovanni Covone, Luca Izzo, Stefano Ettori, Jean Coupon, Maggie Lieu
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 5 September 2017 (MN L A TEX style file v2.2)
PSZ2LenS. Weak lensing analysis of the
Planck clusters in theCFHTLenS and in the RCSLenS
Mauro Sereno , (cid:63) , Giovanni Covone , , Luca Izzo , Stefano Ettori , , Jean Coupon ,Maggie Lieu INAF - Osservatorio Astronomico di Bologna, via Piero Gobetti 93/3, I-40129 Bologna, Italy Dipartimento di Fisica e Astronomia, Università di Bologna, via Piero Gobetti 93/2, I-40129 Bologna, Italy Dipartimento di Fisica, Università di Napoli ‘Federico II’, Compl. Univers. di Monte S. Angelo, via Cinthia, I-80126 Napoli, Italia INFN, Sezione di Napoli, Compl. Univers. di Monte S. Angelo, via Cinthia, I-80126 Napoli, Italia Instituto de Astrofìsica de Andalucìa (IAA-CSIC), Glorieta de la Astronomìa s/n, E-18008 Granada, Spain INFN, Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italia Department of Astronomy, University of Geneva, ch. d’Ecogia 16, CH-1290 Versoix, Switzerland European Space Astronomy Centre (ESA/ESAC), Science Operations Department, E-28691 Villanueva de la Caada, Madrid, Spain
ABSTRACT
The possibly unbiased selection process in surveys of the Sunyaev Zel’dovich effect can unveilnew populations of galaxy clusters. We performed a weak lensing analysis of the PSZ2LenSsample, i.e. the PSZ2 galaxy clusters detected by the
Planck mission in the sky portion coveredby the lensing surveys CFHTLenS and RCSLenS. PSZ2LenS consists of 35 clusters and itis a statistically complete and homogeneous subsample of the PSZ2 catalogue. The
Planck selected clusters appear to be unbiased tracers of the massive end of the cosmological haloes.The mass concentration relation of the sample is in excellent agreement with predictions fromthe Λ cold dark matter model. The stacked lensing signal is detected at σ significance overthe radial range . < R < . Mpc /h , and is well described by the cuspy dark halo modelspredicted by numerical simulations. We confirmed that Planck estimated masses are biasedlow by b SZ = 27 ± (stat) ± (sys) per cent with respect to weak lensing masses. The bias ishigher for the cosmological subsample, b SZ = 40 ± (stat) ± (sys) per cent. Key words: gravitational lensing: weak – galaxies: clusters: general – galaxies: clusters:intracluster medium
The prominent role of clusters of galaxies in cosmology and astro-physics demands for a very accurate knowledge of their propertiesand history. Galaxy clusters are laboratories to study the physics ofbaryons and dark matter in the largest gravitationally nearly viri-alized regions (Voit 2005; Pratt et al. 2009; Arnaud et al. 2010;Giodini et al. 2013). Cosmological parameters can be determinedwith cluster abundances and the observed growth of massive haloes(Mantz et al. 2010; Planck Collaboration et al. 2016c), gas fractions(Ettori et al. 2009), or lensing analyses (Sereno 2002; Jullo et al.2010; Lubini et al. 2014).Ongoing and future large surveys will provide invaluable in-formation on the multi-wavelength sky (Laureijs et al. 2011; Pierreet al. 2016). Large surveys of the Sunyaev Zel’dovich (SZ) sky canfind galaxy clusters up to high redshifts. Successful programs havebeen carried out by the
Planck
Satellite (Planck Collaboration et al.2016a), the South Pole Telescope (Bleem et al. 2015, SPT) and (cid:63)
E-mail: [email protected] (MS) the Atacama Cosmology Telescope (Hasselfield et al. 2013, ACT).SZ surveys should in principle detect clusters regardless of theirdistance. Even though the finite spatial resolution can hamper thedetection of the most distant objects, SZ selected clusters should benearly mass limited. The selection function of SZ selected clusterscan be well determined.Furthermore, SZ quantities are quite stable and not signifi-cantly affected by dynamical state or mergers (Motl et al. 2005;Krause et al. 2012; Battaglia et al. 2012). The relation betweenmass and SZ flux is expected to have small intrinsic scatter (Kayet al. 2012; Battaglia et al. 2012). These properties make the de-termination of cosmological parameters using number counts ofSZ detected clusters very appealing (Planck Collaboration et al.2016a).If confirmed, the mass limited but otherwise egalitarian selec-tion could make the SZ clusters an unbiased sample of the wholemassive haloes in the universe. Rossetti et al. (2016) characterizedthe dynamical state of 132
Planck clusters with high signal to noiseratio using as indicator the projected offset between the peak ofthe X-ray emission and the position of the brightest cluster galaxy c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p Sereno et al. (BCG). They showed that the fraction of dynamically relaxed ob-jects is smaller than in X-ray selected samples and confirmed theearly impression that many
Planck selected objects are dynamicallydisturbed systems. Rossetti et al. (2017) found that the fraction ofcool core clusters is ± per cent and does not show significanttime evolution. They found that SZ selected samples are nearly un-biased towards cool cores, one of the main selection effects affect-ing clusters selected in X-ray surveys.A crucial ingredient to study cluster physics is the mass de-termination. Weak lensing (WL) analyses can provide accurate andprecise estimates. The physics behind gravitational lensing is verywell understood (Bartelmann & Schneider 2001) and mass mea-surements can be provided up to high redshifts (Hoekstra et al.2012; von der Linden et al. 2014a; Umetsu et al. 2014; Sereno2015).The main sources of uncertainty and scatter in WL mass es-timates are due to triaxiality, substructures and projection effects(Oguri et al. 2005; Sereno & Umetsu 2011; Meneghetti et al. 2010;Becker & Kravtsov 2011; Bahé, McCarthy & King 2012; Giocoliet al. 2014). Theoretical predictions based on numerical simula-tions (Rasia et al. 2012; Becker & Kravtsov 2011) and recent mea-surements (Mantz et al. 2015; Sereno & Ettori 2015b) agree on anintrinsic scatter of ∼
15 per cent.More than five hundred clusters with known WL mass aretoday available (Sereno 2015) and this number will explode withfuture large photometric surveys, e.g., Hyper Suprime-Cam Sub-aru Strategic Program (Aihara et al. 2017, HSC-SSP) or Euclid(Laureijs et al. 2011). However, direct mass measurements are usu-ally available only for the most massive clusters. Mass estimates oflesser clusters have to rely on calibrated mass–observable relations(Sereno & Ettori 2017). Due to the low scatter, mass proxies basedon SZ observables are among the most promising.The above considerations motivate the analysis of SZ selectedclusters of galaxies with homogeneous WL data. The relation be-tween WL masses and SZ flux of
Planck selected clusters has beeninvestigated by several groups (Gruen et al. 2014; von der Lin-den et al. 2014b; Sereno, Ettori & Moscardini 2015; Smith et al.2016). The scaling relation between WL mass and integrated spher-ical Compton parameter Y of the 115 Planck selected clusterswith known WL mass was studied in Sereno, Ettori & Moscardini(2015) and Sereno & Ettori (2015a), which retrieved a Y - M in agreement with self-similar predictions, with an intrinsic scatterof ± per cent on the SZ mass proxy.The tension between the lower values of the power spectrumamplitude σ inferred from clusters counts (Planck Collaborationet al. 2016c, σ ∼ . - . and references therein) and higherestimates from measurements of the primary Cosmic MicrowaveBackground (CMB) temperature anisotropies (Planck Collabora-tion et al. 2016b, σ = 0 . ± . ) may be due to the Y - M relation used to estimate cluster masses. Consistency can beachieved if Planck masses, which are based on SZ/X-ray proxies(Planck Collaboration et al. 2014b,a), are biased low by ∼
40 percent (Planck Collaboration et al. 2016c).The level of bias has to be assessed but it is still debated. Gruenet al. (2014) presented the WL analysis of 12 SZ selected clusters,including 5
Planck clusters. The comparison of WL masses andCompton parameters showed significant discrepancies correlatingwith cluster mass or redshift. Comparing the
Planck masses to theWL masses of the WtG clusters (Weighing the Giants, Applegateet al. 2014), von der Linden et al. (2014b) found evidence for a sig-nificant mass bias and a mass dependence of the calibration ratio.The analysis of the CCCP clusters (Canadian Cluster Comparison Project, Hoekstra et al. 2015) confirmed that the bias in the hydro-static masses used by the
Planck team depends on the cluster mass,but with normalization 9 per cent higher than what found in von derLinden et al. (2014b). Smith et al. (2016) found that the mean ratioof the
Planck mass estimate to LoCuSS (Local Cluster SubstructureSurvey) lensing mass is . ± . .An unambiguous interpretation of the bias dependence interms of either redshift or masses can be hampered by the smallsample size. Exploiting a large collection of WL masses, Sereno,Ettori & Moscardini (2015) and Sereno & Ettori (2017) found thebias to be redshift rather than mass dependent.Even though some of the disagreement among competinganalyses can de due to statistical methodologies not properly ac-counting for Eddington/Malmquist biases and evolutionary effects,see discussion in Sereno, Ettori & Moscardini (2015); Sereno &Ettori (2015a, 2017), the mass biases found for different clustersamples do not necessarily have to agree. Different samples coverdifferent redshift and mass ranges, where the bias can differ. Fur-thermore, WL masses are usually available for the most massiveclusters only.In this paper, we perform a WL analysis of a statistically com-plete and homogeneous subsample of the Planck detected clusters,the PSZ2LenS. We analyze all the
Planck candidate clusters inthe fields of two public lensing surveys, the CFHTLenS (CanadaFrance Hawaii Telescope Lensing Survey, Heymans et al. 2012)and the RCSLenS (Red Cluster Sequence Lensing Survey, Hilde-brandt et al. 2016), which shared the same observational instrumen-tation and the same data-analysis tools. PSZ2LenS is homogeneousin terms of selection, observational set up, data reduction, and dataanalysis.The paper is structured as follows. In Section 2 we presentthe main properties of the lensing surveys and the available data. InSection 3, we introduce the second
Planck
Catalogue of SZ Sources(PSZ2, Planck Collaboration et al. 2016a) and the PSZ2LenS sam-ple. In Section 4, we cover the basics of the WL theory. Sec-tion 5 is devoted to the selection of the lensed source galaxies.In Section 6, we detail how we modelled the lenses. The strengthof the WL signal of the PSZ2Lens clusters is discussed in Sec-tion 7. The Bayesian method used to analyze the lensing shearprofiles is illustrated in Section 8. The recovered cluster massesand their consistency with previous results are presented in Sec-tion 9. In Section 10, we measure the mass-concentration relationof the PSZ2LenS clusters. Section 11 is devoted to the analysisof the stacked signal. In Section 12, we estimate the bias of the
Planck masses. A discussion of potential systematics effects andresidual statistical uncertainties is presented in Section 13. Candi-date clusters which were not visually confirmed are discussed inSection 14. Section 15 is devoted to some final considerations. InAppendix A, we discuss the optimal radius to be associated to therecovered shear signal. Appendinx B details the lensing weightedaverage of cluster properties. Appendix C discusses pros and consof some statistical estimators used for the WL mass.
As reference cosmological model, we assumed the concordance flat Λ CDM ( Λ and Cold Dark Matter) universe with density parameter Ω M = 0 . , Hubble constant H = 70 km s − Mpc − , and powerspectrum amplitude σ = 0 . . When H is not specified, h is theHubble constant in units of km s − Mpc − .Throughout the paper, O ∆ denotes a global property of thecluster measured within the radius r ∆ which encloses a mean c (cid:13) , 000–000 SZ2LenS over-density of ∆ times the critical density at the cluster redshift, ρ cr = 3 H ( z ) / (8 πG ) , where H ( z ) is the redshift dependent Hub-ble parameter and G is the gravitational constant. We also define E z ≡ H ( z ) /H .The notation ‘ log ’ is the logarithm to base 10 and ‘ ln ’ is thenatural logarithm. Scatters in natural logarithm are quoted as per-cents.Typical values and dispersions of the parameter distributionsare usually computed as bi-weighted estimators (Beers, Flynn &Gebhardt 1990) of the marginalized posterior distributions. We exploited the public lensing surveys CFHTLenS and RCSLenS.In the following, we introduce the data sets.
