Weak-lensing Mass Calibration of ACTPol Sunyaev–Zel'dovich Clusters with the Hyper Suprime-Cam Survey

The cluster sample used in this work is drawn from the ACTPol two-season cluster catalog (Hilton et al. 2018). The sample was extracted from 148 GHz observations of a 987.5 deg² equatorial field that combined data obtained using the original ACT receiver (MBAC; Swetz et al. 2011) with the first two seasons of ACTPol data. Details of the ACTPol observations and map making can be found in Naess et al. (2014) and Louis et al. (2017). Cluster candidates were detected by applying a spatial matched filter to the map, using the universal pressure profile (UPP; Arnaud et al. 2010) and its associated SZ signal-mass scaling relation to model the cluster signal. Candidates were then confirmed as clusters and their redshifts measured using optical/IR data, principally from the Sloan Digital Sky Survey (SDSS DR13; Albareti et al. 2017). Cluster masses were estimated by applying the profile-based amplitude analysis technique, introduced in Hasselfield et al. (2013). In this paper, we use M_SZ to refer to SZ-based mass estimates that correspond with ${M}_{500{\rm{c}}}^{\mathrm{UPP}}$ as tabulated in Hilton et al. (2018). The full cluster sample in Hilton et al. (2018) are all signal-to-noise ratio (S/N) > 4 with mass range of roughly 2 × 10¹⁴ M_⊙ < M_SZ < 9 × 10¹⁴ M_⊙ with a median mass of M_SZ = 3.1 × 10¹⁴ M_⊙ and redshift range of roughly 0.15 < z < 1.4 with a median redshift of z = 0.49.

Among the HSC first-year data (Aihara et al. 2018b), the XMM field in the HSC wide layer overlaps with the deepest region of the ACTPol maps—the D6 field at 02^h30^m R.A. (Naess et al. 2014; Louis et al. 2017). The sample studied in this work consists of ACTPol clusters in this region that were detected with SNR_2.4 > 5 in the Hilton et al. (2018) catalog, where SNR_2.4 refers to the S/N of the cluster, as measured in a map filtered at an angular scale of 24 (refer to Sections 2.2 and 2.3 of Hilton et al. 2018 for details). Above this threshold, the cluster sample in the HSC S16A region is 100% pure with complete redshift follow-up.

Figure 1 shows the 90% mass completeness limit as a function of redshift across the HSC S16A region, using the UPP model and associated scaling relation to convert the SZ signal into mass, as described in Section 2.4 of Hilton et al. (2018). Averaged over 0.2 < z < 1, the sample is 90% complete for M_SZ > 3.2 × 10¹⁴ M_⊙. This is significantly lower than the equivalent limit of M_SZ > 4.5 × 10¹⁴ M_⊙ obtained when averaging over the whole 987.5 deg² ACTPol field, since the overlapping HSC S16A region lies in a low noise region of the ACTPol survey D6 (see Naess et al. 2014).

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Figure 2 shows the overlap between the ACTPol D6 field and the HSC XMM field. There are eight clusters that span a redshift range of 0.186 ≤ z ≤ 1.004. The average cluster redshift, which is weighted by the source galaxy weight described in Section 3.1, is $\langle {z}_{l}\rangle =0.43$.

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Table 1 lists the properties of the sample. When measuring the shear signal from the HSC data, we define the cluster centers as the brightest cluster galaxy (BCG) locations as determined by Hilton et al. (2018) using a combination of visual inspection and the i, r − i color–magnitude diagram. We match the ACTPol clusters with optically selected clusters in the HSC Wide S16A data set by the CAMIRA algorithm in Oguri et al. (2018), requiring separation <2'. The optical richness derived by CAMIRA, ${\hat{N}}_{\mathrm{mem}}$, is shown in Table 1. Figure 3 shows the HSC color images of our sample together with the SZ S/N contours. The cluster centers, which are defined by the BCG and SZ signals, are consistent for most of the clusters, given that the beam of ACTPol is about 14 at 148 GHz. Only ACT-CL J0229.6-0337 has a significantly large offset between these cluster centers. We will look into how the offset affects our lensing signal in Appendix B.3.

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Table 1.  ACTPol Clusters Overlapping with the HSC XMM Field

Name	SZ R.A.	SZ Decl.	BCG R.A.	BCG Decl.	Redshifta	SNR2.4	MSZ [1014 M⊙]	${\hat{N}}_{\mathrm{mem}}$ b
ACT-CL J0204.8-0303	2:04:49.73	−3:03:38.42	2:04:50.27	−3:03:36.82	0.549	6.84	${3.04}_{-0.48}^{+0.57}$	35.0
ACT-CL J0205.2-0439	2:05:15.90	−4:39:07.50	2:05:16.69	−4:39:19.96	0.968	8.10	${3.12}_{-0.44}^{+0.52}$	38.6
ACT-CL J0215.3-0343	2:15:23.72	−3:43:45.20	2:15:24.01	−3:43:31.98	1.004	5.86	${2.46}_{-0.37}^{+0.44}$	44.8
ACT-CL J0221.7-0346	2:21:44.53	−3:46:19.94	2:21:45.17	−3:46:02.19	0.432	7.29	${3.07}_{-0.50}^{+0.59}$	69.3
ACT-CL J0227.6-0317	2:27:37.77	−3:17:53.48	2:27:38.22	−3:17:57.31	0.838	5.15	${2.19}_{-0.35}^{+0.42}$	50.9
ACT-CL J0229.6-0337	2:29:36.88	−3:37:04.01	2:29:43.97	−3:36:53.52	0.323	5.15	${2.40}_{-0.44}^{+0.54}$	57.0
ACT-CL J0231.7-0452	2:31:43.63	−4:52:56.16	2:31:41.17	−4:52:57.44	0.186	6.85	${3.08}_{-0.58}^{+0.72}$	116.4
ACT-CL J0233.6-0530	2:33:36.27	−5:30:34.52	2:33:35.59	−5:30:21.76	0.435	6.91	${3.11}_{-0.51}^{+0.61}$	46.9

Notes.

^aRedshifts are spectroscopic measurements. ^bRichness is from the HSC CAMIRA cluster catalog (Oguri et al. 2018).

