CLASH: JOINT ANALYSIS OF STRONG-LENSING, WEAK-LENSING SHEAR, AND MAGNIFICATION DATA FOR 20 GALAXY CLUSTERS^*

Our cluster sample stems from the CLASH shear-and-magnification weak-lensing analysis of Umetsu et al. (2014) based primarily on Subaru multi-color imaging. This sample comprises two subsamples, one with 16 X-ray regular clusters and another with four high-magnification clusters, both taken from the CLASH sample of Postman et al. (2012).

Table 1 gives a summary of the properties of 20 clusters in our sample. Following Umetsu et al. (2014), we adopt the brightest cluster galaxy (BCG) position as the cluster center for our mass profile analysis. As discussed by Umetsu et al. (2014), our sample exhibits, on average, a small positional offset between the BCG and X-ray peak, characterized by an rms offset of σ_off ≃ 30 kpc h⁻¹. For the X-ray-selected subsample, σ_off ≃ 11 kpc h⁻¹. This level of offset is negligible compared to the range of overdensity radii of interest for mass measurements (${{\rm{\Delta }}}_{{\rm{c}}}\lesssim 2500$). Hence, smoothing from the miscentering effects will not significantly affect our cluster mass profile measurements (Johnston et al. 2007; Umetsu et al. 2011a).

Table 1.  Properties of the Cluster Sample

Cluster	${z}_{{\rm{l}}}$	R.A.a	decl.a	${k}_{{\rm{B}}}{T}_{X}$ b	${\theta }_{\mathrm{Ein}}$ c	${M}_{2{\rm{D}}}$ (${10}^{13}\;{M}_{\odot }{h}_{70}^{-1}$)d
		(J2000.0)	(J2000.0)	(keV)	('')	$\theta =10^{\prime\prime}$	$\theta =20^{\prime\prime}$	$\theta =30^{\prime\prime}$	$\theta =40^{\prime\prime}$
X-ray Selected:
Abell 383	0.187	02:48:03.40	−03:31:44.9	6.5 ± 0.24	15.1	1.15 ± 0.17	3.04 ± 0.51	4.98 ± 0.96	6.77 ± 1.35
Abell 209	0.206	01:31:52.54	−13:36:40.4	7.3 ± 0.54	8.9	0.93 ± 0.14	2.36 ± 0.45	3.96 ± 0.89	5.60 ± 1.40
Abell 2261	0.224	17:22:27.18	+32:07:57.3	7.6 ± 0.30	23.1	1.91 ± 0.31	4.79 ± 0.69	7.67 ± 1.26	10.42 ± 1.84
RX J2129.7+0005	0.234	21:29:39.96	+00:05:21.2	5.8 ± 0.40	12.9	1.13 ± 0.18	3.36 ± 0.44	5.95 ± 0.74	8.67 ± 1.10
Abell 611	0.288	08:00:56.82	+36:03:23.6	7.9 ± 0.35	18.1	1.84 ± 0.25	5.25 ± 0.83	9.37 ± 1.82	14.26 ± 3.06
MS2137-2353	0.313	21:40:15.17	−23:39:40.2	5.9 ± 0.30	17.1	2.25 ± 0.33	5.23 ± 0.81	7.99 ± 1.38	10.76 ± 2.00
RX J2248.7-4431	0.348	22:48:43.96	−44:31:51.3	12.4 ± 0.60	31.1	2.47 ± 0.45	7.36 ± 1.02	13.14 ± 1.96	19.41 ± 3.07
MACS J1115.9+0129	0.352	11:15:51.90	+01:29:55.1	8.0 ± 0.40	18.1	1.85 ± 0.37	5.79 ± 0.84	11.09 ± 1.50	16.98 ± 2.37
MACS J1931.8-2635	0.352	19:31:49.62	−26:34:32.9	6.7 ± 0.40	22.2	2.91 ± 0.60	7.37 ± 1.12	12.15 ± 1.86	17.21 ± 2.90
RX J1532.9+3021	0.363	15:32:53.78	+30:20:59.4	5.5 ± 0.40	⋯	⋯	⋯	⋯	⋯
MACS J1720.3+3536	0.391	17:20:16.78	+35:36:26.5	6.6 ± 0.40	20.1	2.65 ± 0.35	7.20 ± 1.08	12.39 ± 2.17	17.97 ± 3.36
MACS J0429.6-0253	0.399	04:29:36.05	−02:53:06.1	6.0 ± 0.44	15.7	2.22 ± 0.39	6.80 ± 0.96	12.55 ± 1.86	18.87 ± 3.02
MACS J1206.2-0847	0.440	12:06:12.15	−08:48:03.4	10.8 ± 0.60	26.8	3.37 ± 0.50	9.51 ± 1.39	16.37 ± 2.50	23.18 ± 3.88
MACS J0329.7-0211	0.450	03:29:41.56	−02:11:46.1	8.0 ± 0.50	24.1	3.40 ± 0.62	8.66 ± 1.26	14.36 ± 2.16	21.12 ± 3.17
RX J1347.5-1145	0.451	13:47:31.05	−11:45:12.6	15.5 ± 0.60	33.0	3.56 ± 0.76	11.55 ± 2.14	20.40 ± 3.18	29.25 ± 4.12
MACS J0744.9+3927	0.686	07:44:52.82	+39:27:26.9	8.9 ± 0.80	24.3	4.70 ± 0.92	13.55 ± 2.13	23.64 ± 3.35	34.79 ± 4.57
High Magnification:
MACS J0416.1-2403	0.396	04:16:08.38	−24:04:20.8	7.5 ± 0.80	25.9	1.33 ± 0.18	5.35 ± 0.79	11.32 ± 1.68	17.43 ± 2.51
MACS J1149.5+2223	0.544	11:49:35.69	+22:23:54.6	8.7 ± 0.90	20.4	2.88 ± 0.51	9.29 ± 1.39	17.80 ± 2.87	28.09 ± 5.07
MACS J0717.5+3745	0.548	07:17:32.63	+37:44:59.7	12.5 ± 0.70	55.0	2.01 ± 0.19	8.04 ± 0.74	18.79 ± 2.01	35.80 ± 5.83
MACS J0647.7+7015	0.584	06:47:50.27	+70:14:55.0	13.3 ± 1.80	26.4	3.62 ± 0.77	11.75 ± 1.84	21.59 ± 3.32	31.95 ± 5.45

Notes.

^aThe cluster center is taken to be the location of the BCG when a single dominant central galaxy is found. Otherwise, in the case of MACS J0717.5+3745 and MACS J0416.1−2403, it is defined as the center of the brightest red-sequence-selected cluster galaxies (Umetsu et al. 2014). ^bX-ray temperature from Postman et al. (2012). ^cEffective Einstein radius for a fiducial source at ${z}_{{\rm{s}}}=2$ determined from the HST strong and weak-shear lensing analysis by Zitrin et al. (2015). The reported values are the average of two different models (where available) of Zitrin et al. (2015). The typical model uncertainty in ${\theta }_{\mathrm{Ein}}$ is 10%. ^dLensing estimates of the projected cluster mass ${M}_{2{\rm{D}}}(\lt \theta )$ enclosed within a cylinder of radius θ. The data here are constructed for each cluster by combining two different lens models (where available) of Zitrin et al. (2015). For details, see Section 3.5.

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A careful selection of background galaxies is critical for a cluster weak-lensing analysis, so that unlensed cluster members and foreground galaxies do not dilute the background lensing signal. Umetsu et al. (2014) used the color–color (CC) selection method of Medezinski et al. (2010), typically using the Subaru/Suprim-Cam ${B}_{{\rm{J}}}{R}_{{\rm{C}}}z^{\prime}$ photometry where available (Umetsu et al. 2014, Tables 1 and 2), which spans the full optical wavelength range. The photometric zero points were precisely calibrated to an accuracy of ∼0.01 mag, using the HST photometry of cluster elliptical galaxies and with the help of galaxies with measured spectroscopic redshifts.

For shape measurements, Umetsu et al. (2014) combined two distinct populations that encompass the red and blue branches of background galaxies in CC-magnitude space, having typical redshift distributions peaked around z ∼ 1 and ∼2, respectively (see Figures 1, 5, and 6 of Medezinski et al. 2011). For validation purposes, we have compared our blue+red background samples with spectroscopic samples obtained from the CLASH-VLT large spectroscopic program with VIMOS (Rosati et al. 2014) providing thousands of spectroscopic redshifts for cluster members and intervening galaxies along the line of sight, including lensed background galaxies (e.g., Balestra et al. 2013; Biviano et al. 2013). Combining CLASH-VLT spectroscopic redshifts with the Subaru multi-band photometry available for 10 southern CLASH clusters, we find a mean contamination fraction of (2.4 ± 0.7)% in our blue+red background regions in CC-magnitude space, where the error accounts for Poisson statistics.

Our magnification-bias measurements are based on flux-limited samples of red background galaxies (${R}_{{\rm{C}}}-{z}^{\prime }\gtrsim 0.5$, Medezinski et al. 2011). Apparent faint magnitude cuts (m_lim) were applied for each cluster in the reddest CC-selection band (typically the Subaru z' band) to avoid incompleteness near the detection limit. The threshold m_lim was chosen at the magnitude where the source number counts turn over, and it corresponds on average to the 6σ limiting magnitude within 2'' diameter aperture. Our CC selection does not cause incompleteness at the faint end in the bluer filters (for a general discussion, see Hildebrandt 2015) because we have deeper photometry in the bluer bands (this is by design, so as to detect the red galaxies; see Broadhurst 1995) and our "CC-red" galaxies are relatively blue in B_J − R_C (Figure 1 of Medezinski et al. 2011).

The mean depths $\langle \beta \rangle$ and $\langle {\beta }^{2}\rangle$ of the background samples were estimated using photometric redshifts of individual galaxies determined with the BPZ code (Benítez 2000; Benítez et al. 2004) from our point-spread function (PSF) corrected multi-band photometry (typically with 5 Subaru filters; Table 1 of Umetsu et al. 2014). An excellent statistical agreement was found between the depth estimates $\langle \beta \rangle$ from our BPZ measurements in the cluster fields and those from the COSMOS photometric-redshift catalog (Ilbert et al. 2009), with a median relative offset of 0.27% and an rms field-to-field scatter of 5.0% (Umetsu et al. 2014).

We use the azimuthally averaged radial profile of the reduced tangential shear ${g}_{+}\;=\;{\gamma }_{+}/(1-\kappa )$ as the primary constraint from wide-field weak-lensing observations. We adopt the following approximation for the nonlinear corrections to the source-averaged reduced tangential shear $\langle {g}_{+}\rangle =\left[{\int }_{0}^{\infty }{dz}\;{\bar{N}}_{g}(z){g}_{+}(z)\right]{\left[{\int }_{0}^{\infty }{dz}{\bar{N}}_{g}(z)\right]}^{-1}$ (Seitz & Schneider 1997):

$$\begin{eqnarray}&&\langle {g}_{+}\rangle \approx \displaystyle \frac{\langle W{\rangle }_{g}[{\kappa }_{\infty }(\lt \theta )-{\kappa }_{\infty }(\theta )]}{1-{\kappa }_{\infty }(\theta )\langle {W}^{2}{\rangle }_{g}/\langle W{\rangle }_{g}}=\displaystyle \frac{\langle {\gamma }_{+}\rangle }{1-{f}_{W,g}\langle \kappa \rangle },\end{eqnarray} \tag{ 5 }$$

where $\langle W{\rangle }_{g}$ is the relative lensing strength (Section 2) averaged over the population N_g(z) of source galaxies, ${f}_{W,g}\equiv \langle {W}^{2}{\rangle }_{g}/\langle W{\rangle }_{g}^{2}$ is a dimensionless quantity of the order unity, $\langle \kappa \rangle =\langle W{\rangle }_{g}{\kappa }_{\infty }$, and $\langle {\gamma }_{+}\rangle =\langle W{\rangle }_{g}{\gamma }_{+,\infty }$.

In the present study, we use the weak lensing shear data obtained by Umetsu et al. (2014). The shear analysis pipeline of Umetsu et al. (2014) was implemented based on the procedures described in Umetsu et al. (2010) and on verification tests with mock ground-based observations (Massey et al. 2007; Oguri et al. 2012). The key feature of the shear calibration method of Umetsu et al. (2010) is that we use galaxies detected with very high significance, ν > 30, to model the PSF isotropic correction as function of object size and magnitude. Here ν is the peak significance given by the IMCAT peak-finding algorithm hfindpeaks. Recently, a very similar procedure was used by the Local Cluster Substructure Survey (LoCuSS) collaboration in their Subaru weak-lensing study of 50 clusters (Okabe & Smith 2015). Another important feature is that we select isolated galaxies for the shape measurement to minimize the impact of crowding and blending (Umetsu et al. 2014). To do this, we first identify objects having any detectable neighbor within 3r_g, with r_g the Gaussian scale length given by hfindpeaks. All such close pairs of objects are rejected. After this close-pair rejection, objects with low detection significance ν < 10 are excluded from our analysis. All galaxies with usable shape measurements are then matched with sources in our CC-magnitude-selected background galaxy samples (Section 3.2), ensuring that each galaxy is detected in both the reddest CC-selection band and the shape-measurement band.

Using simulated Subaru/Suprime-Cam images (Massey et al. 2007; Oguri et al. 2012), Umetsu et al. (2010) find that the lensing signal can be recovered with $| m| \sim 0.05$ of the multiplicative shear calibration bias and c ∼ 10⁻³ of the residual shear offset (as defined by Heymans et al. 2006; Massey et al. 2007). Accordingly, Umetsu et al. (2014) included for each galaxy a shear calibration factor of 1/(1 + m) (g → g/0.95) to account for residual calibration. As noted by (Umetsu et al. 2012), the degree of multiplicative bias m depends modestly on the seeing conditions and the PSF quality (Oguri et al. 2012, see Section 3.2 for their image simulations using Gaussian and Moffat PSF profiles with 05–11 FWHM). This variation with the PSF properties limits the shear calibration accuracy to δm ∼ 0.05 (Umetsu et al. 2012, Section 3.3).

From shape measurements of background galaxies, the averaged reduced tangential shear was measured in a set of concentric annuli (i = 1, 2, ..., N_WL) centered on each cluster as

$$\begin{eqnarray}&&\langle {g}_{+,i}\rangle =\left[\displaystyle \sum _{k\in i}\;{w}_{(k)}\;{g}_{+(k)}\right]{\left[\displaystyle \sum _{k\in i}{w}_{(k)}\right]}^{-1},\end{eqnarray} \tag{ 6 }$$

where the index k runs over all objects located within the ith annulus, g_+(k) is an estimate of g₊ for the kth object, and w_k is its statistical weight given by ${w}_{(k)}=1/({\sigma }_{g(k)}^{2}+{\alpha }_{g}^{2})$, with σ_g(k) the uncertainty in the estimate of reduced shear g_k and α_g the softening constant taken to be α_g = 0.4 (Umetsu et al. 2014), a typical value of the mean dispersion ${(\bar{{\sigma }_{g}^{2}})}^{1/2}$ in Subaru observations (e.g., Oguri et al. 2009; Umetsu et al. 2009; Okabe et al. 2010). Here α includes both intrinsic shape and measurement noise contributions. The statistical uncertainty ${\sigma }_{+,i}$ in $\langle {g}_{+,i}\rangle$ was estimated from bootstrap resampling of the background source catalog for each cluster.