The CFHTLS (Canada France Hawaii Telescope Legacy Survey) isa photometric survey performed with MegaCam. The wide surveycovers four independent fields for a total of ∼
154 deg in fiveoptical bands u ∗ , g , r , i , z (Heymans et al. 2012).The survey was specifically designed for weak lensing analy-sis, with the deep i -band data taken in sub-arcsecond seeing condi-tions (Erben et al. 2013). The total unmasked area suitable for lens-ing analysis covers . . The raw number density of lensingsources, including all objects that a shape was measured for, is 17.8galaxies per arcmin (Hildebrandt et al. 2016). The weighted den-sity is 15.1 galaxies per arcmin .The CFHTLenS team provided weak lensing data processedwith THELI (Erben et al. 2013) and shear measurements obtainedwith lensfit (Miller et al. 2013). The photometric redshifts weremeasured with accuracy σ z phot ∼ . z ) and a catastrophicoutlier rate of about 4 per cent (Hildebrandt et al. 2012; Benjaminet al. 2013). The RCSLenS is the largest public multi-band imaging surveyto date which is suitable for weak gravitational lensing measure-ments (Hildebrandt et al. 2016).The parent survey, i.e., the Red-sequence Cluster Survey 2(Gilbank et al. 2011, RCS2) is a sub-arcsecond seeing, multi-bandimaging survey in the griz bands initially designed to optically se-lect galaxy cluster. The RCSLenS project later applied methods andtools already developed by CFHTLenS for lensing studies.The survey covers a total unmasked area of . downto a magnitude limit of r ∼ . (for a point source at σ ). Pho-tometric redshifts based on four bands ( g , r , i , z ) data are avail-able for an unmasked area covering . , where the raw(weighted) number density of lensing sources is 7.2 (4.9) galaxiesper arcmin . The survey area is divided into 14 patches, the largestbeing ×
10 deg and the smallest × .Full details on imaging data, data reduction, masking, multi-colour photometry, photometric redshifts, shape measurements, The public archive is available through the Canadian Astronomy DataCentre at . The data products are publicly available at . tests for systematic errors, and the blinding scheme to allow forobjective measurements can be found in Hildebrandt et al. (2016).The RCSLenS was observed with the same telescope and cam-era as CFHTLS and the project applied the same methods and toolsdeveloped for CFHTLenS. The two surveys share the same obser-vational instrumentation and the same data-analysis tools, whichmake the shear and the photo- z catalogues highly homogeneous,but some differences can be found in the two data sets.CFHTLenS features the additional u band and the co-addeddata are deeper by ∼ mag. The CFHTLenS measured shapesof galaxies in the i band. On the other side, since the i band onlycovers ∼ per cent of the RCS2 area, the r band was used inRCSLenS for shape measurements because of the longest exposuretime and the complete coverage. When available, we exploited ancillary data sets to strengthen themeasurement of photometric redshifts and secure the selection ofbackground galaxies. For some fields partially covering CFHTLS-W1 and CFHTLS-W4, we complemented the CFHTLenS data withdeep near-UV and near-IR observations, supplemented by securespectroscopic redshifts. The full data set of complementary obser-vations was presented and fully detailed in Coupon et al. (2015),who analyzed the relationship between galaxies and their host darkmatter haloes through galaxy clustering, galaxy–galaxy lensing andthe stellar mass function. We refer to Coupon et al. (2015) for fur-ther details.
When available, we used the spectroscopic redshifts collected frompublic surveys by Coupon et al. (2015) instead of the photometricredshift value. Coupon et al. (2015) exploited four main spectro-scopic surveys to collect 62220 unique galaxy spectroscopic red-shifts with the highest confidence flag.The largest spectroscopic sample within the W1 area comesfrom the VIMOS (VIsible MultiObject Spectrograph) Public Ex-tragalactic Survey (VIPERS, Garilli et al. 2014), designed to studygalaxies at . (cid:46) z (cid:46) . . The designed survey covers a total areaof
16 deg in the W1 field and in the W4 field. The first pub-lic data release (PDR1) includes redshifts for 54204 objects (30523in VIPERS-W1). Coupon et al. (2015) only considered the galaxieswith the highest confidence flags between 2.0 and 9.5.The VIMOS-VLT (Very Large telescope) Deep Survey(VVDS, Le Fèvre et al. 2005) and the Ultra-Deep Survey (Le Fèvreet al. 2015) cover a total area of .
75 deg in the VIPERS-W1 field.Coupon et al. (2015) also used the VIMOS-VLT F22 Wide Surveywith 12995 galaxies over down to i < . in the southernpart of the VIPERS-W4 field (Garilli et al. 2008). In total, Couponet al. (2015) collected 5122 galaxies with secure flag 3 or 4.The PRIsm MUlti-object Survey (PRIMUS, Coil et al. 2011)consists of low resolution spectra. Coupon et al. (2015) retained the21365 galaxies with secure flag 3 or 4.The SDSS-BOSS spectroscopic survey based on data releaseDR10 (Ahn et al. 2014) totals 4675 galaxies with Z W ARNING =0within the WIRCam area, see below.
Coupon et al. (2015) conducted a K s -band follow-up of the c (cid:13) , 000–000 Sereno et al.
Table 1.
The PSZ2LenS sample. Column 1: cluster name. Column 2: indexin the PSZ2-Union catalogue. Columns 3 and 4: right ascension and dec-lination in degrees (J2000) of the associated BCG. Column 5: redshift. Astar indicates a photometric redshift. Column 7: lensing survey. Column 8:survey patch. The suffix NIR means that ancillary data were available.
PSZ2 index RA DEC z survey fieldG006.49+50.56 21 227.733767 5.744914 0.078 RCSLenS 1514G011.36+49.42 38 230.466125 7.708881 0.044 RCSLenS 1514G012.81+49.68 43 230.772096 8.609181 0.034 RCSLenS 1514G053.44-36.25 212 323.800386 − − ∗ RCSLenS 2143G059.81-39.09 251 329.035737 1.390939 0.222 RCSLenS 2143G065.32-64.84 268 351.332080 − − ∗ CFHTLenS W3G113.02-64.68 547 8.632500 − − − − − − ∗ RCSLenS 0310G211.31-60.28 955 45.302332 − ∗ RCSLenS 0310G212.25-53.20 956 52.774492 − − − − − VIPERS fields with the WIRCam instrument at CFHT. Noise cor-relation introduced by image resampling was corrected exploitingdata from the deeper UKIDSS Ultra Deep Survey (
K < . Lawrence et al. 2007). Sample completeness reaches 80 per centat K s = 22 .Coupon et al. (2015) also used the additional data set from theWIRCam Deep Survey data (Bielby et al. 2012), a deep patch of0.49 deg observed with WIRCam J , H and K s bands.The corresponding effective area in the CFHTLS after re-jection for poor WIRCam photometry and masked CFHTLenSareas covers ∼ . , divided into 15 and ∼ . inthe VIPERS-W1 and VIPERS-W4 fields, respectively. WIRCAMsources were matched to the optical counterparts based on position. UV deep imaging photometry from the GALEX satellite (Martinet al. 2005) is also available for some partial area. Coupon et al.(2015) considered the observations from the Deep Imaging Sur-vey (DIS). All the GALEX pointings were observed with the NUVchannel and cover ∼ . and ∼ . of the WIRCamarea in VIPERS-W1 and VIPERS-W4, respectively. FUV observa-tions are available for 10 pointings in the central part of W1. The second
Planck
Catalogue of Sunyaev-Zel’dovich Sources(PSZ2, Planck Collaboration et al. 2016a) exploits the 29 monthfull-mission data. The catalogue contains 1653 candidate clusters Δ BCG [ ' ] N Figure 1.
Distribution of displacements between SZ centroid and BCG inthe PSZ2LenS sample. Displacements are in units of arcminutes.
PSZ2LenSAll - log < M Sz > P D F Figure 2.
The mass distribution of the
Planck clusters. The histograms arerescaled to unitary area and show the distribution in mass of all the PSZ2clusters with identified counterpart (green) and the PSZ2LenS subsamplein the fields of the CFHTLenS/RCSLenS (blue). The masses are in units of M (cid:12) . PSZ2LenSAll z P D F Figure 3.
The redshift distribution of the
Planck clusters. The histogramsare rescaled to unitary area and show the distribution in redshift of all thePSZ2 clusters with identified counterpart (green) and the PSZ2LenS sub-sample in the fields of the CFHTLenS/RCSLenS (blue).c (cid:13) , 000–000
SZ2LenS AllPSZ2LenS z M S Z , [ M ⊙ ] Figure 4.
Distribution of the PSZ2 clusters with known redshift in the M SZ - z plane. The black and red points denote all the PSZ2 clusters withidentified counterpart and the PSZ2LenS subsample in the fields of theCFHTLenS/RCSLenS, respectively. and it is the largest, all-sky, SZ selected sample of galaxy clustersyet produced .Only candidates with an SNR (signal-to-noise ratio) above 4.5detected outside the highest-emitting Galactic regions, the Smalland Large Magellanic Clouds, and the point source masks were in-cluded. Out of the total, 1203 clusters are confirmed with counter-parts identified in external optical or X-ray samples or by dedicatedfollow-ups. The mean redshift is z ∼ . and the farthest clusterswere found at z < ∼ , which makes PSZ2 the deepest all-sky cata-logue of galaxy clusters.The Planck team calibrated the masses of the detected clus-ters with known redshift assuming a best fitting scaling relation be-tween M and Y , i.e. the spherically integrated Compton pa-rameter within a sphere of radius r (Planck Collaboration et al.2014b). These masses are denoted as M SZ or M Y z . The cataloguespans a nominal mass range from M SZ ∼ . to × M (cid:12) .We performed the WL analysis of the clusters centred in theCFHTLenS and RCSLenS fields. Out of the 47 PSZ2 sourceswithin the survey fields, we confirmed 40 clusters by visually in-specting the optical images and identifying the BCGs. Five ofthese candidate galaxy clusters are located in regions of the RC-SLenS where photometric redshifts are not available. Even thoughthese galaxy clusters were clearly identified in the optical images,we could not measure the WL signal since we need photomet-ric redshifts for the selection of background galaxies, see Sec-tion 5. These clusters are: PSZ2 G054.95-33.39 (PSZ2 index: 221),G055.95-34.89 (225), G081.31-68.56 (349), G082.31-67.00 (354),and G255.51+49.96 (1177).The final catalogue, PSZ2LenS, includes the confirmed 35galaxy clusters (out of a total of 41 candidates) located in regionswhere photometric redshifts are available and is presented in Ta-ble 1. The cluster coordinates and redshifts correspond to the BCG.We did not confirm 6 candidates. Spectroscopic redshifts were re-covered via the SIMBAD Astronomical Database for 30 out ofthe 35 BCGs. Additional updated redshifts for PSZ2 G053.44-36.25 (212) and G114.39-60.16 (554) were found in Carrasco et al. The union catalogue HFI_PCCS_SZ-
UNION _R2.08.
FITS is availablefrom the
Planck
Legacy Archive at http://pla.esac.esa.int/pla/ . http://simbad.u-strasbg.fr/simbad/ . (2017). For the remaining three clusters, we exploited photometricredshifts. The displacements of the SZ centroid from the BCG arepictured in Fig. 1.Fifteen clusters out of 35 in PSZ2LenS are part of the cosmo-logical subsample used by the Planck team for the analysis of thecosmological parameters with number counts.We could confirm ∼ per cent of the candidate clusters,in very good agreement with the nominal statistical reliability as-sessed by the Planck team (Planck Collaboration et al. 2016a), thatplaced a lower limit of 83 per cent on the purity.The results of our identification process are consistent with thethe validation process by the
Planck team (Planck Collaborationet al. 2016a), who performed a multi-wavelength search for coun-terparts in ancillary radio, microwave, infra-red, optical, and X-raydata sets. 33 out of the 41 candidates were validated by the
Planck team. This subset shares 32 clusters with PSZ2LenS. There are onlya few different assessments by the independent selection processes.We did not include PSZ2 G006.84+50.69 (25), which we identifiedas a substructure of PSZ2 G006.49+50.56 (21), i.e. Abell 2029, seeSection 14. On the other hand, we included PSZ2 G058.42-33.50(243), PSZ2 G198.80-57.57 (902), and PSZ2 G211.31-60.28 (955),which were not validated by the
Planck team.Since we took all the
Planck clusters without any further re-striction, the lensing clusters constitute an unbiased subsample ofthe full catalogue. This is a strength of our sample with respect toother WL selected collections, which usually sample only the mas-sive end of the full population, see discussion in Sereno, Ettori &Moscardini (2015).The mass and redshift distribution of PSZ2LenS is represen-tative of the full population of
Planck clusters, see Figs. 2, 3 and 4.According to the Kolmogorov-Smirnov test, there is a 53 per centprobability that the masses of our WL subsample and of the fullsample are drawn from the same distribution. The redshift distribu-tions are compatible at the 96 per cent level.The cluster catalogue and the shape measurements are ex-tracted from completely different data sets, the PSZ2-Survey andCFHTLenS/RCSLenS data respectively. The distribution of lensesis then uncorrelated with residual systematics in the shape measure-ments (Miyatake et al. 2015).
The reduced tangential shear g + is related to the differential pro-jected surface density ∆Σ + of the lenses (Mandelbaum et al. 2013;Velander et al. 2014; Viola et al. 2015). For a single source redshift, ∆Σ + ( R ) = γ + Σ cr = ¯Σ( < R ) − Σ( R ) , (1)where Σ is the projected surface density and Σ cr is the critical den-sity for lensing, Σ cr = c πG D s D d D ds , (2)where c is the speed of light in vacuum, G is the gravitational con-stant, and D d , D s and D ds are the angular diameter distances to thelens, to the source, and from the lens to the source, respectively.The signal behind the clusters can be extracted by stacking incircular annuli as ∆Σ + ( R ) = (cid:80) i ( w i Σ − cr ,i ) (cid:15) + ,i Σ cr ,i (cid:80) i ( w i Σ − cr ,i ) , (3) c (cid:13) , 000–000 Sereno et al. where (cid:15) + ,i is the tangential component of the ellipticity of the i -thsource galaxy after bias correction and w i is the lensfit weightassigned to the source ellipticity. The sum runs over the galaxiesincluded in the annulus at projected distance R .If the redshifts are known with an uncertainty, as it is thecase for photometric redshifts, the point estimator in Eq. 3 is bi-ased. Optimal estimators exploiting the full information containedin the probability density distribution of the photometric redshifthave been advocated (Sheldon et al. 2004), but these methods canbe hampered by the uncertain determination of the shape of theprobability distribution, which is very difficult to ascertain (Tanakaet al. 2017). However, the level of systematics introduced by theestimator in Eq. 3 for quality photometric redshifts as those of theCFHTLens/RCSLenS is under control and well below the statisti-cal uncertainty, see Sec. 13.6. We can safely use it in our analysis.The raw ellipticity components, e m , and e m , , were calibratedand corrected by applying a multiplicative and an additive correc-tion, e true ,i = e m ,i − c i m ( i = 1 , . (4)The bias parameters can be estimated either from simulated imagesor empirically from the data.The multiplicative bias m was identified from the simulatedimages (Heymans et al. 2012; Miller et al. 2013). The simulation-based estimate mostly depends on the shape measurement tech-nique and is common to both CFHTLenS and RCSLenS. In eachsky area, we considered the average ¯ m , which was evaluated takinginto account the weight of the associated shear measurement (Violaet al. 2015), ¯ m ( R ) = (cid:80) i w i Σ − cr ,i m i (cid:80) i w i Σ − cr ,i . (5)The two surveys suffer for a small but significant additive biasat the level of a few times − . This bias depends on the SNR(signal-to-noise ratio) and the size of the galaxy. The empirical es-timate of the additive bias is very sensitive to the actual propertiesof the data (Heymans et al. 2012; Miller et al. 2013) and it differsin the two surveys (Hildebrandt et al. 2016). The residual bias inthe first component is consistent with zero ( c = 0 ) for CFHTLenS(Heymans et al. 2012; Miller et al. 2013), which is not the case forRCSLenS (Hildebrandt et al. 2016). Furthermore, RCSLenS had tomodel the complex behaviour of the additive ellipticity bias witha two-stage process. The first stage is the detector level correc-tion. Once this is corrected for, the residual systematics attributedto noise bias are removed (Hildebrandt et al. 2016). Our source galaxy sample includes all detected galaxies with a non-zero shear weight and a measured photometric redshift (Miller et al.2013). We did not reject those pointings failing the requirements forcosmic shear but still suitable for galaxy lensing (Velander et al.2014; Coupon et al. 2015).Our selection of background galaxies relies on robust photo-metric redshifts. Photometric redshifts exploiting the ancillary datasets were computed in Coupon et al. (2015) with the template fit-ting code
LEPHARE (Ilbert et al. 2006). The spectroscopic sam-ple described in Section 2.3.1 was used for validation and calibra-tion. These photometric redshifts were retrieved within a dispersion ∼ . – . z ) and feature a catastrophic outlier rate of ∼ - per cent. Main improvements with respect to CFHTLenS rely z phot f O DD S > . Figure 5.