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HSC is the wide-field prime focus camera on the Subaru Telescope (Komiyama et al. 2018; Miyazaki et al. 2018a) located at the summit of Maunakea. The combination of the wide field of view (1.77 deg²), superb image quality (seeing routinely less than 06), and large aperture of the primary mirror (8.2 m) makes HSC one of the best instruments for conducting weak-lensing cosmology. Under the Subaru SSP (Aihara et al. 2018a), HSC started a galaxy imaging survey in 2014 that aims to cover 1400 deg² of the sky down to i = 26 after its fifth year of operation. The first-year galaxy shape catalog (Mandelbaum et al. 2018b) was produced using the data taken from 2014 March through 2016 April with about 90 nights in total. The first-year data consists of six distinct fields (HECTOMAP, GAMA09H, WIDE12H, GAMA15H, XMM, VVDS) and covers 136.9 deg² in total. Note that this catalog is a slight extension of Data Release 1 (Aihara et al. 2018b). As mentioned above, we use the shape catalog in the XMM field (29.5 deg² once we remove the star mask region), which overlaps with the current ACTPol observations. The weighted number density of source galaxies in this field is 22.1 arcmin⁻² and their median redshift is z_m = 0.82.

Here we briefly summarize the HSC shape catalog. For details of the shape catalog production, see Mandelbaum et al. (2018b). The galaxy shapes were estimated on coadded i-band images by the re-Gaussianization technique (Hirata & Seljak 2003), which is a moment-based method with PSF correction. This method was extensively used and characterized in the SDSS (Mandelbaum et al. 2005, 2013; Reyes et al. 2012). The shapes $({e}_{1},{e}_{2})=(e\cos 2\phi ,e\sin 2\phi )$, where ϕ is position angle, are defined in terms of distortion, e = (a²−b²)/(a² + b²), where a and b are the major and minor axes, respectively (Bernstein & Jarvis 2002). The galaxy shapes were calibrated against image simulations generated with GalSim (Rowe et al. 2015), an open source software package, which yields correction factors for the shear measurements. These factors are the multiplicative bias m and additive bias (c₁, c₂), which are defined as g_i,obs = (1 + m)g_i,true + c_i, where (g₁, g₂) is defined in terms of shear, g = (a−b)/(a + b), and must be applied to the shear measurements. Note that the multiplicative bias is shared between the two ellipticity components (for details, see Mandelbaum et al. 2018a). For each galaxy, the shape catalog provides an estimate of the intrinsic shape noise, e_rms, an estimate of measurement noise, σ_e, and inverse weights w from combining e_rms and σ_e. The measurement noise is statistically estimated from the shape measurements performed on simulated images, and the intrinsic shapes are derived by subtracting the measurement noise from the ellipticity dispersion measured from the real data. Note that we use a catalog made with the "Sirius" star mask, which actually includes bright galaxies and thus an extended region around each BCGs may be masked as well. After we performed the lensing measurement, the shape catalog was updated with a more reliable star mask called "Arcturus" (for details, see Coupon et al. 2018). We checked that switching to this new shape catalog changes our fiducial stacked weak-lensing measurements by an amount well within the statistical uncertainty (typically 10%).

We use photometric estimates to select source galaxies based on colors. For this purpose, we use cmodel magnitudes derived by fitting a galaxy's light profile with a composite model of the exponential and de Vaucouleurs profile (Bosch et al. 2018). The HSC SSP catalog has photo-z estimates based on six different methods (Tanaka et al. 2018). Among these methods, we use MLZ, an unsupervised machine-learning method based on the self-organizing map, which is a projection map from multi-dimensional color space to redshift, for our fiducial measurement. We have checked the consistency among lensing signals with different photo-z methods, which is described in Appendix B.2. In this paper, we use the redshift PDFs, P(z), and randomly sampled point estimates that are drawn from the PDFs, z_mc. The latter is specifically used for one of the source galaxy selection methods described below.

Weak gravitational lensing manifests as a coherent distortion of apparent shapes of source galaxies. For a source galaxy at a comoving transverse separation R from the lens center, the lens' gravitational potential induces a tangential distortion, γ_t, which depends on the lens' matter density profile projected along the line of sight, Σ(R), and on the redshifts of the source galaxy, z_s, and the lens, z_l. For the purposes of this work the lenses are galaxy clusters. In terms of the average projected mass density inside R, $\bar{{\rm{\Sigma }}}(\lt R)$, and the critical surface mass density Σ_cr(z_l, z_s) defined below, the tangential distortion can be expressed in terms of the excess surface mass density ΔΣ(R) as follows:

$$\begin{eqnarray}&&{\gamma }_{t}(R)=\displaystyle \frac{\bar{{\rm{\Sigma }}}(\lt R)-{\rm{\Sigma }}(R)}{{{\rm{\Sigma }}}_{\mathrm{cr}}({z}_{l},{z}_{s})}\equiv \displaystyle \frac{{\rm{\Delta }}{\rm{\Sigma }}(R)}{{{\rm{\Sigma }}}_{\mathrm{cr}}({z}_{l},{z}_{s})}.\end{eqnarray} \tag{ 2 }$$

In terms of the angular diameter distances of the source and lens from us, D_A(z_s) and D_A(z_l), and the angular diameter distance between the two, D_A(z_s, z_l), Σ_cr(z_s, z_l) is defined as:

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{cr}}({z}_{l},{z}_{s})=\displaystyle \frac{{c}^{2}}{4\pi G}\displaystyle \frac{{D}_{A}({z}_{s})}{{\left(1+{z}_{l}\right)}^{2}{D}_{A}({z}_{l}){D}_{A}({z}_{l},{z}_{s})}.\end{eqnarray} \tag{ 3 }$$

where the factor (1 + z_l)⁻² comes from our use of comoving coordinates (Mandelbaum et al. 2006).

