<figure><img src="image_20.png" /><figcaption>Figure 18. Same as in Figure 17, but fitting each individual cluster with the two-parameter halo model.</figcaption></figure>

<table><caption>**Table 6**
Systematic Bias in Weak-lensing Mass Modeling</caption><thead><tr><th rowspan="2">M<sub>500, true</sub><sup>a</sup><br>(10<sup>13</sup> h<sup>-1</sup> M<sub>☉</sub>)</th><th rowspan="2">M<sub>200, true</sub><sup>a</sup><br>(10<sup>13</sup> h<sup>-1</sup> M<sub>☉</sub>)</th><th rowspan="2">c<sub>200, true</sub><sup>a</sup></th><th rowspan="2">N<sub>cl</sub></th><th colspan="3">NFW Model</th><th colspan="3">Halo Model</th></tr><tr><th>b<sub>sim, M<sub>500</sub></sub><br>(%)</th><th>b<sub>sim, M<sub>200</sub></sub><br>(%)</th><th>b<sub>sim, c<sub>200</sub></sub><br>(%)</th><th>b<sub>sim, M<sub>500</sub></sub><br>(%)</th><th>b<sub>sim, M<sub>200</sub></sub><br>(%)</th><th>b<sub>sim, c<sub>200</sub></sub><br>(%)</th></tr></thead><tbody><tr><td>0.8</td><td>1.2</td><td>6.1</td><td>100</td><td>-19 ± 8</td><td>-15 ± 8</td><td>-14 ± 8</td><td>-15 ± 8</td><td>-12 ± 9</td><td>-13 ± 8</td></tr><tr><td>1.5</td><td>2.2</td><td>5.7</td><td>100</td><td>-24 ± 7</td><td>-22 ± 7</td><td>-18 ± 7</td><td>-20 ± 7</td><td>-19 ± 7</td><td>-16 ± 8</td></tr><tr><td>2.8</td><td>3.9</td><td>5.6</td><td>100</td><td>-21 ± 6</td><td>-18 ± 6</td><td>-13 ± 8</td><td>-16 ± 6</td><td>-14 ± 6</td><td>-10 ± 8</td></tr><tr><td>4.9</td><td>7.4</td><td>5.0</td><td>100</td><td>-11 ± 5</td><td>-9 ± 5</td><td>-24 ± 6</td><td>-7 ± 5</td><td>-5 ± 5</td><td>-24 ± 6</td></tr><tr><td>8.8</td><td>12.4</td><td>4.6</td><td>100</td><td>-5 ± 3</td><td>-2 ± 3</td><td>-28 ± 4</td><td>-2 ± 3</td><td>2 ± 3</td><td>-30 ± 4</td></tr><tr><td>15.8</td><td>23.1</td><td>4.1</td><td>100</td><td>8 ± 2</td><td>8 ± 2</td><td>-13 ± 4</td><td>10 ± 2</td><td>11 ± 3</td><td>-16 ± 4</td></tr><tr><td>25.5</td><td>37.0</td><td>4.0</td><td>32</td><td>3 ± 3</td><td>6 ± 3</td><td>-24 ± 5</td><td>4 ± 3</td><td>8 ± 4</td><td>-26 ± 5</td></tr><tr><td>55.5</td><td>74.5</td><td>3.2</td><td>7</td><td>1 ± 4</td><td>9 ± 6</td><td>-2 ± 8</td><td>2 ± 4</td><td>10 ± 5</td><td>-3 ± 8</td></tr></tbody></table>

Notes. We characterize the accuracy of our weak-lensing mass measurements using synthetic observations of 639 ΛCDM halos at $z = 0.25$ selected from a DM-only realization of BAHAMAS simulations. We quantify the level of bias in the average cluster mass from weak lensing as $1 + b_{sim,M_Δ}(M_Δ, true) = \langle M_{Δ, WL}/M_{Δ, true} \rangle_g$, where $M_{Δ, true}$ is the true $M_Δ$ mass, $M_{Δ, WL}$ is the $M_Δ$ mass estimated from weak lensing, and those quantities in brackets with subscript “g” denote error-weighted geometric means (Equation (24)). Similarly, we define the bias parameter $b_{sim,c_{200}}$ for the concentration parameter.