The reduced tangential shear profiles analyzed in this study are presented in Figure 2 of Umetsu et al. (2014). For all clusters in our sample, the estimated values for $\langle \beta {\rangle }_{g}$, and f_W,g are summarized in Table 3 of Umetsu et al. (2014). We marginalize over the calibration uncertainty of $\langle \beta {\rangle }_{g}$ in our joint likelihood analysis of multiple lensing probes (Section 3.6.2).

A fundamental limitation of measuring shear only is the mass-sheet degeneracy (Section 2; see also Section 4.2). We can break this degeneracy by using the complementary combination of shear and magnification (Schneider et al. 2000; Umetsu & Broadhurst 2008; Rozo & Schmidt 2010; Umetsu 2013).

Deep multi-color photometry enables us to explore the faint end of the luminosity function of red quiescent galaxies at z ∼ 1 (Ilbert et al. 2010). For such a population, the effect of magnification bias is dominated by the geometric area distortion because few fainter galaxies can be magnified up into the flux-limited sample; this results in a net depletion of source counts (Taylor et al. 1998; Broadhurst et al. 2005; Umetsu & Broadhurst 2008; Umetsu et al. 2010, 2011b, 2012, 2014, 2015; Coe et al. 2012; Ford et al. 2012; Medezinski et al. 2013; Radovich et al. 2015). In the regime of negative magnification bias, a practical advantage is that the effect is not sensitive to the exact form of the source luminosity function (Umetsu et al. 2014).

If the magnitude shift $\delta m=2.5{\mathrm{log}}_{10}\mu$ of an object due to magnification is small compared to that on which the logarithmic slope of the luminosity function varies, the number counts can be locally approximated by a power law at the limiting magnitude m_lim. The magnification bias at redshift z is then given by (Broadhurst et al. 1995)

$$\begin{eqnarray}&&{N}_{\mu }({\boldsymbol{\theta }},z;\lt {m}_{\mathrm{lim}})={\bar{N}}_{\mu }(z)\;{\mu }^{2.5s-1}({\boldsymbol{\theta }},z)\equiv {\bar{N}}_{\mu }(z){b}_{\mu }({\boldsymbol{\theta }},z),\end{eqnarray} \tag{ 7 }$$

where ${\bar{N}}_{\mu }(z)={\bar{N}}_{\mu }(z;\lt {m}_{\mathrm{lim}})$ is the unlensed mean source counts and s is the logarithmic count slope evaluated at m_lim, $s=[d{\mathrm{log}}_{10}{\bar{N}}_{\mu }(z;\lt m)/{dm}{]}_{m={m}_{\mathrm{lim}}}$. In the regime where $2.5\;s\ll 1$, a net count depletion results. Accounting for the spread of ${\bar{N}}_{\mu }(z)$, we express the population-averaged magnification bias as $\langle {b}_{\mu }\rangle =\left[{\int }_{0}^{\infty }{dz}\;{\bar{N}}_{\mu }(z){b}_{\mu }(z)\right]{\left[{\int }_{0}^{\infty }{dz}{\bar{N}}_{\mu }(z)\right]}^{-1}$. Following Umetsu et al. (2014), we interpret the observed number counts on a grid of equal-area cells (n = 1, 2, ...) as (see Appendix A.2 of Umetsu 2013)

$$\begin{eqnarray}&&\langle {b}_{\mu }({{\boldsymbol{\theta }}}_{n})\rangle ={N}_{\mu }({{\boldsymbol{\theta }}}_{n};\lt {m}_{\mathrm{lim}})/{\bar{N}}_{\mu }(\lt {m}_{\mathrm{lim}})\approx \langle {\mu }^{-1}({{\boldsymbol{\theta }}}_{n}){\rangle }^{1-2.5{s}_{\mathrm{eff}}}\end{eqnarray} \tag{ 8 }$$

with ${s}_{\mathrm{eff}}=[d{\mathrm{log}}_{10}{\bar{N}}_{\mu }(\lt m)/{dm}{]}_{m={m}_{\mathrm{lim}}}$ the effective count slope defined in analogy to Equation (7). Equation (8) is exact for s_eff = 0 and gives a good approximation for depleted populations with s_eff ≪ 0.4.

The covariance matrix $\mathrm{Cov}[N{({{\boldsymbol{\theta }}}_{m}),N({{\boldsymbol{\theta }}}_{n})]\equiv ({C}_{N})}_{{mn}}$ of the source counts includes the clustering and Poisson contributions (Hu & Kravtsov 2003) as ${({C}_{N})}_{{mn}}={({\bar{N}}_{\mu })}^{2}{\omega }_{{mn}}+{\delta }_{{mn}}{N}_{\mu }({{\boldsymbol{\theta }}}_{m})$, with ω_mn the cell-averaged angular correlation function of source galaxies (see Equation (14) of Umetsu et al. 2015). The angular correlation length of background galaxies can be small (Connolly et al. 1998; van Waerbeke 2000) compared to the typical resolution ∼1' of cluster weak lensing, so that the correlation between different cells can be generally ignored, whereas the unresolved and nonvanishing correlation on small angular scales accounts for increase of the variance of counts. We thus approximate C_N by (Umetsu et al. 2015)

$$\begin{eqnarray}&&{({C}_{N})}_{{mn}}\approx [\langle \delta {N}_{\mu }^{2}({{\boldsymbol{\theta }}}_{m})\rangle +{N}_{\mu }({{\boldsymbol{\theta }}}_{m})]{\delta }_{{mn}},\end{eqnarray} \tag{ 9 }$$

with $\langle \delta {N}_{\mu }^{2}({{\boldsymbol{\theta }}}_{m})\rangle$ being the total variance of the mth counts. To enhance the signal-to-noise ratio (S/N), we calculate the surface number density n_μ(θ) = dN_μ(θ)/dΩ of source galaxies as a function of clustercentric radius, by averaging the counts in concentric annuli centered on the cluster. The source-averaged magnification bias is then expressed as $\langle {n}_{\mu }(\theta )\rangle ={\bar{n}}_{\mu }\langle {\mu }^{-1}(\theta ){\rangle }^{1-2.5{s}_{\mathrm{eff}}}$ with ${\bar{n}}_{\mu }$ the unlensed mean surface number density.

We measure the magnification bias signal in each annulus (i = 1, 2, ..., N_WL) as (Umetsu et al. 2015)

$$\begin{eqnarray}&&\langle {n}_{\mu ,i}\rangle =\displaystyle \frac{1}{(1-{f}_{\mathrm{mask},i}){{\rm{\Omega }}}_{\mathrm{cell}}}\displaystyle \sum _{m}{{ \mathcal P }}_{{im}}{N}_{\mu }({{\boldsymbol{\theta }}}_{m})\end{eqnarray} \tag{ 10 }$$

with Ω_cell being the solid angle of the cell and ${{ \mathcal P }}_{{im}}={\left({\sum }_{m}{A}_{{mi}}\right)}^{-1}{A}_{{mi}}$ the projection matrix normalized in each annulus as ${\sum }_{m}{{ \mathcal P }}_{{im}}=1$, here, A_mi is the fraction of the area of the mth cell lying within the ith annular bin ($0\leqslant {A}_{{mi}}\leqslant 1$), and f_mask,i is the mask correction factor for the ith annular bin, ${(1-{f}_{\mathrm{mask},i})}^{-1}\equiv {\left[{\sum }_{m}(1-{f}_{m}){A}_{{mi}}\right]}^{-1}{\sum }_{m}{A}_{{mi}}$, with f_m being the fraction of the mask area in the mth cell, due to bad pixels, saturated objects, foreground, and cluster galaxies. The intrinsic clustering plus statistical Poisson contributions to the uncertainty in $\langle {n}_{\mu ,i}\rangle$ are given as

$$\begin{eqnarray}&&{({\sigma }_{\mu ,i}^{\mathrm{int}})}^{2}+{({\sigma }_{\mu ,i}^{\mathrm{stat}})}^{2}=\displaystyle \frac{1}{{(1-{f}_{\mathrm{mask},i})}^{2}{{\rm{\Omega }}}_{\mathrm{cell}}^{2}}\;\displaystyle \sum _{m}{{ \mathcal P }}_{{im}}^{2}{({C}_{N})}_{{mm}},\end{eqnarray} \tag{ 11 }$$

where ${({\sigma }_{\mu }^{\mathrm{int}})}^{2}$ and ${({\sigma }_{\mu }^{\mathrm{stat}})}^{2}$ account for the contributions from the first and second terms of Equation (9), respectively.

In the present work, we use magnification measurements from flux-limited samples of red background galaxies obtained by Umetsu et al. (2014). The analysis procedure used in Umetsu et al. (2014) is summarized as follows: The magnification bias analysis was limited to the $24^{\prime} \times 24^{\prime}$ region centered on the cluster. The clustering error term ${\sigma }_{\mu ,i}^{\mathrm{int}}$ was estimated empirically from the variance in each annulus due to variations of the counts along the azimuthal direction. For the estimation of $\langle {n}_{\mu ,i}\rangle$, a positive tail of >νσ cells with ν = 2.5 was excluded in each annulus by iterative σ clipping to reduce the bias due to intrinsic angular clustering of source galaxies. The Poisson error term ${\sigma }_{\mu ,i}^{\mathrm{stat}}$ was estimated from the clipped mean counts. The systematic change between the mean counts estimated with and without σ clipping was then taken as a systematic error, ${\sigma }_{\mu ,i}^{\mathrm{sys}}=| {n}_{\mu ,i}^{(\nu )}-{n}_{\mu ,i}^{(\infty )}| /\nu$, where ${n}_{\nu ,i}^{(\nu )}$ and ${n}_{\mu ,i}^{(\infty )}$ represent the clipped and unclipped mean counts in the ith annulus, respectively. As noted by Umetsu et al. (2014), the ${\sigma }_{\mu }^{\mathrm{sys}}$ term is sensitive to large-scale variations of source counts and can in principle account for projection effects from background clusters along the line of sight and spurious excess counts due perhaps to spatial variation of the photometric zero point and/or to residual flat-field errors. These errors were combined in quadrature as

$$\begin{eqnarray}&&{\sigma }_{\mu ,i}^{2}={({\sigma }_{\mu ,i}^{\mathrm{int}})}^{2}+{({\sigma }_{\mu ,i}^{\mathrm{stat}})}^{2}+{({\sigma }_{\mu ,i}^{\mathrm{sys}})}^{2}.\end{eqnarray} \tag{ 12 }$$

Since we include the ${\sigma }_{\mu }^{\mathrm{sys}}$ term in the error analysis, our magnification bias measurements are stable and insensitive to the particular choice of ν. The smaller the ν value, the larger the resulting total errors as dominated by the Poisson and systematic terms.

The count normalization and slope parameters $({\bar{n}}_{\mu },{s}_{\mathrm{eff}})$ were estimated from the source counts in the outskirts at $10^{\prime} \leqslant \theta \leqslant {\theta }_{\mathrm{max}}$, with ${\theta }_{\mathrm{max}}=16^{\prime}$ except θ_max = 14' for RX J2248.7−4431 observed with ESO/WFI (Umetsu et al. 2014). The mask-corrected magnification bias profile $\langle {n}_{\mu ,i}\rangle /{\bar{n}}_{\mu }$ is thus proportional to $(1-{f}_{\mathrm{mask},\mathrm{back}})/(1-{f}_{\mathrm{mask},i})\equiv 1+{\rm{\Delta }}{f}_{\mathrm{mask},i}$ with ${f}_{\mathrm{mask},\mathrm{back}}={f}_{\mathrm{mask}}(10^{\prime} \leqslant \theta \leqslant {\theta }_{\mathrm{max}})$ the masked area fraction in the reference background region. Masking of observed sky was accounted for using the method outlined in Umetsu et al. (2011b, Method B of Appendix A). This method can be fully automated to achieve similar performance to conservative approaches (e.g., Method A of Umetsu et al. 2011b) in terms of the masked area fraction once the configuration parameters of SExtractor (Bertin & Arnouts 1996) are optimally tuned. We tuned the SExtractor configuration parameters by setting ${\mathtt{DETECT}}\_{\mathtt{THRESH}}={\mathtt{5}}$ and ${\mathtt{DETECT}}\_{\mathtt{MINAREA}}={\mathtt{300}}$ as found by Umetsu et al. (2011b) to detect foreground objects, cluster galaxies, and defects (e.g., saturated stars and stellar trails) in the coadded images (02 pixel⁻¹ sampling). We marked the pixels that belong to these objects in the ${\mathtt{CHECKIMAGE}}\_{\mathtt{TYPE}}={\mathtt{OBJECT}}$ mode. Recently, Chiu et al. (2015) adopted this method to calculate the masked area fraction for their magnification bias measurements, finding that the SExtractor configuration of Umetsu et al. (2011b) is optimal for their observations with Megacam on the Magellan Clay telescope.

We find that the unmasked area fraction $1-{f}_{\mathrm{mask}}$ is on average 93% of the sky at $\theta \geqslant 10^{\prime}$, decreasing toward the cluster center down to 88% at $1^{\prime} \lesssim \theta \lesssim 2^{\prime}$ (210 kpc ${h}^{-1}\lesssim R\lesssim 420$ kpc h⁻¹ at z = 0.35). The typical variation of the mask correction factor across the radial range is thus $\approx \mathrm{max}({f}_{\mathrm{mask}})-{f}_{\mathrm{mask},\mathrm{back}}\sim 5\%$, which is much smaller than the typical depletion signal $\delta {n}_{\mu }/{\bar{n}}_{\mu }\sim -0.3$ in the innermost radial bin [09, 12]. Hence, the uncertainty in the mask correction is of the second order. Furthermore, the net effect of the mask correction depends on the difference of the f_mask values, ${\rm{\Delta }}{f}_{\mathrm{mask},i}\approx {f}_{\mathrm{mask},i}-{f}_{\mathrm{mask},\mathrm{back}}$ and is insensitive to the particular choice of the SExtractor configuration parameters. Accordingly, the systematic uncertainty on the mask correction is negligible.

The magnification-bias profiles used in this study are presented in Figure 2 of Umetsu et al. (2014). For all clusters, the estimated values and errors for $\langle \beta {\rangle }_{\mu }$, ${\bar{n}}_{\mu }$, and s_eff are summarized in Table 4 of Umetsu et al. (2014). The observed values of s_eff range from 0.11 to 0.20, with a mean of $\langle {s}_{\mathrm{eff}}\rangle =0.153$ and a typical uncertainty of 33% per cluster field. We marginalize over the calibration parameters ($\langle \beta {\rangle }_{\mu },{\bar{n}}_{\mu },{s}_{\mathrm{eff}}$) for each cluster in our joint likelihood analysis of multiple lensing probes (Section 3.6.2).

Detailed strong-lens modeling using many sets of multiple images with known redshifts allows us to determine the critical curves with great accuracy, which then provides accurate estimates of the projected mass enclosed within the critical area A_c of an effective Einstein radius ${\theta }_{\mathrm{Ein}}=\sqrt{{A}_{{\rm{c}}}/\pi }$ (Zitrin et al. 2015).¹⁵ The enclosed projected mass profile

$$\begin{eqnarray}{M}_{2{\rm{D}}}(\lt \theta ) & = & {{\rm{\Sigma }}}_{{\rm{c}}}{D}_{{\rm{l}}}^{2}{\displaystyle \int }_{| {{\boldsymbol{\theta }}}^{\prime }| \leqslant \theta }\kappa ({{\boldsymbol{\theta }}}^{\prime }){d}^{2}{\theta }^{\prime }\\ & = & \pi {({D}_{{\rm{l}}}\theta )}^{2}{{\rm{\Sigma }}}_{{\rm{c}},\infty }{\kappa }_{\infty }(\lt \theta ).\end{eqnarray} \tag{ 13 }$$

at the location θ around θ_Ein is less sensitive to modeling assumptions and approaches (see Coe et al. 2010; Oguri et al. 2012; Umetsu et al. 2012), serving as a fundamental observable quantity in the strong-lensing regime (Coe et al. 2010).