To estimate cluster properties including mass from measurements of ΔΣ, we will start with models of the cluster density radial profile, ρ(r), and integrate to generate modeled lensing profiles ΔΣ(r; M, x), where x signifies other possible parameters of the models (see Section 4). From the data, we can estimate ΔΣ in each radial bin R_i for either a single cluster or a stack of multiple clusters with a weighted sum over the tangential components of the shapes of galaxies, e_t, as follows. We use a weighting based both on the shape catalog weight for the source galaxy, w_s, and on an estimate of the appropriate critical surface mass density for the particular source-lens pair, using the photo-z PDF for each source, P_s(z), to account for the dilution effect of foreground galaxies. (See Section 3.3 for discussion of the impact of possible cluster member contamination, however.) Specifically, we define ${\tilde{w}}_{{ls}}\equiv {w}_{s}\langle {{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{-1}{\rangle }^{2}$, where

$$\begin{eqnarray}&&\left\langle {{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{-1}\right\rangle =\displaystyle \frac{{\int }_{0}^{\infty }{P}_{s}(z){{\rm{\Sigma }}}_{\mathrm{cr}}^{-1}({z}_{l},z){dz}}{{\int }_{0}^{\infty }{P}_{s}(z){dz}}.\end{eqnarray} \tag{ 4 }$$

With two additional calibration factors described below, the lensing profile is estimated as:

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Sigma }}({R}_{i})=\displaystyle \frac{1}{2{ \mathcal R }({R}_{i})}\displaystyle \frac{{\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}{e}_{t,{ls}}{\left\langle {{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{-1}\right\rangle }^{-1}}{(1+K({R}_{i})){\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}}.\end{eqnarray} \tag{ 5 }$$

The factor 1 + K(R) is the shear calibration factor that corrects for multiplicative bias described in Section 2.3;

$$\begin{eqnarray}&&1+K({R}_{i})=\displaystyle \frac{{\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}\left(1+{m}_{s}\right)}{{\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}},\end{eqnarray} \tag{ 6 }$$

where m_s is the multiplicative bias of the source, s. The shear responsivity ${ \mathcal R }({R}_{i})$ is necessary to take into account the summation in non-Euclidean shear space. It is calculated as

$$\begin{eqnarray}&&{ \mathcal R }({R}_{i})=1-\displaystyle \frac{{\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}{e}_{\mathrm{rms},{ls}}^{2}}{{\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}}.\end{eqnarray} \tag{ 7 }$$

We compute ΔΣ in 12 logarithmic bins from 0.1 h⁻¹ Mpc to 10 h⁻¹ Mpc. Note that we do not use all the radial bins for model fitting, as described later.

We estimate the covariance matrix of the lensing signal as

$$\begin{eqnarray}&&C={C}^{\mathrm{stat}}+{C}^{\mathrm{int}}+{C}^{\mathrm{lss}},\end{eqnarray} \tag{ 8 }$$

where C^stat is the statistical uncertainty due to galaxy shapes:

$$\begin{eqnarray}&&\begin{array}{l}{C}_{{ij}}^{\mathrm{stat}}\\ \quad \ =\ \displaystyle \frac{1}{4{{ \mathcal R }}^{2}({R}_{i})}\displaystyle \frac{{\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}^{2}({e}_{\mathrm{rms},{ls}}^{2}+{\sigma }_{e,{ls}}^{2}){\left\langle {{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{-1}\right\rangle }^{-2}}{{\left[1+K({R}_{i})\right]}^{2}{\left[{\displaystyle \sum }_{{ls}\in {R}_{i}}{\tilde{w}}_{{ls}}\right]}^{2}}{\delta }_{{ij}};\end{array}\end{eqnarray} \tag{ 9 }$$

C^int accounts for the intrinsic variations of projected cluster mass profiles such as halo triaxiality, the presence of correlated halos, and the intrinsic scatter of the concentration–mass relation, which is a function of halo mass (Gruen et al. 2015); and C^lss is due to uncorrelated large-scale structure (LSS) along the line of sight. Detailed calculations of the intrinsic and LSS covariances are described in Appendix A. Figure 4 shows the diagonals of the covariance matrix used in our analysis and the correlation matrix, defined as ${C}_{\mathrm{corr},{ij}}={C}_{{ij}}/\sqrt{{C}_{{ii}}{C}_{{jj}}}$. A similar figure was presented in Umetsu et al. (2016), which describes a joint weak and strong lensing analysis of 20 high-mass clusters. Figure 4 shows that the total uncertainty is dominated by the shape noise (C^stat) at r ≲ 3 h⁻¹ Mpc beyond which the relative contribution from LSS noise, uncorrelated with the cluster, becomes important.

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If cluster galaxies are misidentified as background galaxies, they will introduce a systematic dilution of the weak-lensing signal from their galaxy cluster. We look into two distinct source galaxy selection methods which were established in Medezinski et al. (2018b) with the CAMIRA catalog of optically selected clusters in HSC SSP: a selection based on the color–color space (the CC cut) and another based on a cumulative photo-z PDF (the P(z) cut). Note that we use MLZ to define the latter cut and calculate lensing signals.

The CC cut is defined in the g − i versus r − z space to minimize the dilution in lensing signal due to the contamination by cluster members and foreground galaxies. The CC cut is defined differently for a cluster with z_l ≤ 0.4 and z_l > 0.4 to avoid excessively removing galaxies behind low redshift clusters (for the detailed definition, see Appendix A in Medezinski et al. 2018b). Using the CC cut Medezinski et al. (2018b) showed that the dilution is consistent with zero.

The P(z) cut initially proposed in Oguri (2014) is defined by two criteria that each galaxy must satisfy to be identified as a background galaxy. The first criterion is

$$\begin{eqnarray}&&{p}_{\mathrm{cut}}\lt {\int }_{{z}_{\min }}^{\infty }P(z){dz},\end{eqnarray} \tag{ 10 }$$

where p_cut = 0.98, meaning that we require that 98% of the area beneath the P(z) lies beyond z_min. For our analysis z_min = z_l + Δz and we employ Δz = 0.2 for a secure rejection of cluster galaxies, following the investigation by Medezinski et al. (2018b). The second criterion is that each galaxy's randomly drawn point redshift value from its photo-z PDF, z_mc, be less than z_max. This criterion rejects photo-z PDFs that are predominantly above the redshift limit that are considered secure for a given optical survey. This maximum redshift is optimized for HSC and set to z_max = 2.5 (see Medezinski et al. 2018b).

Figure 5 compares the stacked lensing signal from the eight clusters in the sample calculated without source selection cuts, with CC cuts, and with P(z) cuts. We do not find a significant difference among these signals within the error bars. Following the extensive analysis in Medezinski et al. (2018b) that found the CC cuts to be less diluted, we use the CC cut for our fiducial measurement.