BAHAMAS sample. We adopt slightly different priors to model the BAHAMAS sample in the LIRA framework (Section 5.1). First, we set $γ_{Y|Z} = γ_{μ_Z,D} = γ_{σ_Z,D} = 0$ because all halos are sampled at a single redshift of $z = 0.25$. Next, we need $σ_{Z,0} \gtrsim 1$ to approximate $P(Z)$ of the BAHAMAS sample (see Figure 15) with a lognormal distribution in $Z = \log(M_{200}/10^{14} h^{-1} M_☉)$. Since such a solution is strongly disfavored by the assumed prior for $σ_{Z,0}$ (see Equation (39)), we fix the center and width of the lognormal mass PDF to $μ_{Z,0} = 0$ and $σ_{Z,0} = 1$, respectively. We thus have a total of four regression parameters, ($α_{Y|Z}$, $β_{Y|Z}$, $σ_{Y|Z}$, $σ_{X|Z}$).

The results are shown in Figures 14 and 19. The $c_{200}-M_{200}$ relation recovered from the synthetic data is summarized as $c_{200} = (3.7 \pm 0.1) \times (M_{200}/10^{14} h^{-1} M_☉)^{-0.14 \pm 0.02}$ with a logarithmic intrinsic dispersion of $σ(\ln c_{200}) = \ln 10σ_{Y|Z} = (27 \pm 3)\%$. We thus accurately recover the true input values of $β_{Y|Z} = 0.15$ and $σ(\ln c_{200}) = 30\%$ (see Appendix A.2.1) within the statistical uncertainties. On the other hand, we underestimate the normalization of the $c_{200}-M_{200}$ relation by $(18 \pm 2)\%$, as found in Appendix A.2.2. The intrinsic dispersion of the $M_{200, WL}-M_{200}$ relation is found to be $σ(\ln M_{200, WL}) = \ln 10σ_{X|Z} = (6.5 \pm 4.8)\%$, with an upper limit of <23% (33%) at the 95% (99.7%) CL (see Figure 19). In Figure 14, we have applied an upward correction of 16% to the normalization inferred from the regression analysis.

As a sanity check, we perform a similar test for lower fiducial values of the intrinsic dispersion $σ(\ln c_{200})$ in the $c_{200}-M_{200}$ relation.

To this end, we select a subsample of BAHAMAS halos that lie within $1\sigma$ scatter from the mean $c_{200}-M_{200}$ relation, which leaves us with 459 halos. This subsample virtually has a lower dispersion of $σ(\ln c_{200}) = 16\%$, which matches the level expected for X-ray regular clusters (Meneghetti et al. 2014). For this subsample, we obtain $c_{200} = (3.7 \pm 0.1) \times (M_{200}/10^{14} h^{-1} M_☉)^{-0.14 \pm 0.02}$ with $σ(\ln c_{200}) = (21 \pm 4)\%$ from the synthetic data (Figure 19). We thus recover the true input values of $β_{Y|Z} = 0.15$ and $σ(\ln c_{200}) = 16\%$ within the statistical uncertainties. We checked that even lower fiducial values of $σ(\ln c_{200}) \sim 10\%$ can be accurately recovered.

### A.2.4. Halo Modeling

Furthermore, we have tested our shear fitting procedures and pipeline using the standard halo model including the effects of surrounding large-scale structure as a 2-halo term (Equation (26)). We describe the projected halo model with $M_{200}$ and $c_{200}$ as fitting parameters and use the same priors as for the NFW model. As demonstrated in Figure 4, the 2-halo term $\Delta\Sigma_{2h}(R)$ is negligibly small in the comoving radial range $R \in [0.3, 3] h^{-1}$ Mpc. When the 2-halo term is neglected, the halo model reduces to the BMO model that describes a smoothly truncated NFW profile (Section 4.4).