In this work, we use the recent HST results from a joint analysis of CLASH strong- and weak-lensing data presented by Zitrin et al. (2015), who have obtained detailed mass models for each cluster core using two distinct parameterizations (when applicable): One assumes light-traces-mass for both galaxies and DM, while the other adopts an analytical elliptical NFW form for the DM-halo components. Zitrin et al. (2015) performed a detailed comparison of the two models to obtain a realistic, empirical assessment of the true underlying errors, finding that the projected mass enclosed within the critical curves (z_s = 2) agrees typically within ∼15%. Zitrin et al. (2015) recommended replacing the statistical errors of their models with the actual (and much larger) uncertainties that account for model-dependent systematics.

We combine for each cluster the constraints on M_2D(<θ) from two distinct mass models (where available) of Zitrin et al. (2015). To do this, we take a conservative approach that accounts for model-dependent systematic uncertainties. First, at each projected clustercentric distance θ, we improve our estimation of M_2D(<θ) using the average of two different methods as M_2D = (M_Z1 + M_Z2)/2. Next, we combine the 1σ confidence intervals of the two models, by taking the maximum range spanned by their respective confidence limits as an overall total uncertainty, ${\sigma }_{M}=[\mathrm{max}({M}_{{\rm{Z}}1}+{\sigma }_{{\rm{Z}}1},{M}_{{\rm{Z}}2}+{\sigma }_{{\rm{Z}}2})\;$ $-\;\mathrm{min}({M}_{{\rm{Z}}1}-{\sigma }_{{\rm{Z}}1},{M}_{{\rm{Z}}2}-{\sigma }_{{\rm{Z}}2})]/2$. Finally, we rescale the error profile σ_M(θ) such that the fractional uncertainty on M_2D(<θ_Ein) is 15%, as motivated by the findings of Zitrin et al. (2015). For all clusters, this has resulted in increased errors.

Now, the question is how to determine an effective sampling radius Δθ of projected mass constraints M_2D(<θ), avoiding oversampling and reducing correlations between adjacent bins. Following Coe et al. (2010), we estimate here the effective resolution Δθ based on the surface density of observed multiple images as ${N}_{\mathrm{im}}{({\rm{\Delta }}\theta )}^{2}=\pi {\theta }_{\mathrm{Ein}}^{2}$ with N_im the number of multiple images. For our cluster sample, we find a median of ${\bar{N}}_{\mathrm{im}}=17$ multiple images per cluster and a median effective Einstein radius of ${\bar{\theta }}_{\mathrm{Ein}}=22\_\_AMP\_\_farcs;3$ for a fiducial source redshift of z_s = 2 (Table 1), yielding Δθ ≈ 10''. This is consistent with the typical map resolutions in the strong-lensing regime adopted by the SaWLens analysis of Merten et al. (2015, see their Table 5). For the X-ray-selected subsample, we find a median value of ${\bar{\theta }}_{\mathrm{Ein}}=20\_\_AMP\_\_farcs;1$ at z_s = 2. The maximum integration radius is taken to be 40'', which is equal to approximately twice the median Einstein radius (z_s = 2), the region where multiple images form (see Section 3 of Zitrin et al. 2012a).

To summarize, for each cluster except RX J1532.9+3021 for which no secure identification of multiple images has been made (Zitrin et al. 2015), we obtain enclosed projected mass constraints $\{{M}_{2{\rm{D}},i}\}{}_{i=1}^{{N}_{\mathrm{SL}}}\equiv {\{{M}_{2{\rm{D}}}(\lt {\theta }_{i})\}}_{i=1}^{{N}_{\mathrm{SL}}}$ for a set of N_SL = 4 fixed integration radii θ_i = 10'', 20'', 30'', and 40'' from the HST lensing analysis of Zitrin et al. (2015). A summary of the HST lensing mass estimates is given in Table 1.

3.6.1. Lensing Constraints

We consider multiple complementary lensing information available in the cluster regime, namely enclosed projected mass estimates from strong lensing, source-averaged tangential distortion and magnification bias measurements:

$$\begin{eqnarray}&&\{{M}_{2{\rm{D}},i}\}{}_{i=1}^{{N}_{\mathrm{SL}}},{\{\langle {g}_{+,i}\rangle \}}_{i=1}^{{N}_{\mathrm{WL}}},{\{\langle {n}_{\mu ,i}\rangle \}}_{i=1}^{{N}_{\mathrm{WL}}}.\end{eqnarray} \tag{ 14 }$$

Hence, there are a total of ${N}_{\mathrm{GL}}\equiv {N}_{\mathrm{SL}}+2{N}_{\mathrm{WL}}$ independent lensing constraints for each cluster.

Umetsu et al. (2014) measured the shear and magnification effects in N_WL = 10 log-spaced clustercentric radial bins over the range [09, 16'] for all clusters, except [09, 14'] for RX J2248.7−4431 observed with ESO/WFI (Section 3.4). We have ${N}_{\mathrm{SL}}=4$ projected mass estimates (Table 1) from the HST lensing analysis of Zitrin et al. (2015) for all clusters except RX J1532.9+3021 without strong-lensing constraints.

3.6.2. Joint Likelihood Function

In the Bayesian framework of Umetsu (2013, see also Umetsu et al. 2011b), the lensing signal is described by a vector ${\boldsymbol{s}}$ of parameters containing the binned convergence profile ${\{{\kappa }_{\infty ,i}\}}_{i=1}^{N}$ with $N\equiv {N}_{\mathrm{SL}}+{N}_{\mathrm{WL}}$, given by N binned κ values and the average convergence enclosed by the innermost aperture radius θ_min for strong-lensing mass estimates, ${\kappa }_{\infty ,\mathrm{min}}\equiv {\kappa }_{\infty }(\lt {\theta }_{\mathrm{min}})$,¹⁶ so that

$$\begin{eqnarray}&&{\boldsymbol{s}}=\{{\kappa }_{\infty ,\mathrm{min}},{\kappa }_{\infty ,i}\}{}_{i=1}^{N}\equiv {{\rm{\Sigma }}}_{{\rm{c}},\infty }^{-1}{\boldsymbol{\Sigma }}\end{eqnarray} \tag{ 15 }$$

specified by (N + 1) parameters. We have N = 14 for all clusters except N = N_WL = 10 for RX J1532.9+3021 (Section 3.6.1). The number of degrees of freedom (dof) is ${N}_{\mathrm{GL}}-(N+1)={N}_{\mathrm{WL}}-1\quad =\quad 9$ for all clusters.

The joint likelihood function ${{ \mathcal L }}_{\mathrm{GL}}({\boldsymbol{s}})$ for multi-probe lensing observations is given as a product of their separate likelihoods,

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{GL}}={{ \mathcal L }}_{\mathrm{SL}}{{ \mathcal L }}_{\mathrm{WL}}={{ \mathcal L }}_{\mathrm{SL}}{{ \mathcal L }}_{g}{{ \mathcal L }}_{\mu },\end{eqnarray} \tag{ 16 }$$

with ${{ \mathcal L }}_{\mathrm{SL}}$, ${{ \mathcal L }}_{g}$, and ${{ \mathcal L }}_{\mu }$ the likelihood functions for $\{{M}_{2{\rm{D}},i}\}{}_{i=1}^{{N}_{\mathrm{SL}}}$, ${\{\langle {g}_{+,i}\rangle \}}_{i=1}^{{N}_{\mathrm{WL}}}$, and ${\{\langle {n}_{\mu ,i}\rangle \}}_{i=1}^{{N}_{\mathrm{WL}}}$, respectively, defined as

$$\begin{eqnarray}\mathrm{ln}{{ \mathcal L }}_{\mathrm{SL}} & = & -\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{{N}_{\mathrm{SL}}}\displaystyle \frac{{[{M}_{2{\rm{D}},i}-{\hat{M}}_{2{\rm{D}},i}({\boldsymbol{s}})]}^{2}}{{\sigma }_{M,i}^{2}},\\ \mathrm{ln}{{ \mathcal L }}_{g} & = & -\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{{N}_{\mathrm{WL}}}\displaystyle \frac{{[\langle {g}_{+,i}\rangle -{\hat{g}}_{+,i}({\boldsymbol{s}},{\boldsymbol{c}})]}^{2}}{{\sigma }_{+,i}^{2}},\\ \mathrm{ln}{{ \mathcal L }}_{\mu } & = & -\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{{N}_{\mathrm{WL}}}\displaystyle \frac{{[\langle {n}_{\mu ,i}\rangle -{\hat{n}}_{\mu ,i}({\boldsymbol{s}},{\boldsymbol{c}})]}^{2}}{{\sigma }_{\mu ,i}^{2}},\end{eqnarray} \tag{ 17 }$$

with (${\hat{M}}_{2{\rm{D}}},{\hat{g}}_{+},{\hat{n}}_{\mu }$) the theoretical predictions for the corresponding observations (Appendix A) and ${\boldsymbol{c}}$ the calibration nuisance parameters to marginalize over,

$$\begin{eqnarray}&&{\boldsymbol{c}}=\{\langle W{\rangle }_{g},{f}_{W,g},\langle W{\rangle }_{\mu },{\bar{n}}_{\mu },{s}_{\mathrm{eff}}\}.\end{eqnarray} \tag{ 18 }$$

For each parameter of the model ${\boldsymbol{s}}$, we consider a flat uninformative prior with a lower bound of ${\boldsymbol{s}}=0$. Additionally, we account for the calibration uncertainty in the observational parameters ${\boldsymbol{c}}$.

3.6.3. Estimators and Covariance Matrix

We implement our method using a Markov Chain Monte Carlo algorithm with Metropolis–Hastings sampling following the prescription outlined in Umetsu et al. (2011b). The method has been tested (Umetsu 2013) with synthetic weak-lensing catalogs from simulations of analytical NFW lenses performed using the public package glafic (Oguri 2010). The results suggest that, when the mass-sheet degeneracy is broken, both maximum-likelihood (ML) and marginal maximum a posteriori probability (MMAP) solutions provide reliable reconstructions with unbiased profile measurements. Hence, this method is not sensitive to the choice and form of priors. In the presence of a systematic bias in the background-density constraint (${\bar{n}}_{\mu }$), the global ML estimator is less sensitive to systematic effects than MMAP, and provides more accurate reconstructions (Umetsu et al. 2014).

On the basis of our simulations, we thus use the global ML estimator for the determination of the mass profile. In our error analysis we take into account statistical, systematic, cosmic-noise, and intrinsic-variance contributions to the total covariance matrix ${C}_{{ij}}\equiv \mathrm{Cov}({s}_{i},{s}_{j})$ for ${\kappa }_{\infty }={\rm{\Sigma }}/{{\rm{\Sigma }}}_{{\rm{c}},\infty }$ as

$$\begin{eqnarray}&&C={C}^{\mathrm{stat}}+{C}^{\mathrm{sys}}+{C}^{\mathrm{lss}}+{C}^{\mathrm{int}},\end{eqnarray} \tag{ 19 }$$

where C^stat is the posterior covariance matrix that is derived from the data (Equations (16) and (17)) by calculating the sample covariance matrix ${C}_{{ij}}^{\mathrm{stat}}=\langle ({s}_{i}-\langle {s}_{i}\rangle )({s}_{j}-\langle {s}_{j}\rangle )\rangle$ using MCMC-sampled posterior distributions; C^sys accounts for systematic errors due primarily to the mass-sheet uncertainty (Umetsu et al. 2014),

$$\begin{eqnarray}&&{({C}^{\mathrm{sys}})}_{{ij}}=({{\boldsymbol{s}}}_{\mathrm{ML}}-{{\boldsymbol{s}}}_{\mathrm{MMAP}}{)}_{i}^{2}{\delta }_{{ij}},\end{eqnarray} \tag{ 20 }$$

with ${{\boldsymbol{s}}}_{\mathrm{ML}}$ and ${{\boldsymbol{s}}}_{\mathrm{MMAP}}$ the ML and MMAP solutions, respectively; C^lss is due to uncorrelated large-scale structure (LSS) projected along the line of sight (Hoekstra 2003; Umetsu et al. 2011a); C^int accounts for the intrinsic variations of the projected cluster mass profile (Gruen et al. 2015).

The cosmic-noise covariance due to projected uncorrelated LSS is given as (Umetsu et al. 2011a)

$$\begin{eqnarray}&&{({C}^{\mathrm{lss}})}_{{ij}}=\int \displaystyle \frac{{ldl}}{2\pi }\;{C}_{({ij})}^{\kappa \kappa }{\hat{J}}_{0}(l{\theta }_{i}){\hat{J}}_{0}(l{\theta }_{j}),\end{eqnarray} \tag{ 21 }$$

where ${C}_{({ij})}^{\kappa \kappa }$ is the weak-lensing cross power spectrum as a function of angular multipole l evaluated for a given pair of source populations in the ith and jth radial bins; ${\hat{J}}_{0}(/{\theta }_{i})$ is the Bessel function of the first kind and order zero (J₀) averaged over the ith annulus between θ_i,1 and θ_i,2 (>θ_i,1), given as (Umetsu et al. 2011a)

$$\begin{eqnarray}&&{\hat{J}}_{0}(l{\theta }_{i})=\displaystyle \frac{2}{{(l{\theta }_{i,2})}^{2}-{(l{\theta }_{i,1})}^{2}}[l{\theta }_{i,2}{J}_{1}(l{\theta }_{i,2})-l{\theta }_{i,1}{J}_{1}(l{\theta }_{i,1})].\end{eqnarray} \tag{ 22 }$$

We compute the elements of the C^lss matrix for a given pair of source populations, using the nonlinear matter power spectrum of Smith et al. (2003) for the Wilkinson Microwave Anisotropy Probe (WMAP) seven-year cosmology (Komatsu et al. 2011), and then scale to the reference far-background source plane. For the inner radial bins constrained by HST strong lensing, we assume a source plane at z_s = 2, which is a typical redshift of strongly lensed background galaxies (Zitrin et al. 2015). For the outer weak-lensing bins, we use for each cluster the effective mean source redshift (${\bar{z}}_{\mathrm{eff}}$) estimated with multi-band photometric redshifts (see Table 3 of Umetsu et al. 2014). The cross covariance terms between the strong- and weak-lensing bins are computed using the lensing window functions of the respective populations (Takada & White 2004). For a given depth of observations, the impact of cosmic noise is most important where the cluster signal itself is small (Hoekstra 2003).

The intrinsic covariance matrix C^int accounts for the intrinsic variations of the projected cluster density profile (i.e., cluster signal itself) due to the c–M variance, halo asphericity, and the presence of correlated halos (Gruen et al. 2015).¹⁷ In the present work, we consider the diagonal form of the C^int matrix,

$$\begin{eqnarray}&&{({C}^{\mathrm{int}})}_{{ij}}={\alpha }_{\mathrm{int}}^{2}\times ({{\boldsymbol{s}}}_{\mathrm{ML}}{)}_{i}^{2}{\delta }_{{ij}},\end{eqnarray} \tag{ 23 }$$

where the coefficient ${\alpha }_{\mathrm{int}}\approx \sqrt{{C}_{{ii}}^{\mathrm{int}}}/{\kappa }_{\infty ,i}$ represents the fractional intrinsic scatter in κ. On the basis of semi-analytical calculations calibrated by cosmological numerical simulations (Gruen et al. 2015), we find that, for CLASH clusters with a characteristic mass of ${M}_{200{\rm{c}}}\approx {10}^{15}\;{M}_{\odot }{h}^{-1}$ (Umetsu et al. 2014), the diagonal part of the C^int matrix can be well approximated by Equation (23) with α_int ≈ 0.2 in the one-halo regime at $R\lesssim {r}_{200{\rm{m}}}$. In general, the diagonal approximation to C^int can lead to an underestimate of cluster parameters (see Gruen et al. 2015), where the degree of underestimation depends on the radial binning scheme for cluster lensing measurements.

