Weak-lensing mass calibration of \emph{Planck} Sunyaev--Zel'dovich clusters with HSC-SSP Year~3

Technical Summary: Weak-lensing mass calibration of Planck Sunyaev–Zel'dovich clusters with HSC-SSP

Problem and Motivation
Galaxy clusters are critical probes for cosmological parameters, particularly those governing the growth of large-scale structure (e.g., $\sigma_8$ and $\Omega_m$). However, the constraining power of cluster abundance measurements is limited by systematic uncertainties in the mass–observable relation. The Planck satellite provides a large catalog of galaxy clusters selected via the thermal Sunyaev–Zel'dovich (tSZ) effect. While the tSZ signal is nearly redshift-independent, it measures a mass proxy (the Compton-$Y$ parameter) rather than the true halo mass. Converting this proxy to mass typically relies on hydrostatic equilibrium assumptions, which may introduce a systematic bias ($b$) where the inferred mass ($M_{SZ}$) underestimates the true mass ($M_{true}$).

A long-standing tension exists between the mass bias values ($1-b$) derived from weak-lensing (WL) calibrations of Planck clusters and the value required to reconcile Planck cluster counts with primary Cosmic Microwave Background (CMB) anisotropies. Previous WL calibrations have been limited by sample size and depth. This work addresses these limitations by utilizing the deeper and wider coverage of the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) Year 3 (S19A) shape catalog to calibrate a larger sample of Planck clusters.

Methodology
The authors present a WL mass calibration of 19 Planck SZ-selected clusters (from the PSZ2 Union catalog) that overlap with the HSC-SSP S19A footprint. The analysis employs a forward-modeling approach to account for selection effects and statistical biases.

• Data: The study uses the HSC-Y3 shape catalog (based on S19A data), which includes improved PSF modeling and shape measurements for $\sim$35.7 million galaxies. Source galaxies are selected using a "$P(z)$-cut" method to minimize contamination from cluster members and foreground galaxies, utilizing photometric redshifts from the DEmPz algorithm.
• Signal Measurement: The stacked weak-lensing signal, the excess surface mass density $\Delta\Sigma(R)$, is measured using per-cluster lensing weights that match the pipeline's stacking scheme. The analysis includes a null test using random points to identify and subtract additive systematics (e.g., residual PSF leakage).
• Modeling Framework: Unlike simple ratio-based calibrations, the authors use a forward-modeling framework that integrates over the halo mass function weighted by the Planck selection function. This approach self-consistently accounts for:
  • Eddington Bias: The preferential up-scattering of low-mass halos into the SZ-selected sample due to the steeply falling mass function and log-normal scatter in the mass proxy.
  • Miscentering: A fraction of clusters are modeled as having centers offset from the true halo center, following a Rayleigh distribution.
  • Selection Function: The official Planck Union catalog selection function is convolved with the mass scatter.
• Covariance: An analytical covariance matrix is constructed, incorporating contributions from shape noise, uncorrelated projected large-scale structure (LSS), and the noise introduced by subtracting the random-point signal.
• Fitting: Four parameters are fitted over the radial range $0.5\text{--}5.0 \, h^{-1}\text{Mpc}$: the log mass bias $\ln(1-b)$, the miscentered fraction $f_{mis}$, the offset scale $\sigma_{off}$, and the SZ mass scatter $\sigma_{\ln M}$. The analysis uses the Dark Emulator for halo-matter correlation predictions and the Colossus package for NFW profile projections.

Key Results

• Mass Bias Constraint: The primary constraint on the hydrostatic mass bias is $1 - b = 0.73^{+0.10}_{-0.11}$ at an effective redshift $z_{eff} \simeq 0.24$. This corresponds to a $\sim14\%$ statistical uncertainty.
• Scatter and Miscentering: The SZ mass scatter is constrained to $\sigma_{\ln M} = 0.25^{+0.10}_{-0.10}$, consistent with external priors derived from simulations and X-ray studies. The miscentered fraction is $f_{mis} = 0.28^{+0.10}_{-0.10}$ with an offset scale of $\sigma_{off} = 0.28^{+0.11}_{-0.08} \, h^{-1}\text{Mpc}$.
• Effective Mass: The corresponding effective weak-lensing mass for the sample is $\langle M_{500c} \rangle_{WL} \simeq 3.5 \times 10^{14} \, h^{-1}M_{\odot}$.
• Robustness: The results are robust to variations in the fitting range ($R_{max} = 5$ vs. $10 \, h^{-1}\text{Mpc}$), the covariance model (including or excluding LSS terms), and the width of the prior on $\sigma_{\ln M}$. While the data alone provide only a modest constraint on $\sigma_{\ln M}$, the inferred $1-b$ remains stable within $\lesssim 1\sigma$ across different prior assumptions.

Significance and Claims
The paper claims that this measurement is consistent with recent weak-lensing calibrations of SZ-selected clusters from other surveys (e.g., ACT, SPT, DES) and supports the picture that significant mass bias corrections are required. Specifically, the result ($1-b \approx 0.73$) is consistent with previous HSC-based Planck calibrations but with improved precision due to the larger sample size and deeper data.

The authors state that while this result partially alleviates the tension between cluster abundance measurements and primary CMB constraints on cosmological parameters, it does not fully resolve it. The remaining discrepancy suggests that either unaccounted observational systematics (such as baryonic effects, triaxiality, or selection biases) or extensions to the standard $\Lambda$CDM model may be necessary. The paper emphasizes that the current uncertainty is dominated by statistical limits of the $N=19$ sample, and future surveys like the Vera C. Rubin Observatory LSST will be required to push systematics below the percent level and test for redshift-dependent variations in the mass bias.