The results are summarized in Figure 18 and Table 6. Overall, the two-parameter halo modeling of each individual cluster does not significantly improve the accuracy of weak-lensing measurements of cluster mass and concentration,

<footnote><sup>a</sup> True median value in each logarithmic mass bin.</footnote>
although it yields slightly (<1σ) improved levels of accuracy in
the determination of M<sub>500</sub> and M<sub>200</sub>.

Appendix B

Marginalized Posterior Constraints on Regression
Parameters

In Figure 19, we show marginalized one- and two-
dimensional posterior probability distributions of the regression

parameters (α<sub>Y|Z</sub>, β<sub>Y|Z</sub>, σ<sub>Y|Z</sub>, σ<sub>X|Z</sub>) for the c<sub>200</sub>–M<sub>200</sub> relation of
the BAHAMAS sample (Figure 14) recovered from the
synthetic weak-lensing measurements shown in Figure 17
(for details, see Appendix A.2).

In Figures 20 and 21, we show marginalized posterior
probability distributions of the regression parameters for the
c<sub>200</sub>–M<sub>200</sub> relation (Section 5.2) and the T<sub>300 kpc</sub>–M<sub>500</sub> relation
(Section 5.3), respectively, derived for the XXL cluster
population.

<figure><img src="image_21.png" /><figcaption>**Figure 19.** Marginalized one-dimensional (histograms) and two-dimensional (68% and 95% confidence level contour plots) posterior distributions of the regression parameters for the *c*<sub>200</sub>–*M*<sub>200</sub> relation of the BAHAMAS sample at *z* = 0.25, recovered from synthetic weak-lensing measurements (see Figure 17). For each parameter, the blue solid line shows the biweight central location (*C*<sub>BI</sub>) of the marginalized one-dimensional distribution. The left panel is for the full BAHAMAS sample of 639 halos (Figure 15), and the right panel is for a subsample of 459 halos that lie within 1σ scatter from the mean *c*<sub>200</sub>–*M*<sub>200</sub> relation.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Figure 20.** Constraints on the regression parameters for the $c_{200}-M_{200}$ relation of the XXL sample, showing marginalized one-dimensional (histograms) and two-dimensional (68% and 95% confidence level contour plots) posterior distributions. For each parameter, the blue solid line shows the biweight central location ($C_{BI}$) of the marginalized one-dimensional distribution.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>**Figure 21.** Constraints on the regression parameters for the T<sub>300 kpc</sub>–M<sub>500</sub> relation of the XXL sample, showing marginalized one-dimensional (histograms) and two-dimensional (68% and 95% confidence level contour plots) posterior distributions. For each parameter, the blue solid line shows the biweight central location (C<sub>BI</sub>) of the marginalized one-dimensional distribution.</figcaption></figure>

ORCID iDs

Keiichi Umetsu https://orcid.org/0000-0002-7196-4822

Mauro Sereno https://orcid.org/0000-0003-0302-0325

Elinor Medezinski https://orcid.org/0000-0001-7007-2358

Martin Kilbinger https://orcid.org/0000-0001-9513-7138

Mark Birkinshaw https://orcid.org/0000-0002-1858-277X

Nobuhiro Okabe https://orcid.org/0000-0003-2898-0728

Dominique Eckert https://orcid.org/0000-0001-7917-3892

Yutaka Fujita https://orcid.org/0000-0003-0058-9719

Satoshi Miyazaki https://orcid.org/0000-0002-1962-904X

Masamune Oguri https://orcid.org/0000-0003-3484-399X

David Rapetti https://orcid.org/0000-0003-2196-6675

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