We have tested the validity of this approximation in our radial binning scheme by comparing against the semi-analytical model of Gruen et al. (2015). For a cluster halo of ${M}_{200{\rm{c}}}={10}^{15}\;{M}_{\odot }{h}^{-1}$ at z_l = 0.35 (${M}_{200{\rm{m}}}\simeq 13\times {10}^{14}\;{M}_{\odot }{h}^{-1}$) matching approximately the average characteristics of the CLASH sample (Umetsu et al. 2014), we find that the total S/N estimated using the diagonal approximation with α_int = 0.2 is accurate to about 10% in the regime of our cluster lensing observations (${\rm{S}}/{\rm{N}}\sim 10$ per cluster), where the S/N is defined as (Umetsu & Broadhurst 2008)¹⁸

$$\begin{eqnarray}&&{({\rm{S}}/{\rm{N}})}^{2}=\displaystyle \sum _{i,j}\;{s}_{i}{C}_{{ij}}^{-1}{s}_{j}={{\boldsymbol{s}}}^{t}{C}^{-1}{\boldsymbol{s}},\end{eqnarray} \tag{ 24 }$$

with C the total covariance matrix defined in Equation (19). In the present study, we thus adopt ${\alpha }_{\mathrm{int}}=0.2$ in Equation (23) to account for the effects of the intrinsic profile variations in projection space.¹⁹

Following the methodology outlined in Section 3, we analyze our weak- and strong-lensing data presented in Umetsu et al. (2014) and Zitrin et al. (2015) to examine the underlying radial mass distribution for a sample of 20 CLASH clusters (Table 1). We have derived for each cluster a mass-profile solution ${\boldsymbol{\Sigma }}\equiv {{\rm{\Sigma }}}_{{\rm{c}},\infty }{\boldsymbol{s}}={\{{{\rm{\Sigma }}}_{\mathrm{min}},{{\rm{\Sigma }}}_{i}\}}_{i=1}^{N}$ from a joint likelihood analysis of our strong-lensing, weak-lensing shear and magnification data. We find that the minimum ${\chi }^{2}(=-2\mathrm{ln}{{ \mathcal L }}_{\mathrm{GL}})$ values for the best-fit Σ solutions range from χ² = 2.5 (Abell 611) to 14.8 (MACS J0429.6−0253) for 9 dof (a mean reduced χ² of 0.95), indicating good consistency between independent observations having different systematics.

In Appendix B (Figure 11) we show the resulting mass-profile solutions, ${\boldsymbol{\Sigma }}$ (black squares), obtained for our sample along with those by Umetsu et al. (2014, blue circles) and Merten et al. (2015, red dots). Umetsu et al. (2014) derived weak-lensing-only solutions for this sample from a joint likelihood analysis of the shear and magnification measurements. When the inner strong-lensing information (Table 1) is combined with wide-field weak-lensing data, the central weak-lensing bin Σ(<09) (Section 3.6.1) is resolved into (${N}_{\mathrm{SL}}+1$) radial bins, hence improving the determination of the inner mass profile. Merten et al. (2015) performed a two-dimensional SaWLens analysis of 19 CLASH X-ray-selected clusters, by combining the CLASH HST data with the shear catalogs of Umetsu et al. (2014). To break the mass-sheet degeneracy, Merten et al. (2015) assumed outer boundary conditions such that the average convergence vanishes at the edge of the reconstruction. This is a reasonable approximation in wide-field weak-lensing observations entailing the full cluster field well beyond its virial radius, r_vir.

We find overall good agreement between different reconstructions, except for a few systems in the overlap sample, such as MACS J1931.8−2635, RX J1347.5−1145, and MACS J0744.9+3927. For these clusters, the SaWLens reconstructions are systematically lower than those of this work and Umetsu et al. (2014) which include the weak-lensing magnification data. Here MACS J1931.8−2635 (b = −2009) and MACS J0744.9+3927 (b = +2665) are the two lowest Galactic latitude clusters of the overlap sample, implying higher stellar densities, correspondingly large areas masked by bright saturated stars, and hence lower number densities of background galaxies usable for weak lensing (Umetsu et al. 2014, see their Figure 2, Tables 3 and 4). In fact, MACS J1931.8−2635 has the lowest S/N of the weak-lensing observations presented in Umetsu et al. (2014, Table 5). On the other hand, RX J1347.5−1145 (z = 0.451) and MACS J0744.9+3927 (z = 0.686) represent the two highest-z clusters of the overlap sample, both of which exhibit complex mass distributions with a high degree of substructure (see Postman et al. 2012; Merten et al. 2015). For clusters at lower Galactic latitudes and higher redshifts, the color selection of background galaxies is correspondingly more difficult. Therefore, this discrepancy can be attributed in part to systematic uncertainties in the present calibration of magnification measurements for these low Galactic latitude clusters and high-redshift clusters. Weak-lensing mass reconstructions are sensitive to the treatment of boundary conditions if there are massive structures near the data boundaries. Hence, mass profile reconstructions for clusters with high degrees of substructure can be subject to a greater degree of mass-sheet degeneracy. A more quantitative comparison with the SaWLens results from Merten et al. (2015) can be found in Sections 7.3 and 7.4.

Figure 1 shows the average contributions to the total covariance matrix C (Equation (19)) for the convergence profile κ(θ) reconstructed from the joint likelihood analysis of strong-lensing, weak-lensing shear and magnification data. The results are obtained by averaging over 18 clusters observed with both HST and Subaru (i.e., all except RX J1532.9+3021 without strong-lensing constraints and RX J2248.7−4431 based on ESO/WFI data). The diagonal variance terms Var(κ), scaled to the mean depth of Subaru weak-lensing observations, are shown as a function of clustercentric radius θ. At all bins except the innermost bin, the reconstruction uncertainty is dominated by the observational statistical errors (C^stat, red dashed). The relative contribution from intrinsic variance (C^int, gray solid) increases toward the cluster center and becomes important at small cluster radii, $\theta \lesssim 2^{\prime}$. The C^sys term (green dotted) represents the level of residual variance (Equation (20)) due primarily to the mass-sheet degeneracy. In the regime of Subaru weak-lensing (θ ≥ 09), C^sys stays approximately constant at $\sim (2-4)\times {10}^{-4}$ with θ, corresponding to a characteristic mass-sheet uncertainty of ${\sigma }_{\kappa }\sim (1-2)\times {10}^{-2}$ per cluster. A noticeable increase of the cosmic-noise contribution C^lss (blue dotted–dashed) from projected uncorrelated LSS is seen at $\theta \leqslant 40^{\prime\prime}$, within which the reconstruction is dominated by HST strong-lensing measurements with greater depth.

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In Figure 2, we show the cross-correlation coefficients of the ensemble-averaged C^stat and C^lss matrices, whose diagonal elements are shown in Figure 1. We find that adjacent HST bins of the $\langle {C}^{\mathrm{stat}}\rangle$ matrix are anti-correlated at around the 40% level, where the first 4 bins in each panel of Figure 2 correspond to the HST data. This negative covariance arises because they are to satisfy the observed cumulative mass constraints (Equation (13)). We find small correlations of <1% between the HST ($10^{\prime\prime} \leqslant \theta \leqslant 40^{\prime\prime}$) and Subaru ($0\_\_AMP\_\_farcm;9\leqslant \theta \leqslant 16^{\prime}$) radial bins. For the $\langle {C}^{\mathrm{lss}}\rangle$ matrix, since strongly lensed source galaxies at ${z}_{{\rm{s}}}\sim 2$ and weakly lensed source galaxies at z_s ∼ 1 behind a given cluster share the same mass overdensities at $z\lesssim 1$ where the geometrical lensing efficiency for the ${z}_{{\rm{s}}}\sim 2$ sources is large, there are large positive correlations at the ∼70%–20% levels between the HST and Subaru bins, where the degree of correlation increases with decreasing projected separation.

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Tangential shear fitting with a spherical NFW profile is a standard approach for measuring individual cluster masses from weak lensing (Umetsu et al. 2009; Okabe et al. 2010; Applegate et al. 2014). Numerical simulations suggest that these mass estimates tend to be biased low (by ∼5%–10%; see Meneghetti et al. 2010; Becker & Kravtsov 2011; Rasia et al. 2012) because local substructures that are abundant in cluster outskirts dilute the shear tangential to the cluster center. This bias can be reduced if the fitting range is restricted to within $\sim 2{r}_{500{\rm{c}}}\sim {r}_{\mathrm{vir}}$ (Becker & Kravtsov 2011). In the CLASH survey, the availability of multi-probe, multi-scale lensing data allows us to combine weak shear lensing with magnification and/or strong-lensing constraints (Umetsu et al. 2014; Merten et al. 2015). In particular, the complementary combination of shear and magnification data enables to effectively break the mass-sheet degeneracy and reconstruct the total mass distribution Σ (Umetsu et al. 2011b).

Here we revisit the mass estimates for CLASH clusters using our improved mass profile data set (Figure 11) and taking into account the intrinsic contribution (C^int) to the error covariance matrix (Section 3.6.3). We follow the Bayesian approach of Umetsu et al. (2014) to make inference on the NFW halo parameters. We employ the radial dependence of the projected NFW profiles given by Wright & Brainerd (2000), which provides a good description of the projected mass distribution in the one-halo regime, at least in an ensemble-average sense (Oguri & Hamana 2011; Umetsu et al. 2011a; Okabe et al. 2013). We restrict the fitting range to $R\leqslant 2$ Mpc h⁻¹ (Umetsu et al. 2014; Merten et al. 2015), which is close to the virial radius for most CLASH clusters. We specify the two-parameter NFW model using the halo mass M_200c and the halo concentration c_200c = r_200c/r₋₂ with r₋₂ the characteristic radius at which the logarithmic density slope is −2 (Section 5.2.1). We adopt uninformative log-uniform priors in the respective intervals $0.1\leqslant {M}_{200{\rm{c}}}/({10}^{15}\;{M}_{\odot }\;{h}^{-1})\leqslant 10$ and $0.1\leqslant {c}_{200{\rm{c}}}\leqslant 10$ (Umetsu et al. 2014). We check that the mass and concentration estimates for the CLASH sample are not sensitive to the choice of the priors as found by Sereno et al. (2015b, see their Section 2.1). The χ² function for our observations is

$$\begin{eqnarray}&&{\chi }^{2}({\boldsymbol{p}})=\displaystyle \sum _{i,j}\;[{s}_{i}-{\hat{s}}_{i}({\boldsymbol{p}})]{C}_{{ij}}^{-1}[{s}_{j}-{\hat{s}}_{j}({\boldsymbol{p}})],\end{eqnarray} \tag{ 25 }$$

where ${\boldsymbol{p}}=({M}_{200{\rm{c}}},{c}_{200{\rm{c}}})$, and ${\hat{s}}_{i}({\boldsymbol{p}})={\hat{{\rm{\Sigma }}}}_{i}({\boldsymbol{p}})/{{\rm{\Sigma }}}_{{\rm{c}},\infty }$ is the predicted surface mass density averaged over the ith annulus, accounting for the effect of bin averaging (Umetsu et al. 2014).

In Table 2, we give marginalized constraints on the NFW halo parameters (M_200c, c_200c) and the characteristic radius r₋₂. Throughout this paper, we employ the biweight estimators of Beers et al. (1990) for the central location (C_BI) and scale (S_BI) of the marginalized posterior distributions (Sereno & Umetsu 2011; Umetsu et al. 2014, 2015). From the posterior samples, we also derive marginalized constraints on the total enclosed mass M_Δ = M_3D (<r_Δ) at several characteristic interior overdensities Δ (see Section 1). Table 3 summarizes the results of our cluster mass estimates. The median precision on our lensing mass measurements is found to be ∼28% at Δ_c = 200, ∼24% at Δ_c = 500, ∼23% at Δ_c = 1000, and ∼24% at Δ_c = 2500.

Table 2.  NFW Halo Parameters for Individual CLASH Clusters

Cluster	${M}_{200{\rm{c}}}$	${c}_{200{\rm{c}}}$	r−2
	(${10}^{14}\;{M}_{\odot }\;{h}_{70}^{-1}$)		(Mpc ${h}_{70}^{-1}$)
X-ray Selected:
Abell 383	7.98 ± 2.66	5.9 ± 1.8	0.31 ± 0.13
Abell 209	15.40 ± 3.42	2.7 ± 0.6	0.84 ± 0.22
Abell 2261	23.10 ± 5.22	3.7 ± 0.9	0.69 ± 0.20
RX J2129.7+0005	6.14 ± 1.79	5.6 ± 1.6	0.30 ± 0.11
Abell 611	15.76 ± 4.49	3.9 ± 1.2	0.57 ± 0.21
MS2137-2353	13.56 ± 5.27	2.7 ± 1.3	0.80 ± 0.45
RX J2248.7-4431	18.78 ± 6.72	3.6 ± 1.4	0.66 ± 0.32
MACS J1115.9+0129	16.66 ± 3.85	3.0 ± 0.8	0.75 ± 0.23
MACS J1931.8-2635	15.28 ± 7.13	4.4 ± 1.9	0.51 ± 0.30
RX J1532.9+3021	5.98 ± 2.32	5.2 ± 2.8	0.29 ± 0.18
MACS J1720.3+3536	14.50 ± 4.30	4.1 ± 1.3	0.51 ± 0.20
MACS J0429.6-0253	9.76 ± 3.50	4.6 ± 1.7	0.40 ± 0.19
MACS J1206.2-0847	18.17 ± 4.23	3.7 ± 1.1	0.60 ± 0.21
MACS J0329.7-0211	8.65 ± 1.97	6.7 ± 1.6	0.26 ± 0.08
RX J1347.5-1145	34.25 ± 8.78	3.2 ± 0.9	0.85 ± 0.29
MACS J0744.9+3927	18.03 ± 4.96	3.5 ± 1.2	0.58 ± 0.23
High Magnification:
MACS J0416.1-2403	10.74 ± 2.60	2.9 ± 0.7	0.65 ± 0.18
MACS J1149.5+2223	25.02 ± 5.53	2.1 ± 0.6	1.12 ± 0.35
MACS J0717.5+3745	26.77 ± 5.36	1.8 ± 0.4	1.31 ± 0.31
MACS J0647.7+7015	13.90 ± 4.20	4.1 ± 1.5	0.48 ± 0.21

Note. Cluster parameters derived from single spherical NFW fits to individual surface mass density profiles (Figure 11) reconstructed from combined strong-lensing, weak-lensing shear and magnification measurements. We adopt a concordance cosmology of h = 0.7, ${{\rm{\Omega }}}_{{\rm{m}}}=0.27$, and ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.73$. The fitting radial range is restricted to $R\leqslant 2$ Mpc ${h}^{-1}\simeq 2.9$ Mpc ${h}_{70}^{-1}$.