Fraction of galaxies in the W1 field of the CFHTLenS cataloguewith
ODDS (cid:62) . as a function of the photometric redshift. Here, redshiftestimates exploit the optical ugriz bands. The vertical red lines delimit theallowed redshift range for CFHTLenS sources. on the choice of isophotal magnitudes and PSF homogenization(Hildebrandt et al. 2012) at faint magnitude, and the contributionof NIR data above z ∼ . The UV photometry improves the preci-sion of photometric redshifts at low redshifts, z (cid:46) . .As a preliminary step, we identified (as candidate backgroundsources for the WL analysis behind the lens at z lens ) galaxies suchthat z s > z lens + ∆ z lens , (6)where z s is the photometric redshift or, if available, the spectro-scopic redshift. For our analysis, we conservatively set ∆ z lens =0 . . On top of this minimal criterion, we required that the sourcespassed more restrictive cuts in either photometric redshift or colourproperties, which we discuss in the following. As a first additional criterion for galaxies with either spectroscopicredshifts or photometric redshift, z s , we adopted the cuts z > z lens + ∆ z lens AND z min < z s < z max , (7)where z is the lower bound of the region including the 2- σ (95.4per cent) of the probability density distribution, i.e. there is a prob-ability of 97.7% that the galaxy redshift is higher than z .The redshifts z min and z max are the lower and upper limits ofthe allowed redshift range, respectively.For the galaxies with spectroscopic redshift, z min = 0 whereas z max is arbitrarily large. For the sample with only photometric red-shifts, the allowed redshift range was determined according to theavailable bands. For the galaxies exploiting only the CFHTLenSphotometry ( ugriz ), we restricted the selection to . < z phot < . ; for the RCSLenS photometric redshifts, which lack for the u band, we restricted the selection to . < z phot < . ; for galax-ies with additional NIR data, we relaxed the upper limit, i.e. we set z max to be arbitrarily large; for galaxies with ancillary UV data, werelaxed the lower limit, i.e. we set z min = 0 .In case of only optical filters without NIR data, we requiredthat the posterior probability distribution of the photometric red-shift is well behaved by selecting galaxies whose fraction of theintegrated probability included in the primary peak exceeds 80 per c (cid:13) , 000–000 SZ2LenS griAll z phot P D F Figure 6.
Photometric redshift distributions of galaxies in the COSMOScatalogue, before (green) and after (blue) the gri colour-colour cut. cent,
ODDS (cid:62) . . (8)The ODDS parameter quantifies the relative importance of the mostlikely redshift (Hildebrandt et al. 2012). The additional selectioncriterion based on the
ODDS parameter guarantees for a clean se-lection but it is somewhat redundant. In fact, most of the galaxieswith
ODDS < . were already cut by retaining only galaxies inthe redshift range z min < z < z max , see Fig. 5. For sources in theCFHTLenS without ancillary information, a fraction of ∼ percent of the sources in the redshift range . < z < . meet the ODDS requirement.By definition, the constraint z > z lens guarantees that thecontamination is at the . per cent level. The additional ∆ z lens requirement in Eq. (7) makes the contamination even lower. Since ∆ z lens = 0 . is ∼ . σ z phot at z s = 0 . . , we are practicallyrequiring that the contamination is ∼ . (0.6) per cent for galaxiesat z phot = 0 . . .When available, the impact of ancillary UV and mainly NIRdata is significant. Thanks to the increased accuracy in the redshiftestimates, we can include in the background sample more numer-ous and more distant galaxies. In particular, when we could relyon improved photometric redshift estimates based on the NIR ad-ditional data set, we did not have to restrict our redshift sample to z phot < . , increasing the full background source sample by ∼ per cent compared to other CFHTLenS lensing studies, without in-troducing any systematic bias (Coupon et al. 2015). The population of source galaxies can be identified with a colour-colour selection (Medezinski et al. 2010; Formicola et al. 2016).For clusters at z lens < . , we adopted the following criterionexploiting the gri bands, which efficiently select galaxies at z s (cid:38) . (Oguri et al. 2012; Covone et al. 2014): ( g − r < . OR ( r − i > . OR ( r − i > g − r ) . (9)To pass this cut, lensing sources have to be detected in the r bandand in at least one of the filters g or i .Since we use photometric redshifts to estimate the lensingdepth, we required z s > z min , (10) griAll z spec P D F Figure 7.
Spectroscopic redshift distributions of VIPERS/VVDS galaxiesin CFHTLS-W1, W4, before (green) and after (blue) the gri colour-colourcut. as for the z phot selection. The two-colours method may select asbackground sources an overdensity of sources at low photometricredshifts (Covone et al. 2014). Most of these sources are character-ized by a low value of the ODDS parameter, and z phot is not wellconstrained, hinting to possible degeneracies in the photometricredshift determination based only on optical colours. Since z phot still enters in the estimate of the lensing depth, we conservativelyexcluded these galaxies through Eq. (10).The colour cuts in equation (9) were originally proposed byOguri et al. (2012) based on the properties of the galaxies in theCOSMOS photometric catalogue (Ilbert et al. 2009), which pro-vides very accurate photometric redshifts down to i ∼ . Theydetermined the cuts after inspection of the photometric redshift dis-tributions in the g - r versus r - i colour space. The criteria are effec-tive, see Fig. 6. When we analyze the distribution of photometricredshifts, 64.4 per cent of the 385044 galaxies in the COSMOSsurvey with measured photometric redshift have z phot > . , i.e.the highest cluster redshift in our sample. After the colour-colourcut, 92.0 per cent of the selected galaxies have z phot > . . If welimit the galaxy sample to z s > . . , as required in Eq. (10),95.4 (98.3) per cent of the selected galaxies have z phot > . . Infact, a very high fraction of the not entitled galaxies which pass thecolour test ( . per cent) forms an overdensity at z phot (cid:46) . .We can further assess the reliability of the colour-space se-lection considering the spectroscopic samples in CFHTLS-W1 andW4 fields. We considered the 61525 galaxies from the VIPERSand VVDS samples with high quality spectroscopic redshifts andgood CFHTLS gri photometry. Before the cut, 61.6 per cent of thesources have z spec > . . After the cut, 97.0 per cent of the 26711selected galaxies have z spec > . , see Fig. 7. If we only considergalaxies with z s > . . , as required in Eq. (10), 97.7 (98.1)per cent of the selected galaxies have z phot > . .Based on the above results, we can roughly estimate that agalaxy passing the gri cuts has a (cid:38) per cent probability of beingat z > . . When combined with the constraint z phot > z lens ,the combined probability of the galaxy of being behind the highestredshift PSZ2LenS cluster goes up to (cid:38) per cent. The lensing signal is generated by all the matter between the ob-server and the source. For a single line of sight, we can break the c (cid:13) , 000–000 Sereno et al. signal down in three main components: the main halo, the corre-lated matter around the halo, and the uncorrelated matter along theline of sight.The profile of the differential projected surface density of thelens can then be modelled as ∆Σ tot = ∆Σ + ∆Σ ± ∆Σ LSS . (11)The dominant contribution up to ∼ Mpc /h , ∆Σ , comes fromthe cluster; the second contribution is the 2-halo term, ∆Σ , whichdescribes the effects of the correlated matter distribution around thelocation of the main halo. The 2-halo term is mainly effective atscales (cid:38) Mpc. ∆Σ LSS is the noise contributed by the uncorre-lated matter.The cluster can be modelled as a Navarro Frenk White (NFW)density profile (Navarro, Frenk & White 1997), ρ NFW = ρ s ( r/r s )(1 + r/r s ) , (12)where r s is the inner scale length and ρ s is the characteristic density.In the following, as reference halo mass, we consider M , i.e.,the mass in a sphere of radius r . The concentration is defined as c = r /r s .The NFW profile may be inaccurate in the very inner orin the outer regions. The action of baryons, the presence ofa dominant BCG, and deviations from the NFW predictions(Mandelbaum, Seljak & Hirata 2008; Dutton & Macciò 2014;Sereno, Fedeli & Moscardini 2016) can play a role. However,for CFHTLenS/RCSLenS quality data, systematics caused by poormodelling are subdominant with respect to the statistical noise. Fur-thermore, in the radial range of our consideration, . < R < Mpc /h , the previous effects are subdominant.To better describe the transition region between the infallingand the collapsed matter at large radii, the NFW density profile canbe smoothly truncated as (Baltz, Marshall & Oguri 2009, BMO), ρ BMO = ρ NFW ( r ) (cid:18) r t r + r t (cid:19) , (13)where r t is the truncation radius. For our analysis, we set r t =3 r (Oguri & Hamana 2011; Covone et al. 2014).The 2-halo term ∆Σ arises from the correlated matter dis-tribution around the location of the galaxy cluster (Covone et al.2014; Sereno et al. 2015b). The 2-halo shear around a single lensof mass M at redshift z for a single source redshift can be modelledas (Oguri & Takada 2011; Oguri & Hamana 2011) γ + , ( θ ; M, z ) = (cid:90) ldl π J ( lθ ) ¯ ρ m ( z ) b h ( M ; z )(1 + z ) Σ cr D d ( z ) P m ( k l ; z ) , (14)where θ is the angular radius, J n is the Bessel function of n -th or-der, and k l ≡ l/ [(1 + z ) D d ( z )] . b h is the bias of the haloes with re-spect to the underlying matter distribution (Sheth & Tormen 1999;Tinker et al. 2010; Bhattacharya et al. 2013). P m ( k l ; z ) is the lin-ear power spectrum. We computed P m following Eisenstein & Hu(1999), which is fully adequate given the precision needed in ouranalysis.The 2-halo term boosts the shear signal at ∼ Mpc /h butits effect is negligible at R (cid:46) Mpc /h even in low mass groups(Covone et al. 2014; Sereno et al. 2015b). In order to favour a lensmodelling as simple as possible but to still account for the corre-lated matter, we expressed the halo bias b h as a known function ofthe peak eight, i.e. in terms of the halo mass and redshift, as pre-scribed in Tinker et al. (2010). SNR N Figure 8.
Distribution of the signal-to-noise ratio of the shear signal ofthe PSZ2LenS clusters. The white and grey histograms show the combinedRCSLenS plus CFHTLenS or the CFHTLenS sample only, respectively.
The final contribution to the shear signal comes from the un-correlated large scale structure projected along the line of sight.We modelled it as a cosmic noise which we added to the uncer-tainty covariance matrix (Hoekstra 2003). The noise, σ LSS , in themeasurement of the average tangential shear in a angular bin rang-ing from θ to θ caused by large scale structure can be expressedas (Schneider et al. 1998; Hoekstra 2003) σ LSS ( θ , θ ) = 2 π (cid:90) ∞ P k ( l ) g ( l, θ , θ ) dl , (15)where P k ( l ) is the effective projected power spectrum of lensingand the function g ( l, θ , θ ) embodies the filter function U as g ( l, θ , θ ) = (cid:90) θ φU ( φ ) J ( lφ ) dφ . (16)The filter of the convergence power spectrum is specified byour choice to consider the azimuthally averaged tangential shear(Hoekstra et al. 2011). The effects of non-linear evolution on therelatively small scales of our interest were accounted for in thepower spectrum following the prescription of Smith et al. (2003).We computed σ LSS at the weighted redshift of the source distribu-tion. The cosmic-noise contributions to the total uncertainty covari-ance matrix can be significant at very large scales or for very deepobservations (Umetsu et al. 2014). In our analysis, the source den-sity is relatively low and errors are dominated by the source galaxyshape noise. For completeness, we nevertheless considered the cos-mic noise in the total uncertainty budget.
Our lensing sample consists of all the PSZ2 confirmed clusters cen-tred in the CFHTLenS and RCSLenS fields with photometric red-shift coverage. This leaves us with 35 clusters, see Table 1.The lensing properties of the background galaxy samples usedfor the weak-lensing shear measurements are listed in Table 2. Theeffective redshift z back of the background population is defined as η ( z back ) = (cid:80) i w i η i (cid:80) w i , (17)where η = D ds D d /D s . The effective source redshift characterizes c (cid:13) , 000–000 SZ2LenS Table 2.