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Systematic biases in the photometric redshift estimates of source galaxies would propagate to the weak-lensing signal measurement through the calculation of the critical surface density. Following Mandelbaum et al. (2008), this bias in the weak-lensing signal of a cluster at redshift z_l can be estimated as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{\rm{\Sigma }}}{{\rm{\Delta }}{{\rm{\Sigma }}}^{\mathrm{true}}}({z}_{l})=1+b({z}_{l})=\displaystyle \frac{{\sum }_{s}\,{\tilde{w}}_{{ls}}\langle {{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{-1}{\rangle }^{-1}{\left[{{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{\mathrm{true}}\right]}^{-1}}{{\displaystyle \sum }_{s}\,{\tilde{w}}_{{ls}}},\end{eqnarray} \tag{ 11 }$$

where the quantities with a superscript "true" denote the quantity as it would be measured with a spectroscopic sample, the sum over s goes over all source galaxies.

Nominally such photo-z biases are evaluated using a spectroscopic redshift (spec-z) sample that is independent from those used to calibrate the photometric redshifts and has the same population properties (magnitude and color distribution) as our source redshift sample. In practice it is difficult to obtain such a representative spec-z sample given the depth of our source catalog. In principle the difference among the populations of an existing spec-z sample and the weak-lensing source sample can be accounted for by using a clustering and reweighting technique (for assumptions and caveats of this method, see Bonnett et al. 2016; Gruen & Brimioulle 2017). This method decomposes galaxies in the source sample into groups with similar properties. Then the galaxies from the spectroscopic sample in these groups are reweighted to mimic the distribution of the weak-lensing source sample.

The first-year HSC shape catalog has substantial overlaps with public spec-z samples, such as GAMA and VVDS; however, there are not enough galaxies with spec-z to represent the source sample even after reweighting. Instead of the spec-z sample we use the COSMOS-30 band photo-z (Ilbert et al. 2009) sample. We decompose the galaxies in the weak-lensing sample using their i-band magnitude and four colors into cells of a self-organizing map (SOM, S. More et al. 2019, in preparation). Using the HSC photometry of the COSMOS-30 band photo-z sample, we classify them into SOM cells defined by the source galaxy sample and compute their new weights (w_SOM) which adjusts the COSMOS-30 band photo-z sample to mimic our source galaxy sample. We compute the photo-z bias (see Equation (11)) by including w_SOM in the definition of w_ls. Then we average Equation (11) over our cluster sample with the weight defined by Equation (23) in Nakajima et al. (2012). This yields a two percent bias which negligible compared to statistical uncertainties.

In Figure 6, we show the lensing signal, ΔΣ, for each cluster in our sample. We estimate properties for each cluster using the Navarro–Frenk–White (NFW) radial density profile (Navarro et al. 1996, 1997):

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(r)=\displaystyle \frac{{\rho }_{s}}{(r/{r}_{s}){\left(1+r/{r}_{s}\right)}^{2}}.\end{eqnarray} \tag{ 12 }$$

Following Okabe et al. (2010), we convert the parameterization of ρ_NFW from characteristic density and scale, ρ_s and r_s, to mass M_200m, and concentration, defined as c_200m ≡ R_200m/r_s. We integrate the 3D ρ_NFW to generate the fitting function ΔΣ(R; M_200m, c_200m) and fit for both parameters using the Markov chain Monte Carlo (MCMC) sampler called emcee (Foreman-Mackey et al. 2013) with the length of chains of ∼30,000. We restrict the fitting range to 0.3 h⁻¹ Mpc < R < 3 h⁻¹ Mpc, where the lower limit is to avoid using blended images as blending is more prominent toward the cluster center (Medezinski et al. 2018b), and the upper limit is to avoid the fitting being affected by the 2-halo regime (Applegate et al. 2014). We use the full covariance as shown in Equation (8). To calculate C^int, we use the SZ mass of individual clusters in the ACTPol cluster catalog. Despite the mass bias, this choice would not significantly affect the total covariance because C^int is subdominant (see Figure 4). The values of M_200m and c_200m in each chain are converted to M_500c and c_500c using the public, open source software called Colossus (Diemer 2018), and then central values and 1σ uncertainties are calculated, as shown in each panel of Figure 6. The S/N for the weak-lensing measurement and the cluster redshift are also shown in each panel. The three high-redshift clusters, ACT-CL J0205.2-0439, ACT-CL J0215.3-0343, and ACT-CL J0227.6-0317, have low S/Ns which are reflected in their poor best-fit curves and their weakly constrained lensing mass and concentration.

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We obtain the stacked cluster lensing signal (Equation (5)) with a S/N of 11.1 (9.6 for the data used for mass inference, see below). Our goal is to estimate the average mass of the clusters in the stack to compare to the average M_SZ. We introduce an improved method for estimating the average mass in Section 4.2.2, focusing on two models for the density profiles, as described below. For comparison with earlier work, in Section 4.2.1 we also estimate the average mass with the single-mass-bin fit, which is done using only the stacked cluster signal, fitted with a model parameterized by a single mass.

The single-mass-bin fit has been widely performed in the literature, but is limited by the fact that the amplitude of the lensing signal is not linearly proportional to the cluster mass. Our method improves upon it by emulating the stacking process and incorporating the effects of the cluster mass function and selection function.

For estimating the average cluster mass, we use the "Dark Emulator" model and the "Baryonic Simulations" model described in the next few paragraphs. We also present results from the NFW density profile for comparison to earlier work. Both these models provide excess surface density profiles up to the two-halo regime, allowing use of the large-scale information to constrain halo mass. Thus, we extend the fitting range up to 10 h⁻¹ Mpc for those models, rather than the 3 h⁻¹ Mpc limit for the NFW fit, which results in tighter constraints on cluster mass.

The Dark Emulator model is based on a cosmic emulator developed by Nishimichi et al. (2018) and it predicts statistical quantities of halos, including the mass function, the halo-matter cross-correlation, and the halo autocorrelation, as a function of halo mass, redshift, and cosmological model. The Dark Emulator model is based on a large set of N-body simulations, and predicts the lensing profiles with better than 2% precision; details can be found in Murata et al. (2018) and Nishimichi et al. (2018).

The Baryonic Simulations model is based on hydrodynamical simulations of cosmological volumes described in Battaglia et al. (2010). From these simulations we project the mass distributions of the halos in all the simulation snapshots at given redshifts following the methodology in Battaglia et al. (2016).