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Table 3.  Mass Estimates for Individual CLASH Clusters

Cluster	${M}_{2500{\rm{c}}}$	${M}_{1000{\rm{c}}}$	${M}_{500{\rm{c}}}$	Mvira	${M}_{100{\rm{c}}}$	${M}_{200{\rm{m}}}$	$M(\lt 1.5\;\mathrm{Mpc})$
	(${10}^{14}\;{M}_{\odot }$)	(${10}^{14}\;{M}_{\odot }$)	(${10}^{14}\;{M}_{\odot }$)	(${10}^{14}\;{M}_{\odot }$)	(${10}^{14}\;{M}_{\odot }$)	(${10}^{14}\;{M}_{\odot }$)	(${10}^{14}\;{M}_{\odot }$)
X-ray Selected:
Abell 383	2.78 ± 0.63	4.43 ± 1.16	5.88 ± 1.73	9.41 ± 3.33	9.66 ± 3.45	10.34 ± 3.78	6.95 ± 1.63
Abell 209	2.95 ± 0.68	6.18 ± 1.25	9.64 ± 1.97	19.60 ± 4.61	20.49 ± 4.88	22.36 ± 5.44	10.28 ± 1.37
Abell 2261	5.91 ± 1.03	10.85 ± 1.89	15.65 ± 3.05	28.21 ± 6.87	29.39 ± 7.26	31.41 ± 7.94	14.22 ± 1.79
RX J2129.7+0005	2.08 ± 0.44	3.35 ± 0.78	4.48 ± 1.16	7.21 ± 2.23	7.48 ± 2.34	7.87 ± 2.51	5.75 ± 1.23
Abell 611	4.13 ± 0.92	7.48 ± 1.67	10.73 ± 2.65	18.95 ± 5.77	19.99 ± 6.21	20.78 ± 6.54	11.24 ± 1.95
MS2137-2353	2.47 ± 0.72	5.21 ± 1.41	8.28 ± 2.57	16.98 ± 7.31	18.25 ± 8.12	18.91 ± 8.54	9.67 ± 2.12
RX J2248.7-4431	4.45 ± 1.05	8.42 ± 2.08	12.45 ± 3.62	22.54 ± 8.78	24.12 ± 9.68	24.53 ± 9.91	12.67 ± 2.50
MACS J1115.9+0129	3.50 ± 0.82	7.02 ± 1.43	10.67 ± 2.22	20.30 ± 4.97	21.88 ± 5.48	22.24 ± 5.60	11.55 ± 1.61
MACS J1931.8-2635	4.10 ± 1.12	7.36 ± 2.36	10.51 ± 4.05	18.02 ± 9.05	19.18 ± 9.88	19.44 ± 10.07	11.23 ± 3.07
RX J1532.9+3021	1.83 ± 1.01	3.03 ± 1.38	4.17 ± 1.71	7.04 ± 2.79	7.51 ± 3.02	7.58 ± 3.06	5.81 ± 1.63
MACS J1720.3+3536	3.92 ± 0.86	7.00 ± 1.59	9.96 ± 2.53	17.04 ± 5.39	18.29 ± 5.94	18.30 ± 5.94	11.02 ± 2.05
MACS J0429.6-0253	2.85 ± 0.70	4.92 ± 1.32	6.85 ± 2.10	11.35 ± 4.33	12.15 ± 4.76	12.13 ± 4.74	8.40 ± 2.03
MACS J1206.2-0847	4.62 ± 1.01	8.47 ± 1.63	12.24 ± 2.49	21.35 ± 5.29	23.18 ± 5.94	22.82 ± 5.81	12.98 ± 1.84
MACS J0329.7-0211	3.29 ± 0.64	5.02 ± 1.00	6.51 ± 1.37	9.72 ± 2.30	10.35 ± 2.50	10.21 ± 2.45	7.94 ± 1.36
RX J1347.5-1145	7.65 ± 1.63	14.91 ± 2.98	22.33 ± 4.89	40.66 ± 11.13	44.50 ± 12.60	43.59 ± 12.25	19.33 ± 2.61
MACS J0744.9+3927	4.32 ± 1.02	8.12 ± 1.76	11.94 ± 2.81	20.57 ± 5.98	23.21 ± 7.08	21.32 ± 6.29	13.96 ± 2.39
High Magnification:
MACS J0416.1-2403	2.21 ± 0.53	4.48 ± 0.97	6.85 ± 1.52	12.99 ± 3.29	14.13 ± 3.66	14.12 ± 3.66	8.74 ± 1.37
MACS J1149.5+2223	3.73 ± 1.11	8.74 ± 1.98	14.57 ± 3.06	30.45 ± 7.06	34.72 ± 8.36	32.62 ± 7.71	15.31 ± 1.95
MACS J0717.5+3745	3.42 ± 0.87	8.61 ± 1.74	14.98 ± 2.85	32.97 ± 6.88	37.95 ± 8.17	35.46 ± 7.52	15.63 ± 1.75
MACS J0647.7+7015	3.72 ± 0.98	6.65 ± 1.65	9.49 ± 2.51	15.91 ± 5.07	17.59 ± 5.83	16.64 ± 5.40	11.37 ± 2.26

Notes. Cluster mass estimates ${M}_{3{\rm{D}}}(\lt r)$ from single spherical NFW fits to individual surface mass density profiles (Figure 11). All quantities in the table are given in physical units assuming a concordance cosmology of h = 0.7, ${{\rm{\Omega }}}_{{\rm{m}}}=0.27$, and ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.73$. See Table 2 for ${M}_{200{\rm{c}}}$. The fitting radial range is restricted to $R\ \leqslant 2$ Mpc ${h}^{-1}\simeq 2.9$ Mpc ${h}_{70}^{-1}$.

^aVirial overdensity ${{\rm{\Delta }}}_{{\rm{vir}}}$ based on the spherical collapse model (see Appendix A of Kitayama & Suto 1996). For our redshift range $0.187\leqslant z\leqslant 0.686$, ${{\rm{\Delta }}}_{{\rm{vir}}}$ ranges approximately from $\simeq 110$ to 140 with respect to the critical density of the universe at the cluster redshift.

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In the CLASH weak-lensing study of Umetsu et al. (2014), the C^int contribution (Equation (23)) to the total covariance matrix (Equation (19)) was not included in their individual cluster mas measurements at Δ_c ≤ 500. Accordingly, their individual mass measurement errors were underestimated by ∼20% at Δ_c = 200, on average (see also Figure 5 of Gruen et al. 2015), even though C^int is less important at larger cluster radii (Figure 1).

On the other hand, the stacked cluster measurements of Umetsu et al. (2014) empirically account for the cluster-to-cluster profile variations from bootstrap resampling of the cluster sample.

With a given set of surface mass density profiles for individual clusters, we can stack them together to produce an ensemble-averaged radial profile. Following the prescription of Umetsu et al. (2011a), we re-evaluate the ${\boldsymbol{\Sigma }}$ profiles of individual clusters in physical (proper) length units, using the same radial grid for all clusters. Stacking an ensemble of clusters (n = 1, 2, ...) is expressed as (Umetsu et al. 2011a)

$$\begin{eqnarray}&&\langle \langle {\boldsymbol{\Sigma }}\rangle \rangle ={\left(\displaystyle \sum _{n}{{ \mathcal W }}_{n}\right)}^{-1}\left(\displaystyle \sum _{n}\;{{ \mathcal W }}_{n}{{\boldsymbol{\Sigma }}}_{n}\right),\end{eqnarray} \tag{ 26 }$$

where ${{ \mathcal W }}_{n}$ is the sensitivity matrix of the nth cluster,

$$\begin{eqnarray}&&{({{ \mathcal W }}_{n})}_{{ij}}\equiv {{\rm{\Sigma }}}_{({\rm{c}},\infty )n}^{-2}{({C}_{n}^{-1})}_{{ij}},\end{eqnarray} \tag{ 27 }$$

with ${{\rm{\Sigma }}}_{({\rm{c}},\infty ),n}$ the far-background critical surface mass density and C_n the total covariance matrix (Equation (19)) for the nth cluster.²¹ The error covariance matrix for the stacked $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile is given by (Umetsu et al. 2011a)

$$\begin{eqnarray}&&{ \mathcal C }={\left(\displaystyle \sum _{n}{{ \mathcal W }}_{n}\right)}^{-1}.\end{eqnarray} \tag{ 28 }$$

The weight matrix ${{ \mathcal W }}_{n}$ is mass-independent when stacking in physical length units (Okabe et al. 2010, 2013; Umetsu et al. 2011a, 2014), which makes the effective halo mass extracted from the averaged lensing signal a good proxy for the mean population value (see Section 5.4; Johnston et al. 2007; Okabe et al. 2013; Umetsu et al. 2014; Sereno et al. 2015c). On the other hand, stacking in length units scaled to r_Δ weights the contribution of each cluster to each radial bin in a nonlinear and model-dependent manner (Okabe et al. 2013), such that $\mathrm{tr}({ \mathcal W })\propto {r}_{{\rm{\Delta }}}^{2}\propto {M}_{{\rm{\Delta }}}^{2/3}$ when C is dominated by the statistical noise contribution C^stat.

In Figure 3 we show the resulting $\langle \langle {\boldsymbol{\Sigma }}\rangle \rangle$ profile averaged in 13 radial bins. The innermost bin represents the mean density interior to R_min = 40 kpc h⁻¹, which corresponds approximately to the typical resolution limit of our strong-lensing data (θ_min = 10''; Section 3.6.2 and Figure 11). Since R_min is much larger than the rms offset between the BCG and X-ray peak, σ_off ≃ 11 kpc h⁻¹ (Section 3.1), the miscentering effects on the $\langle \langle {\boldsymbol{\Sigma }}\rangle \rangle$ profile are expected to be insignificant for the X-ray-selected subsample (Umetsu et al. 2014). The other bins are logarithmically spaced over the range R = [R_min, R_max] = [40,4000] kpc h⁻¹ ($0.02\lesssim R/{r}_{200{\rm{m}}}\lesssim 2$), spanning two decades in radius. For this sample, we find a sensitivity-weighted average redshift of $\langle \langle {z}_{{\rm{l}}}\rangle \rangle \simeq 0.34$, in close agreement with the median redshift of ${\bar{z}}_{{\rm{l}}}\simeq 0.35$. We detect the stacked lensing signal at a total S/N of ≃33 using the total covariance matrix ${ \mathcal C }$ including the statistical, systematic, projected uncorrelated LSS, and intrinsic-variance contributions (Section 3.6.3).

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We quantify and characterize the ensemble-averaged mass distribution of our X-ray-selected subsample using the $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile (Section 5.1). To interpret the observed averaged lensing signal, we consider the line of sight projected surface mass density around the cluster center, ${\rm{\Sigma }}(R)=\int {dl}\;{\rm{\Delta }}\rho (r)$, with ${\rm{\Delta }}\rho (r)=\rho (r)-{\bar{\rho }}_{{\rm{m}}}$ the mass overdensity. In the regime where $R\lesssim {r}_{200{\rm{m}}}$, ${\rm{\Sigma }}(R)$ is dominated by the cluster halo contribution ρ_h(r), so that ${\rm{\Sigma }}(R)\simeq 2{\int }_{0}^{\infty }{dl}\;{\rho }_{{\rm{h}}}(r)$.

As shown in Figure 3, our weak- and strong-lensing data together cover a wide range of clustercentric distances R, extending out to R_max = 4000 kpc ${h}^{-1}\approx 2{r}_{200{\rm{m}}}$. In the context of the ΛCDM model, the two outermost radial bins lie in the transition between the one-halo and two-halo regimes (Cooray & Sheth 2002), ${r}_{200{\rm{m}}}\lesssim R\lesssim 2{R}_{200{\rm{m}}}$, where the large-scale two-halo contribution to Σ(R) is expected to become important (Oguri & Hamana 2011; Beraldo e Silva et al. 2013; Umetsu et al. 2014).²² In this work, we thus test models both with and without including the two-halo term.

5.2.1. Halo Density Profiles

5.2.2. Projected Halo Model

To interpret the averaged lensing signal in the context of the standard ΛCDM model, we employ as our reference model the halo model prescription of Oguri & Takada (2011). Specifically, we take into account the large-scale clustering contribution ${\rho }_{2{\rm{h}}}(r)$ as

$$\begin{eqnarray}{\rm{\Delta }}\rho (r) & = & {f}_{{\rm{t}}}(r)\;{\rho }_{{\rm{h}}}(r)+{\rho }_{2{\rm{h}}}(r),\\ {f}_{{\rm{t}}}(r) & = & {\left[1+{\left(\displaystyle \frac{r}{{r}_{{\rm{t}}}}\right)}^{2}\right]}^{-2},\end{eqnarray} \tag{ 35 }$$

where f_t(r) describes the steepening of the density profile around a truncation radius r_t (Baltz et al. 2009; Diemer & Kravtsov 2014). We fix the truncation parameter ${\tau }_{200{\rm{c}}}\equiv {r}_{{\rm{t}}}/{r}_{200{\rm{c}}}=3$ (Covone et al. 2014; Umetsu et al. 2014; Sereno et al. 2015c), which is a typical value for cluster-sized halos in ΛCDM cosmology (Oguri & Hamana 2011). When the two-halo term is neglected, the standard NFW halo model (Oguri & Hamana 2011) reduces to the Baltz–Marshall–Oguri (BMO) model that describes a truncated NFW profile, ${f}_{{\rm{t}}}(r){\rho }_{\mathrm{NFW}}(r)$ (Baltz et al. 2009). For an ensemble of clusters with mass M and redshift z, the two-halo term is expressed as ${\rho }_{2{\rm{h}}}(r)={\bar{\rho }}_{{\rm{m}}}{b}_{{\rm{h}}}(M){\xi }_{{\rm{m}}}^{{\rm{L}}}(r)$ with ${\bar{\rho }}_{{\rm{m}}}$ the mean background density of the universe, b_h(M) the linear halo bias, and ${\xi }_{{\rm{m}}}^{{\rm{L}}}(r)$ the linear matter correlation function, all evaluated at $z=\langle \langle {z}_{{\rm{l}}}\rangle \rangle \simeq 0.34$ in the WMAP seven-year cosmology (Section 3.6.3). The two-halo term is proportional to the product ${b}_{{\rm{h}}}{\sigma }_{8}^{2}$, where σ₈ is the rms amplitude of linear mass fluctuations in a sphere of comoving radius 8 Mpc h⁻¹. In the adopted cosmology, σ₈ = 0.81 (Komatsu et al. 2011). We compute the total surface mass density Σ(R) by projecting ${\rm{\Delta }}\rho (r)={f}_{{\rm{t}}}{\rho }_{1{\rm{h}}}+{\rho }_{2{\rm{h}}}$ along the line of sight (Section 2.2 of Oguri & Hamana 2011). To evaluate the halo bias factor b_h(M), we adopt the model of Tinker et al. (2010), which is well calibrated using a large set of N-body simulations. We use their fitting formula with a halo mass definition of Δ_c = 200 (Umetsu et al. 2014), corresponding to Δ_m ≃ 420 in our adopted cosmology.

We constrain model parameters using the $\langle \langle {\boldsymbol{\Sigma }}\rangle \rangle$ profile and its total covariance matrix ${ \mathcal C }$ (Section 5.1). The ${\chi }^{2}$ minimization is performed using the minuit minimization package from the CERN program libraries. The best-fit parameters are reported in Table 4, along with the reduced χ² and corresponding probability to exceed (PTE, hereafter) values.