Background galaxy samples for weak-lensing shear measurements.The signal was collected between . and . Mpc /h . Column 1: PSZ2index of the cluster. Column 2: cluster redshift. Column 3: effective sourceredshift. Column 4: total number of background galaxies. Column 5: rawnumber density of background lensing sources per square arc minute, in-cluding all objects with measured shape. Column 6: WL signal-to-noiseratio. Index z lens z back N g n g SNR21 0.078 0.712 13171 1.61 2.7138 0.044 0.752 71520 3.02 − − − CFHTLenSRCSLenS z S NR Figure 9.
Signal-to-noise ratio of the shear signal versus the cluster redshiftof the PSZ2LenS clusters. The red and black points show the RCSLenS andthe CFHTLenS sample, respectively. the background population. We did not use it in the fitting proce-dure, where we analyzed the differential surface density derivedby considering the individual redshifts of the selected backgroundgalaxies, see Eq. (3).We define the total signal of the detection as the weighted dif-ferential density between 0.1 and 3.16 Mpc /h , (cid:104) ∆Σ+ (cid:105) . L ∝ exp( − χ ) , where the χ function can be written as, χ = (cid:88) i (cid:20) ∆Σ + ( R i ) − ∆Σ + ( R i ; M , c ) δ + ( R i ) (cid:21) ; (20)the sum extends over the radial annuli and the effective radius R i of the i -th bin is estimated as a shear-weighted radius, see Ap-pendix A; ∆Σ + ( R i ) is the differential surface density in the an-nulus and δ + ( R i ) is the corresponding uncertainty also accountingfor cosmic noise.The differential surface density ∆Σ + was measured between0.1 and ∼ . Mpc /h from the cluster centre in 15 radial circu-lar annuli equally distanced in logarithmic space. The binning issuch that there are 10 bins per decade, i.e. 10 bins between 0.1 and Mpc /h . The use of the shear-weighted radius makes the fittingprocedure stabler with respect to radial binning, see Appendix A.The tangential and cross component of the shear were com-puted from the weighted ellipticity of the background sources asdescribed in Section 4.In our reference fitting scheme, we modelled the lens with aBMO profile; alternatively we adopted the simpler NFW profile. c (cid:13) , 000–000 Sereno et al. The probabilities p prior ( M ) and p prior ( c ) are the priors onmass and concentration, respectively. Mass and concentration ofmassive haloes are expected to be related. N -body simulations andtheoretical models based on the mass accretion history show thatconcentrations are higher for lower mass haloes and are smallerat early times (Bullock et al. 2001; Duffy et al. 2008; Zhao et al.2009; Giocoli, Tormen & Sheth 2012). A flattening of the c - M re-lation is expected to occur at higher masses and redshifts (Klypin,Trujillo-Gomez & Primack 2011; Prada et al. 2012; Ludlow et al.2014; Meneghetti & Rasia 2013; Dutton & Macciò 2014; Diemer& Kravtsov 2015).Selection effects can preferentially include over-concentratedclusters which deviate from the mean relation. This effect is verysignificant in lensing selected samples but can survive to some ex-tent even in X-ray selected samples (Meneghetti et al. 2014; Serenoet al. 2015a). Orientation effects hamper the lensing analysis. Asan example, the concentration measured under the assumption ofspherical symmetry can be strongly over-estimated for triaxial clus-ters aligned with the line of sight.In our reference inference scheme, we then considered bothmass and concentration as uncorrelated a priori. As prior for massand concentration, we considered uniform probability distributionsin the ranges . (cid:54) M / (10 h − M (cid:12) ) (cid:54) and (cid:54) c (cid:54) , respectively, with the distributions being null otherwise.There are some main advantages with this non-informative ap-proach: (i) the flexibility associated to the concentration can ac-commodate to deviations of real clusters from the simple NFWmodelling; (ii) we can deal with selection effects and apparentvery large values of c ; (iii) lensing estimates of mass and con-centration are strongly anti-correlated and a misleading strongprior on the concentration can bias the mass estimate; (iv) themass-concentration relation is cosmology dependent with over-concentrated clusters preferred in universes with high values of σ .Since the value of σ is still debated (Planck Collaboration et al.2016c), it can be convenient to relax the assumption on σ and onthe c - M relation.As an alternative set of priors, we adopted uniform distribu-tions in logarithmically spaced intervals, as suitable for positiveparameters (Sereno & Covone 2013): p prior ( M ) ∝ /M and p prior ( c ) ∝ /c in the allowed ranges and null otherwise.These priors avoid the bias of the concentration towards large val-ues that can plague lensing analysis of good-quality data (Sereno& Covone 2013). On the contrary, in shallow surveys such as theRCSLenS, these priors can bias low the estimates of mass and con-centration.As a third prior for the concentration, we considered a log-normal distribution with median value c = 4 and scatter of . in natural logarithms. As before, we considered hard limits < c < . The median value of the prior is approximatelywhat found for massive clusters in numerical simulations. The scat-ter is nearly two times what found for the mass-concentration rela-tion (Bhattacharya et al. 2013; Meneghetti et al. 2014).We did not leave the halo bias as a free parameter, i.e. the prioron the bias is a Dirac delta function δ . In the reference scheme, the1-halo term is described with a BMO profile and the halo bias iscomputed as a function of the peak height ν , b h = b h [ ν ( M , z )] ,as described in Tinker et al. (2010). When we alternatively modelthe main halo as a NFW profile, we set b h = 0 . Gruen + M (< ) [ M ⊙ ] M L C ( < M p c ) [ M ⊙ ] Figure 10. Comparison between the weak lensing masses within 1 Mpcas measured in this analysis and the masses already available in literaturethrough the LC2-single catalogue, M LC2 . Red points, as detailed in the leg-end, refer to the analysis in Gruen et al. (2014). The red full line indicatesthe perfect agreement. Results of the regression procedure for the reference settings ofpriors are listed in Table 3. Virial over-densities, ∆ vir , are basedon the spherical collapse model and are computed as suggested inBryan & Norman (1998).Some Planck clusters in CFHTLenS and RCSLensS have beenthe subject of other WL studies in the past. We collected previousresults from the Literature Catalogs of weak Lensing Clusters ofgalaxies (LC ), the largest compilations of WL masses up to date (Sereno 2015). LC are standardized catalogues comprising 879(579 unique) entries with measured WL mass retrieved from 81bibliographic sources.We identified counterparts in the LC catalogue by matchingcluster pairs whose redshifts differ for less than ∆ z = 0 . andwhose projected distance in the sky does not exceed . Mpc /h .12 PSZ2LenS clusters have already been studied in previousanalyses by Dahle et al. (2002); Dahle (2006); Gruen et al. (2014);Hamana et al. (2009); Kettula et al. (2015); Cypriano et al. (2004);Merten et al. (2015); Okabe et al. (2010); Umetsu et al. (2014,2016); Pedersen & Dahle (2007); Shan et al. (2012); Applegateet al. (2014); Okabe & Smith (2016), for a total of 25 previousmass estimates. For clusters with multiple analyses, we consideredthe results reported in LC -single.We compared spherical WL masses within 1.0 Mpc, seeFig. 10, and within r , see Fig. 11. The agreement with previ-ous results is good, ln M LC2 /M PSZ2LenS ∼ . ± . for masseswithin 1 Mpc and ∼ . ± . for M . The scatter is significantand it is difficult to look for biases, if any.Four clusters in our sample were investigated in Gruen et al. The catalogues are available at http://pico.oabo.inaf.it/~sereno/CoMaLit/LC2/ . c (cid:13) , 000–000 SZ2LenS Table 3. Weak lensing mass measurements. Over-density masses and radii are reported at ∆ = ∆ vir , computedaccording to Bryan & Norman (1998). Spherical masses within fixed physical radii are reported within 0.5, 1.0 and 1.5 Mpc (columns 10, 11, 12). M ∆ isthe mass within the sphere of radius r ∆ . M n Mpc is the mass within the sphere of radius n Mpc. Quoted values are the bi-weight estimators of the posteriorprobability distributions. Masses and radii are in units of M (cid:12) and Mpc, respectively. Index M r M r M r M vir r vir M M M 21 3.2 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± (2014). The analysis of Gruen et al. (2014) was based on the sameCFHTLS images but it is independent from ours for methods andtools. They used different pipelines for the determination of galaxyshapes and photometric redshifts; they selected background galax-ies based on photometric redshift and they did not exploit colour-colour procedures; they considered a fitting radial range fixed inangular aperture ( < θ < (cid:48) ) rather than a range based ona fixed physical length; they measured the shear signal in annuliequally spaced in linear space, which give more weight to the outerregions, rather than intervals equally spaced in logarithmic space;they modelled the lens either as a single NFW profile with a (scat-tered) mass-concentration relation in line with Duffy et al. (2008)or as a multiple component halo. Notwithstanding the very differ-ent approaches, the agreement between the two analyses is good,see Figs. 10 and 11.The most notable difference is in the mass estimate ofPSZ2 G099.86+58.45 (478), when they found M = 18 . +5 . − . × M (cid:12) . Part of the difference, which is however not statisticallysignificant, can be ascribed to the cluster redshift z lens = 0 . as-sumed in Gruen et al. (2014), which was estimated through the me-dian photometric redshift of 32 visually selected cluster membergalaxies and is higher than ours. Sereno & Ettori (2017) estimated the weak lensing calibratedmasses M WLc,500 of the 926 Planck clusters identified throughthe Matched Multi-Filter method MMF3 with measured redshift .Masses were estimated based on the spherically integrated Comp-ton parameter Y . They used as calibration sample the LC -single catalogue and estimated the cluster mass with a fore-casting procedure which does not suffer from selection effects,Malmquist/Eddington biases and time or mass evolution.Weak lensing calibrated masses are available for 29 clustersin the PSZ2LenS sample. The comparison of masses within r isshowed in Fig. 12. The agreement is good, ln M WLc /M PSZ2LenS ∼− . ± . . c (cid:13)000 Planck clusters identified throughthe Matched Multi-Filter method MMF3 with measured redshift .Masses were estimated based on the spherically integrated Comp-ton parameter Y . They used as calibration sample the LC -single catalogue and estimated the cluster mass with a fore-casting procedure which does not suffer from selection effects,Malmquist/Eddington biases and time or mass evolution.Weak lensing calibrated masses are available for 29 clustersin the PSZ2LenS sample. The comparison of masses within r isshowed in Fig. 12. The agreement is good, ln M WLc /M PSZ2LenS ∼− . ± . . c (cid:13)000 , 000–000 Sereno et al. Gruen + M [ M ⊙ ] M L C , [ M ⊙ ] Figure 11. Comparison between the weak lensing masses M as mea-sured in this analysis and the masses M LC2 , reported in the LC -singlecatalogue. Red points, as detailed in the legend, refer to the analysis inGruen et al. (2014). The red full line indicates the perfect agreement. 10 CONCENTRATIONS Masses and concentrations at the standard radius r are reportedin Table 4. PSZ2LenS haloes are well fitted by cuspy models. Thenumber of independent data usually outweighs the χ value.Due to the low SNR of the observations, concentrations canbe tightly constrained only for a few massive haloes. The esti-mated concentrations can be strongly affected by the assumed pri-ors. Whereas the effect of the priors is negligible in massive clusterswith high quality observations (Umetsu et al. 2014; Sereno et al.2015a), it can be significant when the SNR is lower (Sereno &Covone 2013; Sereno et al. 2015a). The prior which is uniformin logarithmic space rather than in linear space favours lower con-centrations. There is no other way to circumvent this problem thandeeper observations.The value of the observed concentrations decreases with mass,see Fig. 13. As customary in analyses of the c - M , we modelled therelation with a power law, c = 10 α (cid:18) z z ref (cid:19) γ (cid:18) M M pivot (cid:19) β ; (21)the intrinsic scatter σ c | M of the concentration around c ( M ) is taken to be lognormal (Duffy et al. 2008; Bhattacharya et al.2013).We performed a linear regression in decimal logarithmic ( log )variables using the R-package LIRA . LIRA performs a Bayesian The catalogue HFI_PCCS_SZ-MMF3_R2.08_MWL C . DAT of Planck masses is available at http://pico.oabo.inaf.it/~sereno/CoMaLit/forecast/ . The package LIRA (LInear Regression in Astronomy) is publicly avail-able from the Comprehensive R Archive Network at https://cran.r-project.org/web/packages/lira/index.html . For furtherdetails, see Sereno (2016). M WL,500 [ M ⊙ ] M W L c , [ M ⊙ ] Figure 12. Comparison between the weak lensing masses M WL , , asmeasured in this analysis, and the masses M WLc , , based on the Comp-ton parameter Y and calibrated through a weak lensing subsample bySereno & Ettori (2017). The red full line indicates the perfect agreement. PSZ2LenSBhattacharya + + + M [ M ⊙ / h ] c Figure 13. The mass–concentration relation of the PSZ2LenS clusters. Thedashed black lines show the median scaling relation (full black line) plus orminus the intrinsic scatter at the median redshift z = 0 . . The shaded greyregion encloses the . per cent confidence region around the median rela-tion due to uncertainties on the scaling parameters. The blue, green, orangeand red lines plot the mass-concentration relations of Bhattacharya et al.