4.2.1. Single Mass-bin Fit

The results of the single-mass-bin fits are summarized in Table 2 and the model fits compared to measurements are shown in Figure 7. When fitting the NFW profile, we again vary both M_200m and c_200m. We assume all the clusters are at a single redshift, which we calculate as a weighted average over the lens-source pairs used in the stacked measurement: $\langle {z}_{l}{\rangle }_{\mathrm{stack}}={\sum }_{{ls}}{\tilde{w}}_{{ls}}{z}_{l}/{\sum }_{{ls}}{\tilde{w}}_{{ls}}=0.43.$ For the Baryonic Simulations model, we average the calculated excess surface density profile as a function of mass in the three redshift outputs that are closest to the value $\langle {z}_{l}{\rangle }_{\mathrm{stack}}$. Note that the fluctuation in ΔΣ at 8 h⁻¹ Mpc that is present in the Baryonic Simulations model is within the 30% intrinsic scatter on the single-mass-bin profile and is not seen in the stacked profile because there are more simulated halos included in that calculation. As Table 2 shows, the single-mass-bin fits have reasonable χ² values (here we fixed cosmological and other models parameters, thus reducing the number of free parameters to just two for the NFW fit, mass and concentration) and yield cluster masses that are within the 1σ errors. However, these models are being applied to the same data, so the differences among the inferred masses cannot be due to statistical error. We interpret these differences to be from systematic errors resulting from modeling uncertainty. Note that since the Dark Emulator is a more complete DM-only model for the mass profile than the NFW profile, we will focus on comparing it to the Baryonic Simulations model. We will continue to show the NFW fit results so that our results can be compared with previous results that did not use emulator fits. We find that using the single-mass-bin fit introduces systematic modeling uncertainty of order 15% for this sample of clusters, which is comparable to the statistical uncertainty for this cluster sample.

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Table 2.  Parameter Constraints from the Single-mass-bin Fit and Stacked Model Method Fit to the Stacked Lensing Data

	Parameter	NFW	Dark Emulator	Baryonic Simulation
Single-mass bin
	${M}_{500{\rm{c}}}^{\mathrm{WL}}$ [1014 M⊙]	${4.26}_{-0.71}^{+0.82}$	${4.22}_{-0.64}^{+0.72}$	${3.67}_{-0.58}^{+0.86}$
	c500c	${2.08}_{-0.71}^{+0.86}$	N/A	N/A
	χ2/dof	1.3/4	2.5/8	3.4/8
Stacked model
	$\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$ [1014 M⊙]	${4.02}_{-0.61}^{+0.65}$	${3.89}_{-0.57}^{+0.61}$	${3.55}_{-0.48}^{+0.63}$
	1−b	${0.71}_{-0.12}^{+0.13}$	${0.74}_{-0.12}^{+0.13}$	${0.80}_{-0.12}^{+0.16}$
	χ2/dof	1.6/5	3.0/8	3.1/8

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4.2.2. Stacked Model Method

Here we describe our improved method for estimating the average cluster mass $\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$ from the cluster sample, which we call the stacked model method. Note that here we use the average weak-lensing mass defined in M_500c to be consistent with the definition of the SZ mass. In this section, we omit the "200m" subscript for simplicity when halo mass is defined in M_200m. The essence of the stacked model method is to model the stacked lensing profile $\langle {\rm{\Delta }}{\rm{\Sigma }}\rangle (R)$, accounting for:

• 1.  
  the weak-lensing weighting, which depends on the cluster z;
• 2.  
  the cluster (halo) mass function, dn(z)/dM;
• 3.  
  the ACTPol cluster selection function, which is known in terms of ${M}_{500{\rm{c}}}^{\mathrm{SZ}}$ and z; and
• 4.  
  the mass-mass scaling relationship between M_SZ and M.

In particular, we define the variables μ_SZ ≡ lnM_SZ and μ ≡ ln M, and we assume

$$\begin{eqnarray}&&{\mu }_{\mathrm{SZ}}=\ B\,\mu +A,\end{eqnarray} \tag{ 13 }$$

but fix the mass-dependent exponent B to unity, so that we can focus on A, which quantifies the constant mass bias between M_SZ and the true mass. We qualitatively discuss mass dependence in Section 4.3, but the S/N of our measurement prevents us from providing an interesting constraint on B. We consider cluster density profiles from the models described in Section 4.2. To construct a model that can be fit to the data as a function of the average cluster mass, we compute the stacked lensing profile and average mass for different values of A, and then interpolate the stacked lensing profile as a function of stacked mass. The details of this process are described below.

The weak-lensing weights are computed for the redshifts z_j of the eight clusters as ${w}_{l}({z}_{j},{R}_{i})={\sum }_{s}{\tilde{w}}_{{ls}}({R}_{i})$, where s includes the subset of source galaxies for the jth cluster after the CC cut (Section 3.2). The weak-lensing weights are a smooth function of z, and so w_l(z, R_i) is estimated by extrapolation from w_l(z_j, R_i) for the Dark Emulator and Baryonic Simulations models.

The inherent dn(z)/dM from the simulations is self-consistently used for Baryonic Simulations models, and for the NFW and Dark Emulator profile modeling the dn(z)/dM from the Dark Emulator is used.

The cluster selection function, S(${M}_{500{\rm{c}}}^{\mathrm{SZ}}$, z), is computed by averaging the ACTPol survey completeness map, which is defined for a given SZ mass and redshift under our selection criteria SNR_2.4 > 5, over the XMM field where the HSC source galaxies exist (see Section 2.3 for details). The selection function is then translated to be a function of M_SZ by converting M_500c to M_200m, assuming the concentration–mass relation derived by Diemer & Kravtsov (2015). Here we again use Colossus for this conversion.

We estimate the conditional probability distribution for M_SZ given M assuming it follows a log-normal distribution, such that

$$\begin{eqnarray}&&P({\mu }_{\mathrm{SZ}}| \mu ;A)=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{{\mu }_{\mathrm{SZ}}| \mu }}\exp \left[-\displaystyle \frac{{\left({\mu }_{\mathrm{SZ}}-\mu -A\right)}^{2}}{2{\sigma }_{{\mu }_{\mathrm{SZ}}| \mu }^{2}}\right],\end{eqnarray} \tag{ 14 }$$

where ${\sigma }_{{\mu }_{\mathrm{SZ}}| \mu }$ is the SZ mass-mass scatter which we fix to ${\sigma }_{{\mu }_{\mathrm{SZ}}| \mu }$ = 0.2 for the following reasons. First, the lensing signal does not constrain the scatter well (e.g., Murata et al. 2018). Second, this is the same scatter assumed to correct the Eddington bias for the M_SZ values quoted in Hilton et al. (2018) and was used in Battaglia et al. (2016). Thus, using this value for ${\sigma }_{{\mu }_{\mathrm{SZ}}| \mu }$ allows for direct comparison to previous results from ACT.