Table 4.  Best-fit Models for the Stacked Mass Profile of the CLASH X-Ray-selected Subsample

Model	${M}_{200{\rm{c}}}$	${c}_{200{\rm{c}}}$	Shape/Structural Parameters	bh	${\chi }^{2}/\mathrm{dof}$	PTEa	Notes
	(${10}^{14}\;{M}_{\odot }\;{h}_{70}^{-1}$)
NFW	${14.4}_{-1.0}^{+1.1}$	${3.76}_{-0.27}^{+0.29}$	${\gamma }_{{\rm{c}}}=1$	⋯	$11.3/11$	0.419	No truncation
gNFW	${14.1}_{-1.1}^{+1.1}$	${4.04}_{-0.52}^{+0.53}$	${\gamma }_{{\rm{c}}}={0.85}_{-0.31}^{+0.22}$	⋯	$10.9/10$	0.366	No truncation
Einasto	${14.7}_{-1.1}^{+1.1}$	${3.53}_{-0.39}^{+0.36}$	${\alpha }_{{\rm{E}}}={0.232}_{-0.038}^{+0.042}$	⋯	$11.7/10$	0.306	No truncation
DARKexp-γb	${14.5}_{-1.1}^{+1.2}$	${3.53}_{-0.42}^{+0.42}$	${\phi }_{0}={3.90}_{-0.45}^{+0.41}$	⋯	$13.5/10$	0.198	No truncation
Pseudo isothermal	⋯	⋯	${V}_{{\rm{c}}}={1762}_{-39}^{+40}$ km s−1, ${r}_{{\rm{c}}}={69}_{-7}^{+7}$ kpc	⋯	$23.6/11$	0.015	No truncation
Burkert	${11.6}_{-0.8}^{+0.8}$	⋯	${r}_{200{\rm{c}}}/{r}_{0}={8.81}_{-0.41}^{+0.42}$	⋯	$29.9/11$	0.002	No truncation
Power-law sphere	${12.5}_{-0.8}^{+0.8}$	⋯	${\gamma }_{{\rm{c}}}={1.78}_{-0.02}^{+0.02}$	⋯	$93.5/11$	0.000	No truncation
Halo Modelc:
NFW+LSS (i)	${14.1}_{-1.0}^{+1.0}$	${3.79}_{-0.28}^{+0.30}$	${\gamma }_{{\rm{c}}}=1$	9.3	$10.9/11$	0.450	ΛCDM ${b}_{{\rm{h}}}(M)$ scaling
NFW+LSS (ii)	${14.4}_{-1.3}^{+1.4}$	${3.74}_{-0.30}^{+0.33}$	${\gamma }_{{\rm{c}}}=1$	${7.4}_{-4.7}^{+4.6}$	$10.8/10$	0.377	bh as a free parameter
Einasto+LSS (i)	${14.3}_{-1.1}^{+1.1}$	${3.69}_{-0.42}^{+0.36}$	${\alpha }_{{\rm{E}}}={0.248}_{-0.047}^{+0.051}$	9.3	$10.7/10$	0.385	ΛCDM ${b}_{{\rm{h}}}(M)$ scaling
Einasto+LSS (ii)	${14.5}_{-1.6}^{+1.9}$	${3.65}_{-0.61}^{+0.47}$	${\alpha }_{{\rm{E}}}={0.245}_{-0.053}^{+0.061}$	${8.7}_{-5.6}^{+5.3}$	$10.6/9$	0.301	bh as a free parameter
DARKexp+LSS (i)	${14.2}_{-1.1}^{+1.2}$	${3.64}_{-0.46}^{+0.44}$	${\phi }_{0}={3.89}_{-0.54}^{+0.51}$	9.3	$11.7/10$	0.308	ΛCDM ${b}_{{\rm{h}}}(M)$ scaling
DARKexp+LSS (ii)	${14.0}_{-1.6}^{+1.8}$	${3.69}_{-0.57}^{+0.53}$	${\phi }_{0}={3.85}_{-0.61}^{+0.57}$	${10.1}_{-5.1}^{+4.9}$	$11.6/9$	0.235	bh as a free parameter

Notes.

^aProbability to exceed the observed ${\chi }^{2}$ value. ^bWe use Dehnen–Tremaine γ-models with the central cusp slope ${\gamma }_{{\rm{c}}}=3{\mathrm{log}}_{10}{\phi }_{0}-0.65$ ($1.7\leqslant {\phi }_{0}\leqslant 6$) as an analytic fitting function for the DARKexp density profile. ^cFor halo model predictions, we decompose the total mass overdensity ${\rm{\Delta }}\rho (r)=\rho (r)-{\bar{\rho }}_{{\rm{m}}}$ as ${\rm{\Delta }}\rho ={f}_{{\rm{t}}}{\rho }_{{\rm{h}}}+{\rho }_{2{\rm{h}}}$ where ${\rho }_{{\rm{h}}}(r)$ is the halo density profile, ${\rho }_{2{\rm{h}}}(r)={\bar{\rho }}_{{\rm{m}}}{b}_{{\rm{h}}}{\xi }_{{\rm{m}}}^{{\rm{L}}}(r)$ is the two-halo term, and ${f}_{{\rm{t}}}{(r)=(1+{r}^{2}/{r}_{{\rm{t}}}^{2})}^{-2}$ describes the steepening of the density profile in the transition regime around the truncation radius r_t, which is assumed to be ${r}_{{\rm{t}}}=3{r}_{200{\rm{c}}}$.

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We first consider halo density profile models without including the two-halo term. In this modeling approach, no truncation is applied when calculating the surface mass density Σ(R), effectively accounting for the large-scale contribution. Figure 4 shows the best-fit model profiles along with the observed $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile. We find that two- or three-parameter cuspy models, namely, the NFW, gNFW, Einasto, and DARKexp-γ models, provide satisfactory fits (PTE > 0.05) to the data. The best-fit gNFW model has a central cusp slope of ${\gamma }_{{\rm{c}}}={0.85}_{-0.31}^{+0.22}$, being consistent with the NFW model (${\gamma }_{{\rm{c}}}=1$). The cored PI and Burkert models provide poor fits with χ²/dof (PTE) values of 23.6/11 (1.5 × 10⁻²) and 29.9/11 (1.7 × 10⁻³), respectively. The power-law sphere model has a reduced χ² value of χ²/dof = 93.5/11 and is strongly disfavored by the $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile having a pronounced radial curvature.

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Now we examine projected halo models following the prescription described in Section 5.2.2. In what follows, we will focus on our best models, specifically, the NFW, Einasto, and DARKexp-γ cuspy density profiles. We note that since the halo bias is given as a function of M_200c, the number of free parameters is unchanged for each case. We find these models (NFW+LSS (i), Einasto+LSS (i), and DARKexp+LSS (i) in Table 4) give statistically comparable fits. In all cases, including the two-halo term improves the fits, while keeping the best-fit parameters essentially unchanged (the difference in each parameter is much less than the 1σ uncertainty). Independent of the chosen profile, we find b_h(M_200c) ∼ 9.3 (${b}_{{\rm{h}}}{\sigma }_{8}^{2}\sim 6.1$) for our best-fit models.

On the basis of the goodness-of-fit statistic, the highest-ranked model among those considered is the NFW halo model (NFW+LSS (i); see the red curve in Figure 4) with ${M}_{200{\rm{c}}}={14.1}_{-1.0}^{+1.0}\times {10}^{14}\;{M}_{\odot }\;{h}_{70}^{-1}$ and ${c}_{200{\rm{c}}}={3.79}_{-0.28}^{+0.30}$, followed by the Einasto and DARKexp halo models (Einasto+LSS (i) and DARKexp+LSS (i)). The best-fit NFW halo model yields an Einstein radius of ${\theta }_{\mathrm{Ein}}={14.0}_{-3.2}^{+3.4}$ arcsec at z_s = 2, which is about 1.7σ lower than the median effective Einstein radius (see Section 3.5) of ${\bar{\theta }}_{\mathrm{Ein}}=20\_\_AMP\_\_farcs;1$ observed for the X-ray-selected subsample (Section 3.5). The best-fit NFW parameters of our projected halo model are in good agreement with those from the stacked shear-only analysis of Umetsu et al. (2014), ${M}_{200{\rm{c}}}={13.4}_{-0.9}^{+1.0}\times {10}^{14}\;{M}_{\odot }\;{h}^{-1}$ and ${c}_{200{\rm{c}}}={4.01}_{-0.32}^{+0.35}$, in which the two-halo term can be safely neglected as the stacked tangential-shear signal, $\langle \langle {\rm{\Delta }}{\rm{\Sigma }}\rangle \rangle$, is only sensitive to the intra-halo mass distribution in our radial range. The Einasto shape parameter is constrained to be ${\alpha }_{{\rm{E}}}={0.248}_{-0.047}^{+0.051}$ from an Einasto halo-model fit to the $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile (Einasto+LSS (i)), in agreement with ${\alpha }_{{\rm{E}}}={0.191}_{-0.068}^{+0.071}$ from the stacked shear-only analysis of Umetsu et al. (2014). Our measurements agree well with predictions from ΛCDM numerical simulations, ${\alpha }_{{\rm{E}}}=0.21\pm 0.07$ (Meneghetti et al. 2014, α_E = 0.24 ± 0.09 when fitted to surface mass density profiles). The fitting formula given by Gao et al. (2008) yields α_E ≃ 0.29 for our X-ray-selected subsample. This is consistent with our results at the 1σ level.

The halo-model interpretation we have adopted is a crude approximation in the transition between the intra-halo and two-halo regimes, neglecting nonlinear effects (Baldauf et al. 2010; Diemer & Kravtsov 2014). In order to effectively account for possible modifications due to nonlinear effects, here we allow the halo bias factor b_h to be a free parameter. We find that allowing b_h to be free does not improve the fits and that the resulting best-fit parameters remain unchanged within the errors (Table 4). The effective halo bias b_h is detected at 1.6σ–2.0σ significance, where the exact level of significance depends on the details of the halo density profile in the transition regime. In Figure 3, we compare the resulting NFW halo model profile (NFW+LSS (ii), red shaded area) and the NFW profile (cyan-shaded area) along with the $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile.

Interpreting the effective mass from stacked lensing requires caution when the cluster sample spans a broad range of masses and redshifts (Mandelbaum et al. 2005; Niikura et al. 2015). If there is a significant scatter in the luminosity–mass relation of halos, then the halo mass distribution of a luminosity-selected sample becomes significantly broader, with the width and asymmetry of the distribution typically increasing for higher luminosity bins. Accordingly, there is no typical mass, and in general the effective mass determination from NFW fits falls between the mean and the median masses of the halo population (Mandelbaum et al. 2005). Alternatively, if a halo sample has a broad redshift distribution, the effective lensing mass determined from stacking with redshift-dependent weights (Equation (27)) is generally different from the mean population mass (Umetsu et al. 2014).

To assess this possibility, we compare the stacked lensing results with the individual cluster masses (Tables 2 and 3). For the former, we consider our best model (NFW+LSS (i) in Table 4) for the X-ray-selected subsample, namely the NFW halo model using the b_h–M relation of Tinker et al. (2010).

Following Umetsu et al. (2014), we construct a composite-halo mass profile $\langle \langle {M}_{{\rm{\Delta }}}\rangle \rangle$ (see also Coupon et al. 2013; Ford et al. 2014) from a sensitivity-weighted average of NFW fits to individual cluster ${\boldsymbol{\Sigma }}$ profiles as

$$\begin{eqnarray}&&\langle \langle {M}_{{\rm{\Delta }}}\rangle \rangle =\displaystyle \frac{{\displaystyle \sum }_{n}\;\mathrm{tr}({{ \mathcal W }}_{n}){M}_{{\rm{\Delta }},n}}{{\displaystyle \sum }_{n}\;\mathrm{tr}({{ \mathcal W }}_{n})}\end{eqnarray} \tag{ 36 }$$

using $\mathrm{tr}({{ \mathcal W }}_{n})\propto {{\rm{\Sigma }}}_{({\rm{c}},\infty )n}^{-2}$ (Equation (27)) as an effective sensitivity weight for each cluster. Hence, the weight depends on cluster redshift z_l through ${{\rm{\Sigma }}}_{({\rm{c}},\infty )}$. The error variance for $\langle \langle {M}_{{\rm{\Delta }}}\rangle \rangle$ is given by ${\sigma }_{{\rm{\Delta }}}^{2}={\left[{\sum }_{n}\mathrm{tr}({{ \mathcal W }}_{n})\right]}^{-2}{\sum }_{n}{\sigma }_{{\rm{\Delta }},n}^{2}{[\mathrm{tr}({{ \mathcal W }}_{n})]}^{2}$ with σ_Δ,n the 1σ uncertainty for M_Δ,n. It was shown by Umetsu et al. (2014) that this composite-halo approach using the trace approximation for the sensitivity matrix can give an adequate description of the observed stacked lensing signal.

The results of this comparison are summarized in Figure 5. At Δ_c = 200, we find a sensitivity-weighted average of $\langle \langle {M}_{200{\rm{c}}}\rangle \rangle =(14.3\pm 1.1)\times {10}^{14}\;{M}_{\odot }\;{h}_{70}^{-1}$ for the X-ray-selected subsample, in excellent agreement with the best-fit halo mass of ${M}_{200{\rm{c}}}={14.1}_{-1.0}^{+1.0}\times {10}^{14}\;{M}_{\odot }\;{h}_{70}^{-1}$ extracted from a halo-model fit to the $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile. On the other hand, the unweighted median masses of the clusters are ∼3%–8% higher than the sensitivity-weighted masses $\langle \langle {M}_{{\rm{\Delta }}}\rangle \rangle$ at each overdensity. This difference is not due to the asymmetry of the distribution of the cluster masses because the unweighted median and mean masses agree to within ∼2% at each overdensity. Our results appear to be robust against different ensemble-averaging techniques once the effects of sensitivity weighting are consistently taken into account, supporting the findings of Umetsu et al. (2014).

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We examine the relationship between halo mass and concentration for our CLASH X-ray-selected subsample. Here we closely follow the Bayesian regression approach of Sereno et al. (2015b), taking into account the correlation between the errors on mass and concentration (e.g., Du & Fan 2014) and the effects of nonuniformity of the mass probability distribution (e.g., Kelly 2007; Andreon & Bergé 2012). We consider a power-law function of the following form:

$$\begin{eqnarray}&&{c}_{200{\rm{c}}}={10}^{\alpha }{\left(\displaystyle \frac{{M}_{200{\rm{c}}}}{{M}_{\mathrm{piv}}}\right)}^{\beta }{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{\gamma },\end{eqnarray} \tag{ 37 }$$

with M_piv and z_piv the pivot mass and redshift, respectively. We set ${M}_{\mathrm{piv}}={10}^{15}\;{M}_{\odot }{h}^{-1}$ and z_piv = 0.34, approximately the effective mean values of the cluster mass and redshift, respectively (Section 5). In the present analysis, we fix γ = −0.668 (Meneghetti et al. 2014) as predicted for the CLASH X-ray-selected population (for details, see Section 6.2) because our data do not have sufficient statistics to constrain the redshift evolution of the c–M relation (Merten et al. 2015).

To perform regression analysis we define new dependent and independent variables in logarithmic form as

$$\begin{eqnarray}Y & \equiv & {\mathrm{log}}_{10}\left[{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-\gamma }{c}_{200{\rm{c}}}\right],\\ X & \equiv & {\mathrm{log}}_{10}({M}_{200{\rm{c}}}/{M}_{\mathrm{piv}}).\end{eqnarray} \tag{ 38 }$$

Equation (37) is then expressed by a linear relation, $Y=\alpha +\beta X$. The observed values of latent variables are denoted with lowercase letters. If the selected clusters in our sample are not uniformly distributed in logarithmic mass X, this can lead to biased estimates of regression parameters (Kelly 2007; Sereno et al. 2015b). We properly account for these effects in Bayesian regression (Kelly 2007; Sereno et al. 2015a, 2015b; Sereno & Ettori 2015a).