(2013), Dutton & Macciò (2014), Ludlow et al. (2016), and Meneghettiet al. (2014), respectively. The dashed red lines enclose the 1- σ scatter re-gion around Meneghetti et al. (2014). hierarchical analysis which can deal with heteroscedastic and cor-related measurements uncertainties, intrinsic scatter, scattered massproxies and time-evolving mass distributions (Sereno 2016). In par-ticular, the anti-correlation between the lensing measured mass andconcentration makes the c - M relation apparently steeper (Augeret al. 2013; Dutton & Macciò 2014; Du & Fan 2014; Sereno et al.2015a). When we correct for this, the observed relation is signifi-cantly flatter (Sereno et al. 2015a). On the other hand, neglecting c (cid:13) , 000–000 SZ2LenS Table 4. Masses and concentrations. Column 1: cluster PSZ2 index; col-umn 2 and 3: bi-weight estimators of M and concentration c , re-spectively. Column 4: minimum χ . Column 5: number of radial annuliwith background galaxies. Masses are in units of M (cid:12) /h .Index M c χ N bins 21 7.0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± the intrinsic scatter of the weak lensing mass with respect to thetrue mass can bias the estimated slope towards flatter values (Rasiaet al. 2012; Sereno & Ettori 2015b). We accounted for both uncer-tainty correlations and intrinsic scatter.A proper modelling of the mass distribution is critical to ad-dress Malmquist/Eddington biases (Kelly 2007). Within the LIRA scheme, the distribution of the covariate is modelled as a mixtureof time-evolving Gaussian distributions, which can be smoothlytruncated at low values to model skewness. The parameters ofthe distribution are found within the regression procedure. Thisscheme is fully effective in modelling both selection effects atlow masses, where Planck candidates with SNR<4.5 are excluded,and the steepness of the cosmological halo mass function at largemasses. We verified that this approach is appropriate for Planck selected objects in Sereno, Ettori & Moscardini (2015); Sereno &Ettori (2015a). For the analysis of the mass-concentration relationof the PSZ2LenS sample we modelled the mass distribution of theselected objects as a time evolving Gaussian function.We found α = 0 . ± . (for z ref = 0 . ), β = − . ± . , γ = 0 . ± . . The relation between mass and concentration isin agreement with theoretical predictions, see Fig. 13, with a very ΔΣ tot ΔΣ h - R [ Mpc / h ] Δ Σ + [ h M ⊙ / p c ] Figure 14. Stacked differential surface density ∆Σ + of the PSZ2LenSclusters. Black points are our measurements. The vertical error bars showthe square root of the diagonal values of the covariance matrix. The hori-zontal error bars are the weighted standard deviations of the distribution ofradial distances in the annulus. The green curve plots the best fitting contri-bution by the central halo; the black curve is the overall best fitting radialprofile including the 2-halo term. marginal evidence for a slightly steeper relation. There is no evi-dence for a time-evolution of the relation. The statistical uncertain-ties make it difficult to distinguish among competing theoreticalpredictions.The estimated scatter of the WL masses, σ M WL | M = 0 . ± . , is in agreement with the analysis in Sereno & Ettori (2015b)whereas the scatter of the c - M relation, σ c | M = 0 . ± . ,is in line with theoretical predictions (Bhattacharya et al. 2013;Meneghetti et al. 2014, σ c | M ∼ . ).The observed relation between lensing mass and concentrationcan differ from the theoretical relation due to selection effects ofthe sample. Intrinsically over-concentrated clusters or haloes whosemeasured concentration is boosted due to their orientation alongthe line of sight may be overrepresented with respect to the globalpopulation in a sample of clusters selected according to their largeEinstein radii or to the apparent X-ray morphology (Meneghettiet al. 2014; Sereno et al. 2015a).The Bayesian method implemented in LIRA can correct forevolution effects in the sample, e.g. massive cluster preferentiallyincluded at high redshift (Sereno et al. 2015a). However, if the se-lected sample consists of a peculiar population of clusters whichdiffer from the global population, we would measure the specific c - M relation of this peculiar sample.Based on theoretical predictions, SZ selected clusters shouldnot be biased, see Section 1. We confirmed this view. We foundno evidence for selection effects: the slope, the normalization, thetime evolution and the scatter are in line with theoretical predic-tions based on statistically complete samples of massive clusters.However, the statistical uncertainties are large and we cannot readtoo much into it. 11 STACKING The low signal-to-noise ratio hampers the analysis of single clus-ters. Some further considerations can be based on the stacked anal-ysis. We followed the usual approach (Johnston et al. 2007; Man-delbaum, Seljak & Hirata 2008; Oguri et al. 2012; Okabe et al. c (cid:13) , 000–000 Sereno et al. - - - R [ Mpc / h ] R Δ Σ X [ M ⊙ / p c ] Figure 15. The renormalized cross-component of the differential shear pro-file of the stacked sample. Errors bars are as in Fig. 14. = 14 . . As typical redshift of the stacked signal, weweighted the redshifts of the clusters by the lensing factor, seeApp. B. The effective lensing weighted redshift is z stack = 0 . ,which is consistent with the median redshift of the sample.The cross-component of the shear profile, ∆Σ × is consistentwith zero at all radii, see Fig. 15. This confirms that the main sys-tematics are under control.We analyzed the stacked signal as a single lens, see Section 6.Since the cluster centres are well determined and we cut the inner kpc /h , we did not model the fraction of miscentred haloes(Johnston et al. 2007; Sereno et al. 2015b), which we assumed tobe null.The stacked signal is well fitted by the truncated BMO haloplus the 2-halo term, χ = 6 . for 15 bins, see Fig. 14. Thecontribution by the 2-halo is marginal even at large radii, i.e. R ∼ Mpc /h , the radial outer limit of the present analysis.Mass, M = (4 . ± . × M (cid:12) /h , and concentra-tion, c = 2 . ± . , of the stacked signal are in line withtheoretical predictions, see Fig. 16.The total stacked signal is mostly driven by very high SNRclusters at low redshifts. We then stacked the signal of thePSZ2LenS clusters in two redshifts bins below or above z = 0 . .The concentrations of both the low (see Fig. 16, middle panel) and Bhattacharya + + + M [ M ⊙ / h ] c Bhattacharya + + + M [ M ⊙ / h ] c Bhattacharya + + + M [ M ⊙ / h ] c Figure 16. Marginalized probability distribution of mass and concentrationof the stacked clusters. The grey shadowed regions include the 1-, 2-, 3- σ confidence region in two dimensions, here defined as the regions withinwhich the probability density is larger than exp[ − . / , exp[ − . / ,and exp[ − . / of the maximum, respectively. The blue, green, orangeand red lines plot the mass-concentration relations of Bhattacharya et al.(2013), Dutton & Macciò (2014), Ludlow et al. (2016), and Meneghettiet al. (2014), respectively, at the effective redshift. The red contours tracethe predicted concentration from Meneghetti et al. (2014) given the ob-served mass distribution and the predicted scatter of the theoretical mass-concentration relation. If needed, published relations were rescaled to ourreference cosmology. Top panel. All clusters were stacked; the effectiveredshift is z = 0 . . Middle panel. Stacking of the clusters at z lens < . ;the effective redshift is z = 0 . . Bottom panel. Stacking of the clusters at z lens > . ; the effective redshift is z = 0 . .c (cid:13) , 000–000 SZ2LenS AllCosmo M WL,500 [ M ⊙ ] M S Z , [ M ⊙ ] Figure 17. Planck SZ masses M SZ versus WL masses M WL for thePSZ2LenS clusters. Red dots mark the cosmological subsample. Massesare in units of M (cid:12) and are computed within r . The red line shows thebisection M SZ = M WL . high (see Fig. 16, bottom panel) redshift clusters are in line withtheoretical predictions.Recently, the CODEX (COnstrain Dark Energy with X-raygalaxy clusters) team performed a stacked weak lensing analysisof 27 galaxy clusters at . (cid:54) z (cid:54) . (Cibirka et al. 2017). Thecandidate CODEX clusters were selected in X-ray surface bright-ness and confirmed in optical richness. They found a stacked signalof M ∼ . × M (cid:12) /h and c = 3 . at a median redshiftof z = 0 . in agreement with theoretical predictions.The LoCuSS clusters were instead selected in X-ray luminos-ity. The analysis of the mass-concentration relation of the samplewas found in agreement with numerical simulations and the stackedprofile in agreement with the NFW profile (Okabe & Smith 2016).Umetsu et al. (2016) analyzed the stacked lensing signal of16 X-ray regular CLASH clusters up to Mpc /h . The profile waswell fitted by cuspy dark-matter-dominated haloes in gravitationalequilibrium, alike the NFW profile. They measured a mean concen-tration of c ∼ . at M ∼ . × M (cid:12) /h .Unlike previous samples, PSZ2Lens was SZ selected. Still,our results fit the same pattern and confirm Λ CDM predictions.To check for systematics, we compared the stacked lens-ing mass to the composite halo mass profile (cid:104) M (cid:105) lw from thesensitivity-weighted average of fits to individual cluster profiles(Umetsu et al. 2014, 2016), see App. B. From Eq. (B3) with Γ =0 . ± . , we obtain (cid:104) M (cid:105) lw = (4 . ± . × M (cid:12) /h ,in excellent agreement with the stacked mass, M = (4 . ± . × M (cid:12) /h . 12 THE BIAS OF PLANCK MASSES The bias of the Planck masses, i.e. the masses reported in the cata-logues of the Planck collaboration, can be assessed by direct com-parison with WL masses. For a detailed discussion of recent mea- Table 5. Bias of the Planck SZ masses with respect to WL masses. Valuesfor calibration samples other than PSZ2LenS are taken from Sereno & Et-tori (2017). Column 1: sample name. Column 2: number of WL clusters, N cl . Columns 3 and 4: typical redshift and dispersion. Columns 5 and 6:typical WL mass and dispersion in units of M (cid:12) . Column 7: mass bias b SZ = ln( M SZ /M WL ) . Typical values and dispersions are computed asbi-weighted estimators. Sample N Cl z σ z M σ M b SZ PSZ2LenS 32 0.20 0.15 4.8 3.4 − ± − ± -single 135 0.24 0.14 7.8 4.8 − ± − ± − ± − ± − ± surements of the bias, we refer to Sereno, Ettori & Moscardini(2015) and Sereno & Ettori (2017). Most of the previous studieshad to identify counterparts of the PSZ2 clusters in previously se-lected samples of WL clusters. This can make the estimate of thebias strongly dependent on the calibration sample and on selectioneffects (Sereno & Ettori 2015b; Battaglia et al. 2016). In fact, WLcalibration clusters usually sample the very high mass end of thehalo mass function. If the mass comparison is limited to the sub-sample of SZ detected clusters with WL observations, the estimatedbias can be not representative of the full Planck sample.Alternatively, Planck measurements can be viewed as follow-up observations of a pre-defined weak lensing sample, see discus-sion in Battaglia et al. (2016). Non-detections can be accounted forby setting the SZ signal of non-detected clusters to values corre-sponding to a multiple of the average noise in SZ measurements.As in the previous case, the calibration sample may be biased byselection effects with respect to the full PSZ2 sample. Here, the in-clusion of non-detections makes the sample inconsistent with the Planck catalogue, which obviously includes only positive detec-tions.The estimate of the bias through the PSZ2LenS sample doesnot suffer from selection effects. It is a faithful and unbiased sub-sample of the whole population of Planck clusters. We can estimatethe bias by comparing SZ to WL masses, see Fig. 17. To directlycompare with the PSZ2 catalogue, we considered M .We followed Sereno & Ettori (2017) and we estimated the bias b SZ by fitting the relation ln (cid:104) M SZ (cid:105) = b SZ + ln (cid:104) M WL (cid:105) . (22)We limited the analysis to the 32 clusters in PSZ2LenS which hada published M SZ mass in the Planck catalogues. We performed theregression with LIRA . We modelled the mass distribution of the se-lected objects as a Gaussian (Sereno & Ettori 2017). Corrections forEddington/Malmquist biases were applied (Sereno & Ettori 2015b;Battaglia et al. 2016) and observational uncertainties and intrin-sic scatters in WL and SZ masses were accounted for. We found b SZ = − . ± . . The bias for the 15 clusters in the cosmologi-cal subsample is b SZ = − . ± . , which is more prominent butstill in good statistical agreement with the result for the full sample.The intrinsic scatter of the WL masses is ± , whereasthe intrinsic scatter of the SZ masses is ± . Planck masses are We define the bias as b SZ = ln M SZ − ln M WL . This definition slightlydiffers from that used in the Planck papers, where the bias is defined as b SZ = M SZ /M − . For low values of the bias, the difference is negli-gible.c (cid:13) , 000–000 Sereno et al. precise (thanks to the small scatter) but they are not accurate (dueto the large bias).Based on mock analyses, Shirasaki, Nagai & Lau (2016)found that enhanced scatter in relations confronting WL mass andthermal SZ effect originates from the combination of the projectionof correlated structures along the line of sight and the uncertaintyin the cluster radius associated with WL mass estimates. Here, weare considering M SZ from the Planck catalogue , which were com-puted in a X-ray based over-density radius. This makes SZ and WLmass measurements uncorrelated but can increase the relation scat-ter (Sereno, Ettori & Moscardini 2015).We determined the bias analyzing the 32 clusters confirmedby both our inspection and the Planck team. Considering the candi-dates confirmed by Planck alone, we should include an additionalcandidate, PSZ2 G006.84+50.69 (PSZ2 index: 25), which is likelya substructure of the nearby larger PSZ2 G006.49+50.56 (21), seeSection 14. Taking as lens position and redshift the PSZ2 catalogueentries, we can estimate a mass lens M = (0 . ± . × M (cid:12) . The mass is compatible with a null signal (as expectedsince we did not find any suitable candidate counter-part) andwould slightly reduce the size of the bias to b SZ = − . ± . .Alternatively, we can assess the level of bias by comparingthe effective weak lensing mass M WL,stack of the stacked lensingprofiles to the sensitivity-weighted average of the Planck masses (cid:104) M SZ (cid:105) lw , see App. B. By assuming Γ = 0 . ± . , we obtain ln( (cid:104) M SZ (cid:105) lw /M WL,stack ) = − . ± . in good agreement withour reference result.Battaglia et al. (2016) argued that if the sample selection pre-serves the original Planck selection, as the case for PSZ2LenS, thefactor b SZ estimated through the Planck catalog masses can sufferby Eddington bias. By comparison with measurements by ACT,they estimated an Eddington bias correction of order of 15 percent. In our reference result based on the linear regression, Ed-dington bias was accounted for by modelling the distribution ofWL masses. The distribution of selected mass is quite symmetric.Assuming a log-normal distribution for the mass distribution, theEddington bias turns out to be negligible when comparing meanvalues too (Sereno & Ettori 2017).Our result is consistent with previous estimates based on WLcomparisons. von der Linden et al. (2014b) found a large bias of b SZ = − . ± . in the WtG sample (Applegate et al. 2014) .Planck Collaboration et al. (2016c) measured b SZ = − . ± . for the WtG sample, b SZ = − . ± . for the CCCP (Hoekstraet al. 2015) sample and b SZ ∼ from CMB lensing. The mean biaswith respect to the LoCuSS sample is b SZ = − . ± . (Smithet al. 2016).The bias measurements reported in Table 5 for samples otherthan PSZ2LenS are taken from Sereno & Ettori (2017), which ho-mogenized the estimates by adopting the same methodology weadopted here. Due to the different methods, the listed values candiffer from the values quoted in the original analyses. 