The stacked model lensing signal used for fitting is

$$\begin{eqnarray}\begin{array}{rcl}\langle {\rm{\Delta }}{\rm{\Sigma }}\rangle ({R}_{i};A) & = & \displaystyle \frac{1}{{n}_{\mathrm{SZ}}(A)}\displaystyle \int {{dzw}}_{l}(z)\displaystyle \frac{{{cr}}^{2}(z)}{H(z)}\displaystyle \int d\mu M\displaystyle \frac{{dn}}{{dM}}(\mu ,z)\\ & & \times \ \displaystyle \int d{\mu }_{\mathrm{SZ}}\,{M}_{\mathrm{SZ}}S({M}_{\mathrm{SZ}},z)P({\mu }_{\mathrm{SZ}}| \mu ;A)\\ & & \times \,{\rm{\Delta }}{\rm{\Sigma }}({R}_{i},\mu ,z).\end{array}\end{eqnarray} \tag{ 15 }$$

Here r²(z)c/H(z) is the comoving volume per unit redshift interval and per unit steradian, ΔΣ(M, z, R_i) depends on the cluster profile model considered, and n_SZ is the expected number density per unit steradian of ACTPol clusters given the weak-lensing weights:

$$\begin{eqnarray}\begin{array}{rcl}{n}_{\mathrm{SZ}}(A) & = & \displaystyle \int {{dzw}}_{l}(z)\displaystyle \frac{{{cr}}^{2}(z)}{H(z)}\displaystyle \int d\mu M\displaystyle \frac{{dn}}{{dM}}(z)\\ & & \times \ \displaystyle \int d{\mu }_{\mathrm{SZ}}\,{M}_{\mathrm{SZ}}S({M}_{\mathrm{SZ}},z)P({\mu }_{\mathrm{SZ}}| \mu ;A).\end{array}\end{eqnarray} \tag{ 16 }$$

Finally, we estimate the average mass for the cluster sample as

$$\begin{eqnarray}\begin{array}{rcl}\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle (A) & = & \displaystyle \frac{1}{{n}_{\mathrm{SZ}}(A)}\displaystyle \int {{dzw}}_{l}(z)\displaystyle \frac{{{cr}}^{2}(z)}{H(z)}\displaystyle \int d\mu M\displaystyle \frac{{dn}}{{dM}}(z)\\ & & \times \ \displaystyle \int d{\mu }_{\mathrm{SZ}}\,{M}_{\mathrm{SZ}}S({M}_{\mathrm{SZ}},z)P({\mu }_{\mathrm{SZ}}| \mu ;A)\\ & & \times \,{M}_{500{\rm{c}}}(M,z),\end{array}\end{eqnarray} \tag{ 17 }$$

where M_500c(M, z) is again the halo mass converted mass from M_200m to M_500c at redshift z. Now that we have both $\langle {\rm{\Delta }}{\rm{\Sigma }}\rangle$ and $\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$ as a function of A, we interpolate $\langle {\rm{\Delta }}{\rm{\Sigma }}\rangle$ as a function of $\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$. We fit our stacked lensing model to the stacked lensing measurement using $\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$ as a free parameter.

For the Baryonic Simulations model we calculate the stacked weak-lensing signal as calculated in Battaglia et al. (2016) for a given average sample mass (Equations (15) and (17)) for each simulated halo surface density profile by the weak-lensing weight, the volume factor (comoving distance squared) associated with each simulation snapshot, the scatter in the scaling relation, and the ACTPol selection function to the simulated halos, described in Hilton et al. (2018).

We summarize the results of fits for $\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$ in Table 2 and show the best-fit profiles in Figure 7. Similar to the single-mass-bin fit, we restrict the fitting range of the NFW fit to 0.3 h⁻¹ Mpc < R < 3 h⁻¹ Mpc, but using the larger radii up to R ∼ 10 h⁻¹ Mpc changes the constraint well within the 1σ statistical uncertainty. The resulting masses from the Dark Emulator and Baryonic Simulations models are within 10% of each other, which we interpret as the systematic modeling uncertainty for the stacked model. This systematic error is still below our 15% statistical uncertainties on the mass, but with roughly eight more clusters they will become comparable. Looking ahead, if we want to use the full potential statistical power of the HSC survey or other comparable imaging surveys we need to reduce systematic modeling uncertainties.

Comparing the masses from the single-mass-bin fit and the stacked model method, we find that the single-mass-bin masses are systematically high by 3%–7%, depending on the profile model. We interpret this as a systematic bias from the single-mass-bin fitting technique that results from its lack of accounting for the mass and selection functions. Such a bias will become more important as samples of clusters with weak-lensing measurements increase.

Using the constraints on $\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$ obtained above, we derive the values for $1-b=\langle {M}_{500{\rm{c}}}^{\mathrm{SZ}}\rangle /\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$, which is shown in Table 2. We estimate the average SZ mass using the lensing weight to be consistent with the stacking process in lensing measurement; $\langle {M}_{500{\rm{c}}}^{\mathrm{SZ}}\rangle ={\sum }_{{ls}}{w}_{{ls}}{M}_{500{\rm{c}},l}^{\mathrm{SZ}}$/${\sum }_{{ls}}{w}_{{ls}}={2.87}_{-0.20}^{+0.25}\times {10}^{14}\ {\text{}}{M}_{\odot }$. We then propagate the errors in $\langle {M}_{500{\rm{c}}}^{\mathrm{WL}}\rangle$ and $\langle {M}_{500{\rm{c}}}^{\mathrm{SZ}}\rangle$ into 1−b. With only eight clusters the precision on the 1−b measurement we present in this work has comparable precision to previous measurements which have comparable or larger sample sizes. In Figure 8 we compiled from the literature previous measurements of 1−b on ACT and Planck selected clusters from various weak-lensing measurements and include this measurement for comparison. In this work and previous work we directly compare ACT and Planck SZ masses because both ACT and Planck use the same SZ mass scaling relations and pressure profiles (for more details see Battaglia et al. 2016). However, in the recent ACTPol cluster catalog (Hilton et al. 2018), we found evidence of mass-dependent bias at the 2σ level when we compared the Planck and ACT SZ masses. We are ignoring that fact here because the mass dependence may only be the result of selection effects at the intersection of the Planck and ACT samples. With a much larger sample from ACT in the near future we will be able to address this further.