The clusters in our X-ray-selected subsample are selected to be X-ray hot (>5 keV) and regular in X-ray morphology (Postman et al. 2012). In general, if we select a cluster sample by imposing certain thresholds on cluster observables, then the steepening at the high end of the intrinsic mass function, combined with the selection effects, makes the resulting mass probability distribution of selected clusters approximately lognormal (Appendix A of Sereno & Ettori 2015a).

In this study, we model the intrinsic probability distribution ${ \mathcal P }(X)$ of logarithmic mass X with a single Gaussian function characterized by two parameters, namely the mean μ and the dispersion τ (Kelly 2007). This approach alleviates the problem of assuming a uniform prior distribution on the independent variable X and, in general, provides a good approximation for a regular unimodal distribution (Kelly 2007; Andreon & Bergé 2012; Sereno & Ettori 2015a). A uniform prior distribution can be recovered in the limit of $\tau \to \infty$.

The conditional probability distribution ${ \mathcal P }({\boldsymbol{y}}| {\boldsymbol{x}})={\prod }_{n}{ \mathcal P }({y}_{n}| {x}_{n})$ of ${\boldsymbol{y}}=\{{y}_{n}\}$ given ${\boldsymbol{x}}=\{{x}_{n}\}$ is then written as (Kelly 2007)

$$\begin{eqnarray}&&\mathrm{ln}{ \mathcal P }({\boldsymbol{y}}| {\boldsymbol{x}})=-\displaystyle \frac{1}{2}\displaystyle \sum _{n}\left[\mathrm{ln}(2\pi {\sigma }_{n}^{2})+{\left(\displaystyle \frac{{y}_{n}-\langle {y}_{n}| {x}_{n}\rangle }{{\sigma }_{n}}\right)}^{2}\right],\end{eqnarray} \tag{ 39 }$$

where $\langle {y}_{n}| {x}_{n}\rangle$ and ${\sigma }_{n}^{2}\equiv \mathrm{Var}({y}_{n}| {x}_{n})$ are the conditional mean and variance of y_n given x_n, respectively:

$$\begin{eqnarray}\langle {y}_{n}| {x}_{n}\rangle & = & \alpha +\beta \mu +\displaystyle \frac{\beta {\tau }^{2}+{C}_{{xy},n}}{{\tau }^{2}+{C}_{{xx},n}}({x}_{n}-\mu ),\\ {\sigma }_{n}^{2} & = & {\beta }^{2}{\tau }^{2}+{\sigma }_{Y| X}^{2}+{C}_{{yy},n}-\displaystyle \frac{{(\beta {\tau }^{2}+{C}_{{xy},n})}^{2}}{{\tau }^{2}+{C}_{{xx},n}},\end{eqnarray} \tag{ 40 }$$

where ${\sigma }_{Y| X}$ is the intrinsic scatter in the Y–X relation; ${C}_{{xx},n}$, ${C}_{{yy},n}$, and ${C}_{{xy},n}={C}_{{yx},n}$ are the elements of the covariance matrix between the observables x_n and y_n of the nth cluster. If ${ \mathcal P }({X}_{n})$ is assumed to be uniform, we have $\langle {y}_{n}| {x}_{n}\rangle =\alpha +\beta {x}_{n}$ and ${\sigma }_{n}^{2}={\sigma }_{Y| X}^{2}+{C}_{{yy},n}+{\beta }^{2}{C}_{{xx},n}-2\beta {C}_{{xy},n}$ (e.g., Kelly 2007; Okabe & Smith 2015).

We use uninformative priors for all parameters. We adopt uniform priors for the intercept α, the slope β, and the mean μ. For the variance parameters ${\sigma }_{Y| X}^{2}$ and τ², we use the inverse Gamma distribution, ${\rm{I}}{\rm{\Gamma }}(\epsilon ,\epsilon )$, with $\epsilon \ll 1$ a small number (Andreon & Hurn 2010; Sereno & Ettori 2015a, 2015b). In our analysis, we choose  = 10⁻³ (Sereno & Ettori 2015a, 2015b).

Figure 6 shows the two-dimensional marginalized posterior distributions derived for all pairs of the regression parameters $(\alpha ,\beta ,{\sigma }_{Y| X},\mu ,\tau )$. The marginalized constraints (${C}_{\mathrm{BI}}\pm {S}_{\mathrm{BI}};$ Section 4.2) on the intercept, slope, and intrinsic scatter are α = 0.60 ± 0.04, β = −0.44 ± 0.19, and ${\sigma }_{Y| X}=0.056\pm 0.026$ (or, a natural logarithmic scatter of 0.13 ± 0.06). The posterior distributions show a long tail extending toward positive values of β, corresponding to shallower slopes for the c–M relation. This tail is associated with small values of the τ parameter that describes the width of the mass probability distribution ${ \mathcal P }(X)$. Hence, accounting for the nonuniformity of the mass probability distribution is crucial for making inference of the underlying c–M relation for the CLASH sample. We have checked that the tail in the posterior distribution of β disappears if ${ \mathcal P }(X)$ is assumed to be uniform. We see from Figure 6 that the constraint on the τ parameter is dominated by the prior especially at $\tau \gtrsim 0.5$ and that the data are not informative enough to determine the dispersion of the intrinsic mass distribution.

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In Figure 7, we summarize in the c–M plane our regression results obtained for our 16 CLASH X-ray-selected clusters. A detailed comparison with the SaWLens results (Merten et al. 2015) is presented in Section 7.4.2.

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Understanding the selection function and observational biases arising from projection effects is crucial when interpreting the c–M relation derived from lensing observations. For the CLASH sample, this has been addressed in detail by Meneghetti et al. (2014) using a sample of ∼1400 cluster-sized halos ($0.25\leqslant z\leqslant 0.67$) selected from the MUSIC-2 nonradiative hydrodynamical N-body simulations (Sembolini et al. 2013) in a ΛCDM universe ($h=0.7,{{\rm{\Omega }}}_{{\rm{m}}}=0.27,{{\rm{\Omega }}}_{{\rm{\Lambda }}}=1-{{\rm{\Omega }}}_{{\rm{m}}},{\sigma }_{8}=0.82$).

Meneghetti et al. (2014) characterized a sample of halos that follows the CLASH selection function based on X-ray morphological regularity. Their results suggest that the CLASH X-ray-selected subsample is prevalently composed of relaxed clusters (∼70%) and largely free of orientation bias (Section 5). Another important implication is that this subsample is expected to have a very small scatter in concentration because of the high degree of regularity in their X-ray morphology.

Meneghetti et al. (2014) find that the mean two-dimensional concentration of the CLASH X-ray-selected sample is expected to be ∼11% higher than that for the full population of clusters (Meneghetti et al. 2014). According to their simulations, we expect that the NFW concentrations of this sample determined from noise-free Σ profiles are in the range $3\lesssim {c}_{200{\rm{c}}}\lesssim 6$, with a mean (median) value of 3.87 (3.76) and a standard deviation of 0.61. The distribution follows a power-law relation of the form $c\propto {M}^{-0.160\pm 0.108}\times {(1+z)}^{-0.668\pm 0.341}$ with a normalization of $c{| }_{z=0.34}=3.96\pm 0.14$ at ${10}^{15}\;{M}_{\odot }\;{h}^{-1}$. This model, shown as the red solid line in Figure 7, is in excellent agreement with our regression results (gray shaded area) with $c{| }_{z=0.34}=3.95\pm 0.35$ at 10¹⁵ M_⊙ h⁻¹, as well as with the stacked lensing measurements: ${c}_{200{\rm{c}}}={3.79}_{-0.28}^{+0.30}$ (blue triangle; Section 5) from our stacked lensing analysis and ${c}_{200{\rm{c}}}={4.01}_{-0.32}^{+0.35}$ (yellow contours) from the stacked shear-only analysis of Umetsu et al. (2014). The derived slope of β = −0.44 ± 0.19 is somewhat steeper than, but consistent within errors with, the predicted values of β = −0.160 ± 0.108 (Meneghetti et al. 2014). The intrinsic scatter, ${\sigma }_{Y| X}=0.056\pm 0.026$, is in agreement with the expected standard deviation of 0.61 around the mean 3.87 (Meneghetti et al. 2014), corresponding to $\sigma ({\mathrm{log}}_{10}{c}_{200{\rm{c}}})\sim 0.07$, or $\sigma (\mathrm{ln}{c}_{200{\rm{c}}})\sim 0.16$. This is significantly smaller than the typical intrinsic scatter predicted for the full (relaxed) population of halos, $\sigma ({\mathrm{log}}_{10}{c}_{200{\rm{c}}})\sim 0.14$–0.16 (0.1–0.12) (Neto et al. 2007; Duffy et al. 2008; Bhattacharya et al. 2013).

As described in Section 3, we have accounted for various sources of errors associated with our strong-lensing, weak-lensing shear and magnification measurements. All these errors are encoded in the measurement uncertainties $({\sigma }_{+},{\sigma }_{\mu },{\sigma }_{M})$ that enter the joint likelihood analysis (Section 3.6) and contribute to the posterior covariance matrix C^stat of the mass profile solution ${\boldsymbol{s}}={\boldsymbol{\Sigma }}/{{\rm{\Sigma }}}_{\infty ,{\rm{c}}}$. In particular, our magnification bias analysis (Section 3.4) accounted for spurious large-scale variations of the red galaxy counts (${\sigma }_{\mu }^{\mathrm{sys}}$), as well as the angular clustering (${\sigma }_{\mu }^{\mathrm{int}}$) and Poisson error (${\sigma }_{\mu }^{\mathrm{stat}}$) contributions. The fractional area f_mask lost to cluster members, foreground objects, and defects was calculated as a function of clustercentric radius and corrected for in the source counts according to Equation (10). In our mass profile analysis, the total covariance matrix C includes additional contributions from the residual mass-sheet uncertainty C^sys, the cosmic noise C^lss, and the intrinsic variations C^int of the cluster signal due to the c–M variance, halo asphericity, and the presence of correlated halos (Section 3.6.3).

Additionally, we quantified potential sources of systematic errors in our weak-lensing shear+magnification measurements as follows (Section 3): (1) dilution of the weak-lensing signal by cluster members (2.4%, Section 3.2), (2) photometric-redshift bias in the mean depth estimates $\langle \beta \rangle$ (0.27%, Section 3.2), and (3) shear calibration uncertainty (5%, Section 3.3). These systematic errors scale approximately linearly with the cluster mass and add to 5.6% in quadrature. For the absolute calibration of cluster masses, one needs to take into account the systematic bias due to mass model uncertainties. Meneghetti et al. (2014) find that spherical NFW masses of cluster-sized halos estimated from their surface mass densities are biased low on average by 5% and that the bias is significantly reduced for relaxed halos (1%–2%) because they are more spherical than unrelaxed ones. According to their simulations, the degree of negative bias expected for our sample dominated by relaxed clusters is 3% (Massimo Meneghetti, private communication). Hence, the systematic uncertainty in the absolute mass calibration is estimated to be ≃6%. This implies that the total uncertainty in the absolute mass calibration with the sample of 20 clusters is $\sqrt{{0.28}^{2}/20+{0.06}^{2}}\simeq 9\%$ at ${{\rm{\Delta }}}_{{\rm{c}}}=200$ (see Section 4.2).

A joint analysis of multi-scale, multiple lensing probes makes it possible to improve the precision of cluster mass profile reconstructions over a wide range of clustercentric radii and to self-calibrate systematics as well as observational parameters that describe the intrinsic properties of source populations (Rozo & Schmidt 2010; Umetsu 2013). When the wide-field CLASH weak-lensing data are combined with the inner HST lensing constraints (N_SL = 4), the central weak-lensing bin Σ(<09) of Umetsu et al. (2014) is resolved into 5 radial bins of the surface mass density. We find a significant improvement of ∼45% on average in terms of the total S/N (Equation (24)) from adding the HST lensing information to the wide-field weak-lensing data.

The improved sensitivity and resolution at 10''–40'' have also allowed us to determine the inner characteristic radius r₋₂ and the halo concentration c_200c = r_200c/r₋₂ for each individual cluster (Tables 2). Umetsu et al. (2014) did not attempt to determine c_200c for each cluster because the weak-lensing profiles of individual clusters are highly degenerate in M_200c and c_200c, which can potentially bias the slope of the c–M relation determined from weak lensing (Hoekstra et al. 2011; Du & Fan 2014). This is particularly the case for high-z, low-mass systems, for which the characteristic profile curvature around r₋₂ is unconstrained by noisy and sparse weak-lensing data. For such clusters, the constraints on c_200c are essentially imposed by prior information. We note again that the C^int contribution to the total covariance matrix C was not included in the individual cluster mass measurements of Umetsu et al. (2014, Table 6), so that their mass measurement errors were underestimated (Section 4.2) with respect to this work.

A multi-probe approach combining complementary probes of cluster lensing allows us to test the consistency and robustness of cluster mass measurements (Umetsu 2013). Now we compare our weak+strong lensing mass estimates derived for our full sample of 20 CLASH clusters with the weak-lensing masses of Umetsu et al. (2014) to assess the level of systematic uncertainties in the ensemble mass calibration.

In Figure 8, we show for each cluster the weak+strong to weak lensing mass ratio, ${M}_{3{\rm{D}}}^{\mathrm{WL}+\mathrm{SL}}/{M}_{3{\rm{D}}}^{\mathrm{WL}}$, as a function of spherical radius r. The results are shown in the range r = [200, 2000] kpc h⁻¹ ($0.1\lesssim r/{r}_{200{\rm{m}}}\lesssim 1$) where the weak-lensing mass measurements were obtained by Umetsu et al. (2014, see their Figure 4). At each cluster radius, we compute the unweighted average of the mass ratios for our sample using geometric averaging (Donahue et al. 2014; Umetsu et al. 2014).²⁴ Figure 8 shows that the ensemble-averaged mass ratio $\langle {M}_{3{\rm{D}}}^{\mathrm{WL}+\mathrm{SL}}/{M}_{3{\rm{D}}}^{\mathrm{WL}}\rangle$ is consistent with unity within the errors at all cluster radii. In particular, the mass offset is within 2% at $r\gtrsim 600$ kpc ${h}^{-1}\sim 1.5{r}_{2500{\rm{c}}}$. We see a trend for the average ratio to decrease toward the center, reaching 0.95 ± 0.06 at r = 200 kpc h⁻¹. This indicates that the HST analysis (Zitrin et al. 2015) favors relatively low central masses and hence low halo concentrations although the difference is not significant, $| \langle {M}_{3{\rm{D}}}^{\mathrm{WL}+\mathrm{SL}}/{M}_{3{\rm{D}}}^{\mathrm{WL}}\rangle -1| \leqslant 5\%\pm 6\%$, compared to the sensitivity limit with 20 clusters. This level of mass offset is smaller than, but consistent with, the value 8% ± 9% based on the shear–magnification consistency test of Umetsu et al. (2014).

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On the basis of these consistency checks, the residual systematic uncertainty in the ensemble mass calibration is estimated to be of the order ∼5%–8% in the one-halo regime, r = [200, 2000] kpc h⁻¹. This however implies that, on an individual cluster basis, there is a large scatter of ∼20%–40% between different reconstruction methods that use different combinations of data. The level of uncertainty in the ensemble mass calibration (5%–8%) empirically estimated using different combinations of lensing probes is in agreement with the systematic uncertainty in the absolute mass calibration (6%) assessed in Section 7.1.