13 SYSTEMATICS Weak lensing measurements of masses are very challenging. Infact, masses reported by distinct groups may differ by ∼ - - - ( ΔΣ gri - ΔΣ zphot )/ δΔΣ zphot N Figure 18. Distribution of the differences between the differential surfacedensity as measured with a background source population of galaxies se-lected with the photometric redshifts, ∆Σ zphot , or with the g − r − i colour-colour method, ∆Σ gri . The difference is in units of the statistical uncertain-ties on ∆Σ zphot . We plotted the shear signals generated by the PSZ2LenSclusters in the outer annulus, . (cid:54) R (cid:54) . Mpc /h . The presence of systematics may be tested by comparing re-sults obtained with different methodologies and under different as-sumptions. Our results are consistent over a variegated sets of cir-cumstances, see Tables 6 and 7. Systematic errors on the amplitudeof the lensing signal ∆Σ + are approximated as mass uncertaintiesthrough M ∼ ∆Σ / , see App. B. The purity of the selected background galaxies is crucial to a properWL analysis. Cluster members or foreground galaxies not prop-erly identified can dilute the lensing signal. Contamination by fore-ground galaxies is most severe in the inner regions. We tried toovercome this by considering conservative selection criteria basedon either photometric redshifts or colour-colour cuts. Our selectioncriteria suffer by a nominal (cid:38) per cent contamination. The pricefor a conservative selection procedure is the low number of retainedbackground galaxies. We checked for consistency by redoing the analysis and consider-ing the selection procedures separately, see Table 6. The two selec-tion criteria, i.e. either cuts in z phot or in g − r − i colours, arecomplementary. On average, only per cent of the total num-ber of retained galaxies is selected by both methods. The percent-age is slightly higher ( ∼ per cent) for low redshift clusters( z lens < . ).The colour-colour cuts are very effective in selecting back-ground galaxies at z (cid:38) . whereas the z phot method can also sam-ple lower redshifts sources. As a consequence, the effective sourceredshift of the galaxies selected by the g − r − i cuts is larger. Onone side, these galaxies have a large lensing depth due to the geo-metrical distance factor. On the other side, the z phot method selectsnearer and brighter galaxies, whose shape is better determined andwhich have a larger shear weight.The comparison of the estimated differential surface densityas obtained with the two different selection methods is showed in c (cid:13) , 000–000 SZ2LenS Table 6. WL analyses exploiting different methods for the selection of background galaxies. Column 1: cluster PSZ2 index. Column 2: cluster redshift.Column 3: raw number density of background lensing sources per square arc minute, including objects with measured shape. Sources were selected with eitherthe colour cuts or the photometric redshift methods. Column 4: effective source redshift. Column 5: WL signal-to-noise ratio. Column 6: M in units of M (cid:12) /h . Quoted values are the bi-weight estimators of the posterior probability distributions. Columns 7, 8, 9, 10: same as columns 3, 4, 5, and 6 butfor sources selected with the colour-colour cuts only. Columns 11, 12, 13, 14: same as columns 3, 4, 5, and 6 but for sources selected with the photometricredshifts only. gri OR z phot gri z phot Index z n g z back SNR M n g z back SNR M n g z back SNR M 21 0.078 1.61 0.71 2.7 7.0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Fig. 18. Measurements are very well consistent within the statisticaluncertainties.We also re-estimated the masses adopting the reference fittingscheme for each selection method. Results are listed in Table 6. Forclusters with high SNR, the agreement between the masses esti-mates is excellent. For lower SNR lenses, the statistical agreementis still good but the uncertainties affecting the mass estimates arevery large and the comparison is not so significant.The complementarity and the consistent results justify thecombined use of the two selection methods. Cluster members can dilute the lensing signal mostly in central re-gions. Thanks to our conservative background selection, this effectis not significant in our analysis and we preferred not to introducecorrective boosting factors. We checked the dilution effects in two ways. Firstly, the radial distribution of the number density profileis constant to a good degree, with no bump in the inner regions, seeFig. 19.Secondly, the mass measurement do not change significantlyif we excise a larger inner region, see columns 2 and 4 of Table 7.In particular, if we set R min = 0 . Mpc /h , the estimated mass M of the stacked profile changes by ∼ per cent, well be-low the uncertainty of ∼ per cent. The variation is due to thelower statistical power of the data sets (excluding the inner bins,the SNR is 12.1) and the lower capability of breaking the mass-concentration degeneracy rather than being significant of a system-atic uncertainty. In Sec. 5, we showed that the contamination affecting the sampleof selected background galaxies is contained to the (cid:46) per cent c (cid:13) , 000–000 Sereno et al. - R [ Mpc / h ] n s e l e c t ed [ ga l / a r c m i n ] Figure 19. Mean number density profile of the selected background sourcesas a function of the radial distance from the cluster centre. level. Since in absence of intrinsic alignments foreground galaxiesdo not contribute a net shear signal, the contamination depletes theshear signal by the same amount, which causes an under-estimationof the mass by (cid:46) per cent. The effect of priors on mass and concentration is usually negligiblebut it can play a role when the signal-to-noise ratio of the obser-vations is low. Regression results obtained under different assump-tions are summarized in Table 7.Differences among prior schemes are smaller than statisticaluncertainties. The only scheme which gives systematically lowermasses, mostly at the low mass tail, is that exploiting priors whichare uniform in logarithmic units, see column 5 of Table 7. For lowSNR systems, these priors can bias the results towards lower values.This has to be counterbalanced by a careful choice of the lowermass limit , which can make the prior informative again. For thisreason, we preferred uniform priors in linear space.We verified that a more informative prior on the concentra-tion inspired by numerical simulations, see columns 3 and 7 of Ta-ble 7, significantly improves neither the accuracy nor the precision,compare with columns 2 and 6 of Table 7, which makes the lessinformative priors preferable. A careful choice of the estimator is crucial. The choice has to betuned to the quality of the data (Beers, Flynn & Gebhardt 1990).In particular, the maximum likelihood method can be less stablein low SNR systems, see Appendix C. At the low mass end, thebest-fitting value can underestimate the mass with respect to thebi-weight estimator, see column 2 of of Table 7. However, differ-ences are smaller than the statistical uncertainties. For larger massclusters, differences are negligible. Recent N -body simulations have showed that the traditional NFWfunctional form may fail to describe the structural properties of cosmic objects at the percent level required by precision cosmol-ogy (Dutton & Macciò 2014; Klypin et al. 2016; Meneghetti et al.2014).The Einasto radial profiles can provide a more accurate de-scription of the main halo. Sereno, Fedeli & Moscardini (2016)computed the systematic errors expected for weak lensing analy-ses of clusters of galaxies if one wrongly models the lens densityprofile. At the typical mass of the PSZ2LenS clusters, M ∼ . × M (cid:12) /h , the systematic error is below the per cent levelwhereas the viral masses and concentrations of the most massivehalos at M ∼ M (cid:12) /h can be over- and under-estimated by ∼ per cent, respectively.The inclusion of the inner regions, R min = 0 . Mpc /h in col-umn 2 of Table 7, does not significantly improve the statistical ac-curacy of the results with respect to fitting procedure neglectingthem, R min = 0 . Mpc /h in column 4, but it can make the resultsmore accurate thanks to a much better determination of the concen-tration and the breaking of the related degeneracyThe proper modelling of the outer parts of the shear profile canbe crucial in high SNR systems. For analyses that include the outerregions, i.e., R (cid:38) Mpc /h , the effect of correlated matter maybe significant and the use of the NFW profile can be worrisome(Oguri & Hamana 2011). The truncation of the profile can removethe unphysical divergence of the total mass of the NFW halo andpartially removes systematic errors. However, only accounting forthe 2-halo term can accurately describe the transition between thecluster and the correlated matter which occurs beyond the virialradius in the transition region from the infalling to the collapsedmaterial (Diemer & Kravtsov 2014).Thanks to our treatment of the 2-halo term, we could fit theshear profile up to large radii, R = 3 . Mpc /h . Even thoughdifferences are smaller than statistical uncertainties, some featuresemerge. The inclusion of the outer regions improves both accuracy,i.e. the size of the systematic error, and precision, i.e. the size of thestatistical uncertainty.If we do not truncate the main halo and we do not considerthe 2-halo term, fitting up to large radii can underestimate the haloconcentration and bias the mass high (Oguri & Hamana 2011). Wefound that the NFW fitting out to large radii, see column 6 of ofTable 7, can overestimate masses with respect to the more completemodelling based on the truncated BMO density profile plus the 2-halo term, see column 2 of of Table 7. By proper modelling theouter regions, we correct a potential systematic error of ∼ percent.Inclusion of outer regions can significantly improve the pre-cision too. As can be seen from the comparison of the case incolumn 6 of Table 7 where R min = 3 . Mpc /h with the case R min = 2 . Mpc /h in column 8, the statistical uncertainty de-creases by ∼ per cent. This feature is crucial in low SNR sys-tems where most of the signal is collected in the outer regions.In summary, residual systematic bias due to halo modelling isat the per cent level if we properly model the deviations from theNFW profile, mostly at large radii. Locating the centres of dark matter haloes is critical for the unbi-ased analysis of mass profiles (George et al. 2012). Miscentreingleads to underestimate ∆Σ + at small scales and to bias low themeasurement of the concentration (Johnston et al. 2007).We identified the centre of the cluster as the BCG. Brightgalaxies or X-ray emission from hot plasma can be used to trace c (cid:13) , 000–000 SZ2LenS Table 7. Masses determined assuming different halo modellings, priors or radial ranges. The setting is specified in the first five rows before the line break,where we list the density profile of the main halo (either NFW or BMO in row 1), the priors for mass (row 2), concentration (row 3) and halo bias (row 4),and the radial range (row 5). The symbols U , log U , log N and δ denote the uniform prior in linear space, the uniform prior in log-intervals, the lognormaldistribution, and the Dirac delta, respectively. Mass priors are renormalized between . and × M (cid:12) /h , concentration priors between c = 1 and . For the halo bias, the function b h [ ν ( M , z )] follows Tinker et al. (2010). For the reference case (column 2), we also report the best-fitting value inround brackets. Cluster PSZ2 indexes are listed in Column 1. Masses are in units of M (cid:12) /h , lengths in units of Mpc /h . Bi-weight estimators of centrallocation and scale of the posterior distributions are reported.1-halo BMO BMO BMO BMO NFW NFW NFW p prior ( M ) U U U log U U U U p prior ( c ) U log N U log U U log N U p prior ( b h ) δ [ b h ( ν )] δ [ b h ( ν )] δ [ b h ( ν )] δ [ b h ( ν )] δ [0] δ [0] δ [0] R -range [0 . , 3] [0 . , 3] [0 . , 3] [0 . , 3] [0 . , 3] [0 . , 3] [0 . , 21 7.0( 7.8 ) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± the halo centre. George et al. (2012) investigated the consequencesof miscentring on the weak lensing signal from a sample of 129X-ray-selected galaxy groups in the COSMOS field with redshifts (cid:46) z (cid:46) and halo masses in the range - M (cid:12) . By mea-suring the stacked lensing signal around different candidate centres,they found that massive galaxies near the X-ray centroids trace thecentre of mass to (cid:46) kpc, whereas the X-ray position or the cen-troids based on the mean position of member galaxies have largeroffsets primarily due to the statistical uncertainties in their positions(typically ∼ - kpc).In complex clusters, the BCG defining the cluster centre mightbe misidentified or it might not coincide with the matter centroid, but this second effect is generally small and negligible at the weaklensing scale (George et al. 2012; Zitrin et al. 2012).Our choice to identify the BCG as the cluster centre and to cutthe inner R < . Mpc /h region makes the effects of miscentringof second order in our analysis. We checked this by re-extractingthe shear signal of the clusters around the SZ centroid and by re-computing the masses as described in Section 8. Masses are con-sistent and differences are well below the statistical uncertainties,see Fig. 20 and Table 8. By comparing the stacked profiles, wefound that the systematic error in mass due off-centering is negli-gible ( ∼ . per cent).In fact, the typical displacement between the BCG and the SZcentroid is of the order of the arcminute, well below the maximum c (cid:13) , 000–000 Sereno et al. Table 8. Masses and SNR determined assuming different centres, i.e. eitherthe BCG or the SZ centroid. Column 1: cluster PSZ2 index. Columns 2,3 and 4: displacement between the BCG and the SZ centroid in units ofarcminutes, kpc /h , and r , respectively. Columns 5 and 6: mass and SNRassuming that haloes are centred in the BCG. Columns 7 and 8: same ascolumns 5 and 6 but assuming the SZ centroid as centre. Masses are inunits of M (cid:12) /h . ∆ BCG BCG SZIndex [ (cid:48) ] [ kpc /h ] [ r ] M SNR M SNR21 1.42 88 0.06 7.0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± - - - - ( M SZ - cen,200 - M BCG,200 )/ δ M BCG,200 N Figure 20. Distribution of the differences between masses measured aroundthe BCG, M BCG , , or around the SZ centroid, M SZ-cen , . Differencesare in units of the statistical uncertainties on M BCG , . δ z phot = ( + z ) δ z phot = ( + z ) δ z phot = ( + z ) z cl δ Δ Σ + / Δ Σ + Figure 21. Systematic relative difference of lensing weighted differentialdensity due to scatter in the photometric redshifts of the background galax-ies. Δ z phot =- ( + z ) Δ z phot =- ( + z ) z cl δ Δ Σ + / Δ Σ + Figure 22. Systematic relative difference of lensing weighted differentialdensity due to bias in the photometric redshifts of the background galaxies. radius considered in the WL analysis. The displacement is also verysmall with respect to r . Most of the times, the inner cut encom-passes the shift. This makes the estimate of the total SNR within Mpc /h mostly insensitive to the accurate determination of thecentre, see Table 8. Photometric redshift systematics can impact weak lensing analysesby biasing the estimation of the surface critical density Σ cr . For ourestimator of the differential density, we computed the critical den-sity for each source at the peak of the photometric redshift prob-ability density. This is justified since we limited the selection ofbackground galaxies to redshift ranges where the photo- z probabil-ity density distribution is mostly well behaved and single peaked,see Fig. 5.Coupon et al. (2015) tested the impact of including high-redshift sources and the reliability of the point estimator for thecritical density. They verified that photometric redshifts and shapemeasurements in CFHTLenS with additional NIR data are robustenough beyond z s > . . They selected an arbitrary sample oflow-redshift lens galaxies with a spectroscopic redshift and theymeasured the galaxy-galaxy lensing signal using all sources with c (cid:13) , 000–000 SZ2LenS . < z s < . and all sources with z s > . and they found nosignificant difference between the two signals.To quantitatively estimate the systematic error, we performeda simulation. We approximated the true redshift distribution of thefield galaxies as the distribution of the measured photometric red-shifts in a CFHTLenS field. The distribution of photometric red-shifts was then simulated by considering a Gaussian error δz phot ∝ z phot . We simulated the lens as a NFW toy model and extractedas background galaxies the sub-sample with z phot > z lens + ∆ z lens ,with ∆ z lens = 0 . , see Sec. 5. We assigned to each source the trueshear distortion and its real lensing weight from the shear catalog.We finally computed the ∆Σ + estimator for the simulated input orthe scattered redshifts. Results are summarized in Fig. 21 for dif-ferent lens redshift and photo- z uncertainties.For a redshift uncertainty of δz phot (cid:46) . z phot ) , as typicalof the CHFTLenS survey in the range . (cid:46) z phot (cid:46) . or for theRCSLenS in the range . (cid:46) z phot (cid:46) . , the systematic error onthe differential density is below the percent level for lenses up to z cl ∼ . . For the highest redshift clusters in our sample at z cl (cid:38) . , the uncertainty is ∼ per cent.Together with the scatter, a bias in the estimated z phot can af-fect the mass calibration. The bias, defined as the mean ( z phot − z spes ) / (1 + z spec ) including the outliers, in RCSLenS for sourceswith ODDS > . (as by our selection) is of order of ∼ . forredshifts in the range . (cid:46) z phot (cid:46) . , and it is stays well below5 per cent even relaxing the selection criteria (Hildebrandt et al.2016).To quantitatively estimate the related systematic error in themass calibration, we performed a simulation as before but we ap-plied a constant bias rather than scattering the distribution of trueredshifts. Results are summarized in Fig. 22. For a bias of − . ,the systematic error on the shear signal is (cid:46) per cent in an amplelens redshift range.Our treatment did not explicitly consider catastrophic outliersas a secondary population in the source redshift distribution. Out-liers are defined as objects with ∆ z = ( z phot − z spes ) / (1 + z spec ) larger than an arbitrary threshold. In CFHTLenS, less than 4 percent of estimated redshifts are regarded as outliers ( | ∆ z | > . ,Hildebrandt et al. 2012). The fraction of outliers is significantlylower if galaxies are selected by the ODDS parameter (Hildebrandtet al. 2016).However, we accounted for outliers in two ways, which canreproduce their main effects. Firstly, the bias estimates includesoutliers. Secondly, we considered Gaussian distributions with quiteextended tails. For δz phot / (1 + z phot ) = 0 . (0.06), ∼ (1.2) percent of the sources are seen as outliers.The systematic error on the mass, accounting for both scatterand bias, can be derived from the amplitude error of the lensing sig-nal by using δM ∝ ∆Σ . , see App. B. We can then estimatea mass uncertainty of ∼ per cent. Our result is in good agreementwith the analysis in Melchior et al. (2017), who investigated howthe estimate of the mean critical density varies as a function of lensredshift among different photometric redshift algorithms. A small calibration uncertainty in the shape measurements at thelevel of a few per cents can severely limit the accuracy on the mass(von der Linden et al. 2014a; Umetsu et al. 2014).Multiplicative and additive biases in shape measurement forthe CFHTLenS (Heymans et al. 2012; Miller et al. 2013) and theRCSLens (Hildebrandt et al. 2016) were identified on simulated images. The multiplicative bias mostly depends on the shape mea-surement technique rather than on the actual properties of the dataand can be well assessed with a simulation-based estimate. The av-erage calibration correction to the RCSLenS ellipticities is of orderof ∼ per cent (Hildebrandt et al. 2016).Liu, Ortiz-Vazquez & Hill (2016) proposed a data driven ap-proach to calibrate the multiplicative bias m by cross-correlatingCFHTLenS galaxy density maps with CFHTLenS shear maps and Planck CMB lensing maps. The additional correction for faintergalaxies may be relevant for cosmic shear analysis, but we couldneglect it for our analysis.Whereas simulation-testing shows that the multiplicative biasis well controlled, detailed comparison of separate shape cataloguesof actual data can find that the residual systematic is larger. Jarviset al. (2016) performed a detailed comparison of two independentshape catalogues from the Dark Energy Survey Science Verificationdata and found a systematic uncertainty of δm ∼ . . We canconservatively assume that this is the shear systematics affectingour analysis too, which entails a related mass uncertainty of ∼ . per cent. Two neighbouring clusters that fall along the line of sight may beblended by the SZ cluster finder into a single, apparently largercluster. Whereas the Compton parameters add approximately lin-early, projection effects can severely impact the weak lensing mass.The lensing amplitude ∆Σ + is a differential measurement and theestimated mass of the blended cluster can be well below the sum ofthe masses of the aligned halos. Then, the blended object deviatesfrom the mean scaling relation between SZ signal and mass.To estimate this effect, we follow Simet et al. (2017). The sys-tematic uncertainty due to projection effects can be approximatedas δM/M ∼ p ( (cid:15) − . p ( (cid:15) − . , (23)where p is the fraction of aligned clusters and (cid:15) is an effective pa-rameter which characterizes the effective mass contribution of theprojected halo. The parameter (cid:15) depends on the relative position ofthe two blended haloes along the line of sight, and on their shape,elongation and concentration. If (cid:15) = 0 . , we correctly estimate thetotal mass; if (cid:15) = 0 , the second halo is hidden and contributes nomass. Planck objects are rare and the chance to have two or more ofthem aligned is small, < ∼ per cent considering their tendency to becorrelated. The systematic error on mass due to projection effectsis then negligible ( (cid:46) per cent). Residual systematic and statistical uncertainties on the mass cali-bration not corrected for in our analysis are listed in Table 9. Weassumed that systematics related to priors, mass estimators, and ra-dial range were properly accounted for and eliminated in our anal-ysis. The main contributors to the systematic error budget are thecalibration uncertainties of the multiplicative shear bias, the photo- z performance and the selection of the source galaxies. We esti-mated that the total level of systematic uncertainty affecting ourmass calibration and estimate of the Planck mass bias if ∼ percent.Even though the systematics are specific to the data set and c (cid:13)000 CMB lensing maps. The additional correction for faintergalaxies may be relevant for cosmic shear analysis, but we couldneglect it for our analysis.Whereas simulation-testing shows that the multiplicative biasis well controlled, detailed comparison of separate shape cataloguesof actual data can find that the residual systematic is larger. Jarviset al. (2016) performed a detailed comparison of two independentshape catalogues from the Dark Energy Survey Science Verificationdata and found a systematic uncertainty of δm ∼ . . We canconservatively assume that this is the shear systematics affectingour analysis too, which entails a related mass uncertainty of ∼ . per cent. Two neighbouring clusters that fall along the line of sight may beblended by the SZ cluster finder into a single, apparently largercluster. Whereas the Compton parameters add approximately lin-early, projection effects can severely impact the weak lensing mass.The lensing amplitude ∆Σ + is a differential measurement and theestimated mass of the blended cluster can be well below the sum ofthe masses of the aligned halos. Then, the blended object deviatesfrom the mean scaling relation between SZ signal and mass.To estimate this effect, we follow Simet et al. (2017). The sys-tematic uncertainty due to projection effects can be approximatedas δM/M ∼ p ( (cid:15) − . p ( (cid:15) − . , (23)where p is the fraction of aligned clusters and (cid:15) is an effective pa-rameter which characterizes the effective mass contribution of theprojected halo. The parameter (cid:15) depends on the relative position ofthe two blended haloes along the line of sight, and on their shape,elongation and concentration. If (cid:15) = 0 . , we correctly estimate thetotal mass; if (cid:15) = 0 , the second halo is hidden and contributes nomass. Planck objects are rare and the chance to have two or more ofthem aligned is small, < ∼ per cent considering their tendency to becorrelated. The systematic error on mass due to projection effectsis then negligible ( (cid:46) per cent). Residual systematic and statistical uncertainties on the mass cali-bration not corrected for in our analysis are listed in Table 9. Weassumed that systematics related to priors, mass estimators, and ra-dial range were properly accounted for and eliminated in our anal-ysis. The main contributors to the systematic error budget are thecalibration uncertainties of the multiplicative shear bias, the photo- z performance and the selection of the source galaxies. We esti-mated that the total level of systematic uncertainty affecting ourmass calibration and estimate of the Planck mass bias if ∼ percent.Even though the systematics are specific to the data set and c (cid:13)000 , 000–000 Sereno et al. Table 9. Systematic error budget on the mass calibration of the PSZ2LenSclusters. Sources of systematics (col. 1) are taken as uncorrelated.Source Mass error [%]Shear measurements 5Photometric redshifts 5Line-of-sight projections 1Contamination and membership dilution 3Miscentering 0.5Halo modelling 1Total 8 to the analysis, our systematic assessment is comparable to Mel-chior et al. (2017), who performed a weak lensing mass calibrationof redMaPPer galaxy clusters in Dark Energy Survey Science Ver-ification data, and to Simet et al. (2017), who measured the weaklensing mass-richness relation of the SDSS (Sloan Digital Sky Sur-vey) redMaPPer clusters.We did not consider as systematics triaxiality, orientation andsubstructures. The presence of substructures can dilute or enhancethe tangential shear signal (Meneghetti et al. 2010; Becker &Kravtsov 2011; Giocoli et al. 2012, 2014), and lensing effects de-pend on the cluster orientation (Oguri et al. 2005; Sereno 2007;Sereno & Umetsu 2011; Limousin et al. 2013). For systems whosemajor axis points toward the observer, WL masses derived underthe standard assumption of spherical symmetry are typically over-estimated. The opposite occurs for clusters elongated in the planeof the sky, which are in the majority if the selected sample is ran-domly oriented.We treated these effects as sources of intrinsic scatter, whichquantifies the difference between the deprojected WL mass mea-surement and the true halo mass (Sereno & Ettori 2015b), ratherthan as systematic errors. In our regression scheme, we modelledthe scatter of the WL mass, which we found to be ± percent from the analysis of the mass-concentration relation, see Sec-tion 10, and ± per cent from the analysis of the Planck massbias, see Section 12. 14 NOT CONFIRMED CLUSTERS We could not confirm seven out of the 47 candidate PSZ2 clustersin the CFHTLenS and RCSLenS fields. Out the subsample suitablefor our WL analysis, i.e. the 41 candidate PSZ2 sources in the fieldswith photometric redshifts, we could not find evident counterpartsfor six candidates.By visual inspection, we could detect neither any evidentgalaxy overdensity in the optical images nor an extended X-ray sig-nal from archive ROSAT (Röntgensatellit) or XMM -Newton (X-rayMulti-Mirror) images near the candidates PSZ2 G006.84+50.69(PSZ2 index: 25), G098.39+57.68 (463), and G233.46+25.46(1062). We also did not find any galaxy cluster in the SIMBAD As-tronomical Database within the uncertainty region associated withthe PSZ2 source.The analysis of the WL shear around them could notsupport the presence of a counterpart either. In particular,PSZ2 G006.84+50.69 (25) may be a substructure of the nearbylarger PSZ2 G006.49+50.56 (21), i.e. Abell 2029. The SNR aroundthe SZ centroid is . and the WL signal is compatible with nomass lens, M = (0 . ± . × M (cid:12) /h .For PSZ2 G098.39+57.68 (463), we estimated a SNR = − . by assuming as lens redshift the median redshift of the PSZ2clusters, i.e z = 0 . .The median redshift of the galaxies nearbyPSZ2 G233.46+25.46 (1062) is z phot = 0 . . This supposedlens redshift is too high to perform a reliable WL analysis. Wefound just 3 source galaxies passing our criteria behind thiscandidate.For two candidates, PSZ2 G084.69-58.60 (371) and G201.20-42.83 (912), a galaxy overdensity is seen in the photometric redshiftdistribution, but we could not assign a clear-cut BCG based on vi-sual inspection or available information from the public catalogs.The WL signal around these candidates can be tentatively measuredby locating the halo at the SZ centroid and estimating the redshiftas the peak of the distribution of measured z phot along the line ofsight. The SNR of PSZ2 G084.69-58.60 (371), and G201.20-42.83(912) is . and − . , respectively. There is no indication fromWL alone of the presence of a massive halo.For two more candidate PSZ2 clusters, the identification wasambiguous because more than one counterpart could be assignedwithin the uncertainty region: PSZ2 G092.69+59.92 (421) andG317.52+59.94 (1496). For the source PSZ2 G092.69+59.92 (421),the closest candidate BCG, at z spec = 0 . , is located 3.8 (cid:48) awayfrom the SZ centroid. The weak lensing SNR around this positionis 1.72, with a mass of M = (3 . ± . × M (cid:12) /h . ForPSZ2 G317.52+59.94 (1496), there is a possible identification witha galaxy cluster at z phot = 0 . , but photometric redshifts for thebackground sources in the RCSLenS are not available in that re-gions, and so it is excluded from our final catalog. 