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Previous weak-lensing measurements of Planck clusters von der Linden et al. (2014b) and Hoekstra et al. (2015) show marginal evidence for a mass dependence in 1−b. Combining these results with the previous ACT clusters result (Battaglia et al. 2016) qualitatively strengthens this evidence. Since Battaglia et al. (2016) was published, several new measurements of 1−b were made using Planck SZ clusters. The combination of previous measurements with our new measurement indicates that the observational case for a mass dependence in 1−b is weaker (see Figure 8). Simply fitting 1−b as a function of mass yields an exponent of 1−b ∝ M^{−0.01±0.02}. Here we have excluded the measurement from LoCuSS (Smith et al. 2016) as their X-ray measurement using spectroscopic-like temperatures (Martino et al. 2014) gives 10% higher X-ray masses in contrast to other measurements and they have a factor of two smaller errors than any other measurements of 1−b. We caution against any strong conclusions from our simple analysis above, as a proper analysis requires compilation of all the Planck and ACT SZ clusters with weak-lensing measurements, selecting and accounting for multiple weak-lensing measurements of these clusters, and precise Eddington bias corrections that account for the selection functions for each of these surveys convolved with the Planck and ACT selection functions. Comparisons of 1−b across different measurements should include sample variance errors, but such errors are typically not included. We calculate the sample variance contribution to $\langle {\rm{\Delta }}{\rm{\Sigma }}\rangle (R)\rangle$ by randomly sampling halos from the simulations (Battaglia et al. 2010) that satisfy the selection function and find this increases the diagonal of our covariance matrix by 20%–30%. Performing similar analyses on all the measurements in Figure 8 will strengthen our conclusions that currently these measurements are consistent and do not show evidence of a mass dependence in 1−b.

We calculate the covariance due to the projection effect of LSS based on Oguri & Takada (2011). For the nth cluster at redshift z_n we calculate the covariance as

$$\begin{eqnarray}\begin{array}{rcl}{C}_{{ij},n}^{\mathrm{lss}} & = & \langle {{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{-1}{\rangle }^{-2}\displaystyle \int \displaystyle \frac{{\ell }d{\ell }}{2\pi }{C}_{{\ell }}^{\kappa \kappa }{J}_{2}\left(\displaystyle \frac{{\ell }{R}_{i}}{\chi ({z}_{n})}\right){J}_{2}\\ & & \times \ \left(\displaystyle \frac{{\ell }{R}_{j}}{\chi ({z}_{n})}\right),\end{array}\end{eqnarray} \tag{ 18 }$$

where J₂(x) is the second-order Bessel function and χ(z) is the comoving distance at redshift z. We approximate the inverse critical surface mass density averaged over source galaxies as $\langle {{\rm{\Sigma }}}_{\mathrm{cr},{ls}}^{-1}\rangle \sim {{\rm{\Sigma }}}_{\mathrm{cr}}^{-1}({z}_{n},\langle {z}_{s}\rangle )$, where $\langle {z}_{s}\rangle$ is the mean redshift calculated using the photo-z PDF stacked over source galaxies with the weak-lensing weight, i.e.,

$$\begin{eqnarray}&&\langle {z}_{s}\rangle =\displaystyle \frac{\int {dz}\ z\ {P}_{\mathrm{stacked}}(z)}{\int {{dzP}}_{\mathrm{stacked}}(z)},\end{eqnarray} \tag{ 19 }$$

where ${P}_{\mathrm{stacked}}(z)={\sum }_{s}{\tilde{w}}_{{ls}}P(z)/{\sum }_{s}{\tilde{w}}_{{ls}}$ and s runs over source galaxies in all the radial bins after the color–color cut as a function of lens redshift (See Section 3.3 for details). The weak-lensing power spectrum ${C}_{{\ell }}^{\kappa \kappa }$ is defined as

$$\begin{eqnarray}&&{C}_{{\ell }}^{\kappa \kappa }=\int d\chi \displaystyle \frac{{\left[{W}^{\kappa }(z(\chi ))\right]}^{2}}{{\chi }^{2}}{P}_{m}^{\mathrm{NL}}\left(k=\displaystyle \frac{{\ell }}{\chi };z\right),\end{eqnarray} \tag{ 20 }$$

where W^κ(z) is the lensing weight function defined by

$$\begin{eqnarray}&&{W}^{\kappa }(z)=\displaystyle \frac{{\rho }_{m}(z)\langle {{\rm{\Sigma }}}_{\mathrm{cr}}^{-1}\rangle }{(1+z)}.\end{eqnarray} \tag{ 21 }$$

Here we again use the same approximation $\langle {{\rm{\Sigma }}}_{\mathrm{cr}}^{-1}\rangle \sim {{\rm{\Sigma }}}_{\mathrm{cr}}^{-1}(z,\langle {z}_{s}\rangle )$. The nonlinear matter power spectrum ${P}_{m}^{\mathrm{NL}}(k)$ is calculated by CAMB using the Halofit prescription (Smith et al. 2003) with the fitting parameter derived by Takahashi et al. (2012). We then calculate the covariance of the stacked lensing signal as

$$\begin{eqnarray}&&{C}_{{ij}}^{\mathrm{lss}}=\displaystyle \frac{{\displaystyle \sum }_{n}{\tilde{v}}_{n,i}{\tilde{v}}_{n,j}{C}_{{ij},n}^{\mathrm{lss}}}{{\displaystyle \sum }_{n}{\tilde{v}}_{n,i}{\displaystyle \sum }_{l}{\tilde{v}}_{n,j}},\end{eqnarray} \tag{ 22 }$$

where ${\tilde{v}}_{n,i}$ is the sum of the weak-lensing weight within the ith bin of the nth cluster.