CLASH provides a sizable sample (20 clusters at 0.19 < z < 0.69, $\bar{z}=0.377$) for the calibration of the high end of the cluster mass function. In principle, weak lensing can yield unbiased mass estimates (assuming sphericity) for a sample of clusters that is largely free of orientation bias (Meneghetti et al. 2014). In practice, however, lensing mass measurements can be subject to various (known and unknown) systematic effects, as discussed in Section 7.1. In our early work (e.g., Broadhurst et al. 2005; Umetsu & Broadhurst 2008), we established that the dominant source of systematic effects in cluster weak lensing is the contamination of background galaxy samples by cluster members, which can lead to a substantial underestimation of the true lensing signal. This point has been supported by recent observations (Okabe et al. 2013; Applegate et al. 2014; Hoekstra et al. 2015) through systematic mass comparisons between different surveys that use different approaches to measuring weak lensing.

Here we compare our cluster mass estimates (Tables 2 and 3) with those obtained from other cluster lensing surveys that overlap with our sample, namely the Weighing the Giants program (WtG; Applegate et al. 2014), the Canadian Cluster Cosmology Project (CCCP; Hoekstra et al. 2015), and the LoCuSS (Okabe & Smith 2015, LoCuSS),²⁵ as well as with those from the SaWLens analysis of Merten et al. (2015). The WtG sample is a representative X-ray luminous subset of the ROSAT All-sky Survey (RASS) clusters at 0.15 < z < 0.7, with a median redshift of $\bar{z}=0.387$. The WtG clusters span a wide range of dynamical states, as well as of redshifts (von der Linden et al. 2014a). The CCCP sample is a mixture of X-ray luminous clusters for which archival B- and R-band observations made with the CFH12k camera on the 3.6 m Canada–France–Hawaii Telescope (CFHT) were available (Hoekstra 2007) and a temperature-selected subset of clusters drawn from the ASCA survey (${k}_{{\rm{B}}}{T}_{X}\gt 5\;{\rm{keV}}$, Hoekstra et al. 2012), spanning the range 0.15 < z < 0.55 ($\bar{z}=0.233$). The LoCuSS sample is drawn from the RASS catalogs at 0.15 < z < 0.3 ($\bar{z}=0.229$) and is approximately X-ray luminosity limited (Okabe & Smith 2015). The observed X-ray temperatures of the LoCuSS clusters are ${k}_{{\rm{B}}}{T}_{X}\gtrsim 5\;{\rm{keV}}$ (Martino et al. 2014, see their Figure 5). The LoCuSS sample is selected purely on the X-ray luminosity, ignoring other physical parameters and relaxation properties (Smith et al. 2015).

In all cases, the cluster masses are measured assuming a spherical NFW halo (Section 4.2). The WtG and LoCuSS mass measurements are based on weak-lensing observations with Subaru/Suprime-Cam, while the CCCP survey uses weak-lensing data taken with CFHT. The CCCP and LoCuSS surveys used the BCG as the cluster center as done in our work (Section 3.1), whereas the WtG survey adopted the X-ray centroid as the cluster center (von der Linden et al. 2014a). We note that, for all clusters in our sample, the mass measurements presented in Tables 2 and 3 are insensitive to the choice of the cluster center as discussed in Section 3.1. Another key difference is that the LoCuSS and CLASH surveys controlled contamination of their background galaxy samples at the $\lesssim 2\%$ level by imposing stringent color cuts (Section 3.2), albeit with increased shot noise, whereas the WtG and CCCP surveys boosted the diluted shear signal to correct statistically for contamination, by assuming that the observed number density profile of a pure background galaxy sample is flat. This correction, referred to as a boost factor, is not valid in general as it ignores the depletion or enhancement of the number density of background galaxies due to magnification bias (Umetsu et al. 2014; Chiu et al. 2015; Okabe & Smith 2015; Ziparo et al. 2015).

In the following, we compare cluster masses between two studies by using the same aperture radii to avoid aperture mismatch problems. These comparisons are limited to those overdensity radii (r_Δ) where the fitting ranges typically overlap. The results of the comparisons are shown in Figure 9. For each case, we calculate the mass ratios for the overlap sample using the unweighted geometric mean (Section 7.2), unless otherwise noted.

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7.3.1. CLASH: Merten et al. (2015)

7.3.2. The WtG Project

7.3.3. The CCCP

7.3.4. The LoCuSS

7.4.1. Comparison with ΛCDM Predictions from the Literature

We compare our individual cluster mass and concentration measurements (Table 2; Figure 7) with predictions from numerical simulations in the literature. To statistically quantify the level of agreement with a given predicted c–M relation, we use frequentist measures of goodness of fit. Specifically, for a given fixed c(M, z) function, we evaluate the χ² goodness of fit to the null hypothesis that the sample data are derived from the model population. The χ² statistic is then defined as

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{n}\;{\left[\displaystyle \frac{{\mathrm{log}}_{10}{c}_{n}-{\mathrm{log}}_{10}c({M}_{n},{z}_{n})}{{\sigma }_{n}}\right]}^{2},\end{eqnarray} \tag{ 41 }$$

where σ_n is the total statistical uncertainty of the nth cluster,

$$\begin{eqnarray}&&{\sigma }_{n}^{2}={\sigma }_{\mathrm{int},0}^{2}+{C}_{{yy},n}+{\beta }_{0}^{2}{C}_{{xx},n}-2{\beta }_{0}{C}_{{xy},n}\end{eqnarray} \tag{ 42 }$$

with β₀ the mass slope of the intrinsic c–M relation under consideration and σ_{int, 0} the intrinsic scatter in ${\mathrm{log}}_{10}{c}_{200{\rm{c}}}$ at fixed mass and redshift. For all models, we fix ${\sigma }_{\mathrm{int},0}={\sigma }_{Y| X}\simeq 0.057$ according to our regression results (Section 6.1) and assume that the effective number of parameters for the null model is three (i.e., the intercept, mass slope, and redshift evolution).

Table 5 lists for each model the values of χ² and PTE, along with the weighted geometric average²⁸ and the standard deviation of observed-to-predicted concentration ratios ${c}^{(\mathrm{obs})}/{c}^{(\mathrm{pred})}$. The theoretical predictions from Meneghetti et al. (2014) are based on nonradiative simulations of DM and baryons, and those from the others are based on DM-only simulations. In all cases, halo masses and concentrations are defined using the overdensity Δ_c = 200 and measured assuming the spherical NFW profile, either in projection (2D) or in three-dimensions (3D). For peak-height-based c(ν) relations (Prada et al. 2012; Diemer & Kravtsov 2015), we assume the WMAP seven-year cosmology of Komatsu et al. (2011) to calculate the relationship between peak height and halo mass at each redshift.²⁹

Table 5.  Comparison of Measured and Predicted Concentrations for the CLASH X-Ray-selected Subsample

Author	Sample	3D/2D	Functiona	${c}^{(\mathrm{obs})}/{c}^{(\mathrm{pred})}$		${\chi }^{2}$	PTEb
				Averagec	σd
Theory:
Duffy et al. (2008)	full	3D	c–M	1.331 ± 0.108	0.334	22.6	0.046
Duffy et al. (2008)	relaxed	3D	c–M	1.165 ± 0.094	0.290	13.6	0.399
Prada et al. (2012)	full	3D	c–ν	0.733 ± 0.065	0.244	24.6	0.026
Bhattacharya et al. (2013)	full	3D	c–ν	1.169 ± 0.095	0.292	14.1	0.369
Bhattacharya et al. (2013)	relaxed	3D	c–ν	1.131 ± 0.092	0.277	12.4	0.494
Dutton & Macciò (2014)	full	3D	c–M	1.061 ± 0.086	0.262	10.4	0.659
Meneghetti et al. (2014)	full	3D	c–M	1.061 ± 0.089	0.279	10.2	0.675
Meneghetti et al. (2014)	relaxed	3D	c–M	0.990 ± 0.083	0.249	9.2	0.760
Diemer & Kravtsov (2015)	full (median)	3D	c–ν	1.021 ± 0.083	0.330	14.4	0.349
Diemer & Kravtsov (2015)	full (mean)	3D	c–ν	1.060 ± 0.086	0.326	13.8	0.391
Meneghetti et al. (2014)	full	2D	c–M	1.087 ± 0.092	0.336	13.5	0.413
Meneghetti et al. (2014)	relaxed	2D	c–M	1.040 ± 0.086	0.283	10.8	0.628
Meneghetti et al. (2014)	CLASH	2D	c–M	0.988 ± 0.078	0.227	9.6	0.730
Observations:
Merten et al. (2015)	CLASH	2D	c–M	1.133 ± 0.087	0.209	9.2	0.754

Notes.

^ac–M: power-law $c(M,z)$ relation; c–ν: halo concentration given as a function of peak height $\nu (M,z)$. ^bProbability to exceed the measured ${\chi }^{2}$ value assuming the standard ${\chi }^{2}$ probability distribution function. ^cWeighted geometric average of observed-to-predicted concentration ratios. ^dStandard deviation of the distribution of observed-to-predicted concentration ratios.

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We first consider models for the full population of halos based on the three-dimensional characterization of the halo density profile. We find that recent c–M relations from Bhattacharya et al. (2013), Dutton & Macciò (2014), Meneghetti et al. (2014), and Diemer & Kravtsov (2015) are in satisfactory agreement with the data. The Meneghetti et al. (2014) model (PTE = 0.675), which is calibrated for a cosmology with a relatively high normalization (h = 0.7, Ω_m = 0.27, σ₈ = 0.82; Section 6.2), yields the highest goodness of fit among those considered here, followed by the Dutton & Macciò (2014) model (PTE = 0.659) calibrated for the Planck cosmology (h = 0.671, Ω_m = 0.3175, σ₈ = 0.8344). Our measurements are 33% ± 11% higher than the predictions of Duffy et al. (2008) based on the WMAP five-year cosmology ($h=0.742,{{\rm{\Omega }}}_{{\rm{m}}}=0.258,{\sigma }_{8}=0.796$). This is in line with the findings of Dutton & Macciò (2014), who showed that the c–M relation in the WMAP five-year cosmology has a 20% lower normalization at z = 0 than in the Planck cosmology. On the other hand, the observed concentrations are 27% ± 7% lower than the predictions of Prada et al. (2012). Their model exhibits a flattening and upturn of the c–M relation in the high ν regime, so that their concentrations derived for cluster halos are substantially higher than those of others. We refer the reader to Meneghetti & Rasia (2013) and Diemer & Kravtsov (2015) for detailed discussions on the possible origin of this discrepancy.

Next, we consider models derived for relaxed populations of cluster halos (Duffy et al. 2008; Bhattacharya et al. 2013; Meneghetti et al. 2014). Numerical simulations suggest that relaxed subsamples have concentrations that are on average ∼10% higher than for the full samples. In all cases, we find improved agreement with the data compared to the full-sample comparison (Table 5). This is consistent with the expectation that the CLASH X-ray-selected subsample is largely composed of relaxed clusters (Section 6.2). For the Meneghetti et al. (2014) model, the agreement is particularly excellent, with a PTE of 0.760.

Finally, we examine the two-dimensional c–M relations of Meneghetti et al. (2014) obtained from fitting Σ profiles of simulated halos (Section 6.2).³⁰ As summarized in Table 5, we find a better agreement with the model that explicitly takes into account the CLASH selection function based on X-ray morphology (PTE = 0.730).

In summary, we find that, overall, cluster c–M relations that are calibrated for recent cosmologies yield good agreement with the observations (Bhattacharya et al. 2013; Dutton & Macciò 2014; Meneghetti et al. 2014; Diemer & Kravtsov 2015). An improved agreement is achieved when selection effects are taken into account in the models (Duffy et al. 2008; Bhattacharya et al. 2013; Meneghetti et al. 2014), matching the characteristics of the CLASH clusters in terms of the overall degree of relaxation and X-ray morphological regularity.

7.4.2. Comparison with the SaWLens Results

7.4.3. Revisiting the Overconcentration Problem

In contrast to the CLASH X-ray-selected subsample, clusters selected to have large Einstein radii (θ_Ein) represent a highly biased population with their major axis preferentially aligned with the line of sight. A population of such superlenses might also be biased toward halos with intrinsically higher concentrations (Hennawi et al. 2007). In the context of ΛCDM, the projected mass distributions of superlens clusters have ∼40%–60% higher concentrations than typical clusters with similar masses and redshifts (Oguri & Blandford 2009).

Prior to the CLASH survey, Umetsu et al. (2011a, hereafter U11) performed a strong-lensing, weak-lensing shear and magnification analysis of four strong-lensing-selected clusters of similar masses (A1689, A1703, A370, and Cl0024+17) at $\langle \langle {z}_{{\rm{l}}}\rangle \rangle \simeq 0.32$ using high-quality HST and Subaru observations. These clusters display prominent strong-lensing features, characterized by ${\theta }_{\mathrm{Ein}}\gtrsim 30^{\prime\prime}$ (${z}_{{\rm{s}}}=2$). U11 show that the stacked $\langle \langle {\rm{\Sigma }}\rangle \rangle$ profile of the four clusters is well described by a single NFW profile in the one-halo regime, with an effective concentration of ${c}_{\mathrm{vir}}={7.68}_{-0.40}^{+0.42}$ (${c}_{200{\rm{c}}}\simeq 6.29$) at ${M}_{\mathrm{vir}}={22.0}_{-1.4}^{+1.6}\times {10}^{14}\;{M}_{\odot }\;{h}_{70}^{-1}$ (${M}_{200{\rm{c}}}\;\simeq 19.4\times {10}^{14}\;{M}_{\odot }{h}_{70}^{-1}$), corresponding to an Einstein radius of ${\theta }_{\mathrm{Ein}}\simeq 36^{\prime\prime}$ (z_s = 2). Semianalytical simulations of ΛCDM incorporating idealized triaxial halos yield a ∼40%–60% bias correction for a strong-lensing cluster population (Oguri & Blandford 2009). After applying a 50% superlens correction, U11 found a discrepancy of ∼2σ with respect to the Duffy et al. (2008) c–M relation based on the WMAP five-year cosmology. U11 conclude that there is no significant tension between the concentrations of their clusters and those of CDM halos if large lensing biases are coupled to a sizable intrinsic scatter in the c–M relation.

Figure 10 compares in the c–M plane the stacked full-lensing constraints for our CLASH X-ray-selected subsample (blue contours; NFW+LSS (i) in Table 4) and those for the strong-lensing-selected sample of U11 (gray contours). In the figure we overplot theoretical c–M relations of Duffy et al. (2008), Bhattacharya et al. (2013), Dutton & Macciò (2014), Meneghetti et al. (2014), and Diemer & Kravtsov (2015, their mean relation; see Table 5), all evaluated for the full population of halos at the mean redshift $\langle \langle {z}_{{\rm{l}}}\rangle \rangle \simeq 0.32$ of the U11 sample. Figure 10 demonstrates that c–M relations that are calibrated for more recent cosmologies and simulations provide better agreement with our CLASH measurements (see Section 7.4.1).

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To account for the superlens bias in the U11 sample, we plot each of the c–M models with a maximal 60% correction (Oguri & Blandford 2009). We find that, once the effects of selection and orientation bias are taken into account, the full-lensing results of U11 come into line with the models of Dutton & Macciò (2014), Meneghetti et al. (2014), and Diemer & Kravtsov (2015), the three most recent c–M models studied in this work. Hence, the discrepancy found by U11 can be fully reconciled by the higher normalization of the c–M relation as favored by recent WMAP and Planck cosmologies.