15 CONCLUSIONS Ongoing and future surveys are providing deep and accurate multi-wavelength observations of the sky. SZ selected samples of clus-ters of galaxies have some very coveted qualities. In principle, theyshould provide unbiased and mass limited samples representativeof the full population of cosmic haloes up to high redshifts.To date, quality multi-probe coverage is still restricted to lim-ited areas. We performed a WL analysis of the clusters of galaxieswhich were SZ selected by the Planck mission in the fields coveredby the CFHTLenS and the RCSLenS. The surveys are not deep butthe sample, which we named PSZ2LenS, is statistically completeand homogeneous in terms of observing facilities, and data acqui-sition, reduction, and analysis.Clusters are selected in SZ nearly independently of their dy-namical and merging state. They should sample all kinds of clus-ters. In fact, we found that the Planck selected clusters are standardhaloes in terms of their density profile, which is well fitted by cuspyhalo models, and in terms of their concentrations, which nicely fitthe Λ CDM prediction by numerical simulations. This suggests thatthe SZ detection does not suffer from over-concentration biases, asalso inferred by Rossetti et al. (2017) based on the comparison ofthe X-ray properties of the highest SNR Planck clusters with X-rayselected samples.Thanks to the statistical completeness of the PSZ2LenS sam-ple, which is a faithful subsample of the whole population of Planck clusters, we could asses the bias of the SZ Planck massesby comparison with the WL masses. We found a mass bias of − . ± . (stat.) ± . (sys.). We could estimate the effectivebias over the full mass and redshift range of the Planck clusters.Most of the previous analyses considered small mass ranges, i.e.the massive end of the mass function, or they were limited to in- c (cid:13) , 000–000 SZ2LenS termediate redshifts, where the WL signal is optimized. The mostsensible comparison is with Sereno & Ettori (2017) who extendedtheir analysis to lower masses and higher redshifts by exploiting aheterogeneous data set. Our results are in full agreement.By comparison with WL masses, we confirmed that Planck masses are precise, i.e. the statistical uncertainties and the intrinsicscatter is small, but they are not accurate, i.e. they are systemati-cally biased. ACKNOWLEDGEMENTS MS thanks Keiichi Umetsu and Peter Melchior for useful discus-sions. SE and MS acknowledge the financial contribution from con-tracts ASI-INAF I/009/10/0, PRIN-INAF 2012 ‘A unique dataset toaddress the most compelling open questions about X-Ray GalaxyClusters’, and PRIN-INAF 2014 1.05.01.94.02 ‘Glittering Kaleido-scopes in the sky: the multifaceted nature and role of galaxy clus-ters’. SE acknowledges the financial contribution from contractsNARO15 ASI-INAF I/037/12/0 and ASI 2015-046-R.0.This research has made use of NASA’s Astrophysics Data Sys-tem (ADS) and of the NASA/IPAC Extragalactic Database (NED),which is operated by the Jet Propulsion Laboratory, California In-stitute of Technology, under contract with the National Aeronauticsand Space Administration.This work is based on observations obtained withMegaPrime/MegaCam, a joint project of CFHT and CEA/IRFU, atthe Canada-France-Hawaii Telescope (CFHT) which is operatedby the National Research Council (NRC) of Canada, the InstitutNational des Sciences de l’Univers of the Centre National de laRecherche Scientifique (CNRS) of France, and the Universityof Hawaii. 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M., 2005, Reviews of Modern Physics, 77, 207von der Linden A. et al., 2014a, MNRAS, 439, 2von der Linden A. et al., 2014b, MNRAS, 443, 1973Zhao D. H., Jing Y. P., Mo H. J., Börner G., 2009, ApJ, 707, 354Zitrin A., Bartelmann M., Umetsu K., Oguri M., Broadhurst T.,2012, MNRAS, 426, 2944 APPENDIX A: RADIUS Different recipes for the effective radius of a radial annulus havebeen proposed. A simple estimator is the mean of the inner R min and outer R max radii, (cid:104) R (cid:105) mean = R min + R max . (A1) c (cid:13) , 000–000 SZ2LenS R / r α Figure A1. Logarithmic slope of the reduce shear profile as a function ofradius for a NFW lens of mass M = 5 × M (cid:12) /h and concentrationin agreement with the relation found in Dutton & Macciò (2014) for the Planck cosmology, i.e. c ∼ . . We placed the lens at z d = 0 . andthe background sources at z s = 1 . . The radius is in units of r . Area - weightedMean ( α = ) α = α ( R min ) - - - R [ Mpc / h ] Δ g + / g + Area - weightedMean ( α = ) α = α ( R min ) - - R [ Mpc / h ] Δ R / R Figure A2. Top panel : relative difference between the average reducedshear and the reduced shear computed at different radii. The vertical redlines delimit the radial annuli, which are ten bins equally spaced in loga-rithmic units between 0.1 and Mpc /h . The blue, green and orange pointsrefers to effective radii computed as the mean, (cid:104) R (cid:105) mean , the area-weightedmean (cid:104) R (cid:105) aw , or the shear-weighted mean (cid:104) R (cid:105) gw with logarithmic slopecomputed at the inner radius. The lens properties are as in Fig. A1. Bot-tom panel : same as the top panel but for the relative difference betweendifferent estimators of the radius and the true shear-weighted radius. Alternatively, for a spatially uniform number density of back-ground galaxies, the effective radius can be estimated as the area-weighted mean, (cid:104) R (cid:105) aw = (cid:82) R max R min R dR (cid:82) R max R min RdR = 23 R min + R max + R min R max R min + R max . (A2)The area-weighted mean is higher than the simple mean, since mostof the area is near the outer radius.Here, we define the effective radius (cid:104) R (cid:105) gw as the shearweighted radius, g + ( (cid:104) R (cid:105) gw ) = (cid:82) R max R min g + ( R ) RdR (cid:82) R max R min RdR . (A3)For a power-law shear profile, g + ∼ R − α , (cid:104) R (cid:105) gw = (cid:18) − α R − α max − R − α min R max − R min (cid:19) − /α . (A4)Equations (A1) and (A2) are particular cases of equation (A4) for α = 1 and − , respectively.In general, the logarithmic slope of the reduced shear profile, α , varies with the radius. The notable exception is the singularisothermal sphere with α SIS = 1 . For most profiles the slope isclose to one, see Fig. A1 for the case of the NFW halo.In Fig. A2, we show that the simple mean, i.e. the shear-weighted radius with α = 1 , provides a very good approximationof the effective radius. The area weighted radius, whereat the meanshear is under-estimated, is larger than the effective radius. In fact,the area-weighting scheme accounts for most of the galaxies beingnear the upper radius but does not account for their lower shear.Since the lens properties are not known when we stack the sig-nal, the shear-weighted radius with α = 1 is an acceptable choice.More elaborate schemes, as that fixing the slope at its value at theinner radius, which would nevertheless require some knowledge ofthe profile, do not improve the radius estimates significantly, seeFig. A2.The previous discussion relied on the continuous limit wherethe background galaxy distribution is uniform and lie at a singleredshift. For sparse populations which are redshift distributed, wehave to compute the effective radius as (cid:104) R (cid:105) gw = (cid:32) (cid:80) i w i Σ − cr ,i R − αi (cid:80) i w i Σ − cr ,i (cid:33) − /α , (A5)where we exploited the power-law approximation for the shear pro-file. The shear-weighted radius makes the fitting procedure to shearprofiles less dependent on the binning scheme. APPENDIX B: LENSING WEIGHTED AVERAGE Stacking and combining lensing data or results are a highly non-linear process. A sensible way to define the central estimate of acluster property O for a lensing sample is the lensing weightedaverage (Umetsu et al. 2014), (cid:104)O(cid:105) lw = (cid:80) i W i O i (cid:80) i W i , (B1)where the sums runs over the cluster sample and the weight W ofthe i -th cluster is W i = (cid:88) j w i,j Σ − cr ,j , (B2) c (cid:13) , 000–000 Sereno et al. R = / hR = / hR = / h M [ M ⊙ / h ] Γ Figure B1. Logarithmic slope of the differential density profile of a NFWhalo, Γ = d ln ∆Σ + /d ln M , as a function of mass, at differentredshifts and radial distances from the cluster centre. Concentrations areassigned through the mass-concentration relation from Meneghetti et al.(2014). The full and dashed lines are for lenses at z cl = 0 . and . , re-spectively. The blue, green, and orange lines (from top to bottom) are forradii R = 0 . , , and Mpc /h , respectively. where the sum run over the selected background galaxies behindthe i -th cluster. The weight W accounts for the total shear weight ofthe cluster and accounts for the shear weight w of each lens-sourcepair, the lens and source redshifts through the critical density, andthe angular size of the clusters, since lower redshift clusters subtenda larger angle in the sky for a fixed physical length and hence alarger number of background galaxies.We verified, for example, that the definition in Eq. (B1) isappropriate to assign a redshift to the stacked profile, i.e. the re-covered mass M stack of a stacking sample of clusters with thesame mass M = M cl but at different redshifts is equal to (cid:104) M (cid:105) lw (cid:39) M cl if z stack = (cid:104) z cl (cid:105) lw .The lensing average in Eq. (B1) can be modified for some ob-servables to account for the fact that we stack the density profiles ∆Σ + . In practice, we have to recover the mean observable fromthe stacked profile. If ∆Σ + ∝ O Γ , then (Melchior et al. 2017) (cid:104)O(cid:105) lw = (cid:32) (cid:80) i W i O Γ i (cid:80) i W i (cid:33) / Γ . (B3)If we consider the mass as the observable, the exponent Γ candiffer from 1. The dependence of the mass on the density profilecan be approximated with a power low ∆Σ + ∝ M Γ200 with Γ = d ln ∆Σ + d ln M ; (B4)for an isothermal model, Γ = 1 . For a NFW halo, the logarithmicdensity slope for a range of radii and redshifts is shown in Fig. B1.The slope is larger at small radii or large redshifts and spans a rangefrom ∼ . to . Based on some toy model simulations mimickingour stacking analysis, we found that Γ ∼ . is appropriate forour range of masses and redshifts and for our fitting procedure. APPENDIX C: ROBUST ESTIMATOR The posterior probability density function of the mass of lowsignal-to-noise ratio systems can be asymmetric or peaked near oneof the imposed borders. As an extreme example, the more likelymass of a low mass group detected with a negative signal-to-noise MLCBI M true [ M ⊙ / h ] M f i t [ M ⊙ / h ] Figure C1. Fitted mass versus true input mass M for simulated NFWlenses. The green points and the associated error bars denote the centralbiweight estimator. The blue points denote the maximum likelihood esti-mator. ratio will coincide with the lower limit of the allowed parameterrange. The problem is then to identify a reliable and stable massestimator. The median (Gott et al. 2001) or the bi-weight locationestimator C BI (Beers, Flynn & Gebhardt 1990) are regarded as ro-bust choices for the central location and have been considered inWL analyses (Sereno & Umetsu 2011). Here, we want to comparethe performances of the bi-weight location estimator against themaximum likelihood estimator.We simulated the shear profile of clusters with shallow qualitydata. Lenses were modeled as NFW haloes at redshift z d = 0 . .We assumed a shape noise error dominated by the intrinsic distri-bution of ellipticities, with a dispersion of σ e = 0 . , and we alsoconsidered the noise from the large scale structure. We considered abackground population at z s = 0 . with a source density of n g = 2 background galaxies per square arc minute.We simulated 100 lens masses with a constant logarithmicspacing from M = 5 × to × M (cid:12) /h . Concentra-tion were associated assuming the scattered relation from Dutton& Macciò (2014). The shear profiles were finally simulated in 10equally spaced logarithmic radial annuli between 0.1 and Mpc /h .We fitted the simulated profiles as in Section 8. Resultsare summarized in Fig. C1. At the high mass end ( M (cid:38) M (cid:12) /h ), the signal to noise is high enough and the estimatedmass is stable whatever the estimator. At the low mass end, fluc-tuations are larger and differences can be significant. Results areusually consistent within the errors but the maximum likelihoodestimator is more prone to outliers and often attracted towards theextremes of the allowed mass range. For our simulation, this prob-lem is under control since we could fit the toy-clusters with theright NFW profile. However, the problem can be exacerbated withreal clusters which can deviate from the halo modelling we assumefor the fit.The bi-weight estimator is stabler but it can be influenced bythe prior. Assuming a uniform prior, masses can be biased high at c (cid:13) , 000–000 SZ2LenS the low mass end. This would not be the case assuming a prioruniform in log space, which however could be inadequate at inter-mediate masses. Since most of the Planck clusters are expected tobe (cid:38) M (cid:12) /h , the prior uniform in mass has to be preferred.For the simulated lenses with M > M (cid:12) /h , the dis-tribution of the relative deviations expressed as ln( M fit /M true ) hasmean . ( . ) and standard deviation equal to . ( . ) forthe bi-weight (maximum likelihood) estimator. c (cid:13)000