We estimate the covariance that accounts for intrinsic variations of projected cluster mass profile, which includes the scatter in concentration and the triaxiality effect, based on Umetsu et al. (2016). They found that the intrinsic covariance of the lensing convergence κ = Σ(R)/Σ_cr can be well approximated by

$$\begin{eqnarray}&&{C}_{{ij}}^{\kappa ,\mathrm{int}}={\alpha }_{\mathrm{int}}^{2}{\kappa }^{2}{\delta }_{{ij}},\end{eqnarray} \tag{ 23 }$$

where α_int = 0.2 for M_200c ∼ 10¹⁵ h⁻¹ M_⊙ clusters. This formalism excludes the external contribution from C^lss, which was formally included in C^int covariance by Gruen et al. (2015). We convert Equation (23) to the covariance of excess surface density ${C}_{{ij}}^{{\rm{\Delta }}{\rm{\Sigma }},\mathrm{int}}$, assuming the NFW profile with concentration–mass relation derived in Diemer & Kravtsov (2015). Therefore, the covariance ${C}_{{ij}}^{{\rm{\Delta }}{\rm{\Sigma }},\mathrm{int}}$ depends on cluster mass. Note that, however, covariances with difference cluster mass are actually similar in their shapes and have the same form once they are scaled by $r\to r/{r}_{200{\rm{m}}}$. We then approximately calculate the covariance of the stacked lensing signal by

$$\begin{eqnarray}&&{C}_{{ij}}^{\mathrm{int}}=\displaystyle \frac{{C}_{{ij}}^{{\rm{\Delta }}{\rm{\Sigma }},\mathrm{int}}(\langle M\rangle )}{{N}_{\mathrm{cl}}},\end{eqnarray} \tag{ 24 }$$

where $\langle {M}_{\mathrm{WL}}\rangle$ (in M_500c) is the typical mass of our cluster sample and N_cl is the number of clusters. We vary the typical mass within $1.0\lt \langle {M}_{\mathrm{WL}}\rangle /{10}^{14}\,{M}_{\odot }\lt 7.0$ and see how the 1−b constraints are affected. We find the change in 1−b is within 1%, and thus use a fixed mass $\langle {M}_{\mathrm{WL}}\rangle =3.5\times {10}^{14}\,{M}_{\odot }$ for the intrinsic covariance in the main text.

B.1.1. B-mode Signal

Since weak lensing is caused by a scalar potential, a 45° rotated component from tangential shear, or B-mode, should be statistically consistent with zero. The left panel of Figure 9 shows the stacked B-mode signal around our ACTPol cluster sample. For our fitting range (0.3 h⁻¹ Mpc < R < 10 h⁻¹ Mpc), χ²/dof = 7.05/9, and thus our B-model signal is consistent with zero.

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B.1.2. Random Signal

We use MLZ photo-z estimates for our fiducial measurement as described in Section 2.3. In this section we describe the details of other photo-z methods in the HSC SSP catalog and check the consistency between ACTPol cluster lensing signals based on different photo-z methods.

The HSC SSP catalog has photo-z estimates based on the following methods; DEmP, Ephor, Franken-Z, Mizuki, and NNPZ (Tanaka et al. 2018), in addition to MLZ. DEmP is a method designed to minimize major issues of conventional empirical methods, such as how to choose a proper fitting function and biases due to the population of a training data set, by introducing regional polynomial fitting and uniformly weighted training set (Hsieh & Yee 2014). Ephor is a neural network photo-z code fed with de Vaucouleur flux and exponential flux. Franken-Z is a hybrid approach that combines the data-driven nature of machine-learning and statistical rigor from template fitting. Mizuki is a Bayesian template fitting method which allows for simultaneously constraining physical properties of galaxies such as star formation and photo-z (Tanaka 2015). NNPZ follows the method introduced by Cunha et al. (2009), a nearest neighbor method that finds nearest neighbors around an unknown object in the color/magnitude space from a reference sample and uses the reference redshift histogram as the PDF.

The top panel of Figure 10 shows lensing signals measured with different photo-zs. The fractional residuals between MLZ and other photo-z methods are shown in the bottom panel. When calculating error bars of fractional residuals, we account for the correlation between lensing signals based on different photo-zs. To estimate the correlation, we generate 18 realizations of galaxy shape catalogs with randomly rotated shapes. We then measure lensing signals around clusters in the same manner as described in Section 3, using different photo-zs. Although the signal itself has no tangential shear signal, we can still compute the correlation between lensing signals computed with different photo-zs. The inverse-variance weighted average of the fractional residual ranges from −0.05 ± 0.01 (Mizuki) to 0.01 ± 0.01 (Ephor), which is smaller than the expected deviation due to statistical uncertainties of the stacked lensing signal, i.e., ${\delta }_{{\rm{\Delta }}{\rm{\Sigma }}}={\left[{\sum }_{i}{\left({\rm{\Delta }}{\rm{\Sigma }}({R}_{i})/{\sigma }_{{\rm{\Delta }}{\rm{\Sigma }}({R}_{i})}\right)}^{2}\right]}^{-1/2}$, where ${\sigma }_{{\rm{\Delta }}{\rm{\Sigma }}({R}_{i})}$ is the shape noise. Thus we conclude that the relative bias between photo-z methods are within the statistical uncertainties.

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If the cluster center used in the lensing measurement (in this case, the BCG position) is offset from the gravitational potential minimum, the lensing signal at inner radii around and below the scale of off-centering is diluted. Previous studies showed that the positions of BCGs are better tracers of potential minima than other optical tracers such as the luminosity-weighted centers (Viola et al. 2015), and than X-ray centers because of the large statistical uncertainties in the X-ray center (George et al. 2012; von der Linden et al. 2014a). We expect this is also the case for our measurement. The ACT beam at 148 GHz is 14 and with a 5σ SZ detection we expect the astrometric uncertainties to be around 20'' at z ≳ 0.5, which is similar to the typical off-centering except for the cluster J0229.6-0337.

We check the impact of off-centering by comparing the lensing signal calculated with the BCG center to that with the SZ center. The comparison is shown in Figure 11. We do not find a significant difference between these signals, especially at the scales we use when fitting the models (R > 0.3 h⁻¹ Mpc), and thus it does not matter which center we use at this regime of S/N for the stacked signal. Even if the lensing shear profile is affected by the off-centering effect, the enclosed mass should be properly extracted if we use the lensing signals to sufficiently large radii compared to the off-centering distance (see Oguri & Takada 2011). For our fiducial measurement we use the BCG positions as centers.

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