The Hydrostatic Mass Bias and the ${\sigma }_8$ Tension: A Multi-Probe Forecast for Stage-IV/V Surveys

Abstract

The hydrostatic mass bias ($b_{\mathrm{HSE}}$) is a leading systematic uncertainty in cluster cosmology and a principal source of degeneracy with ${\sigma }_8$ and ${\Omega }_m$. We investigate the capability of Stage-IV CMB and optical surveys to calibrate $b_{\mathrm{HSE}}$ using tomographic cross-correlations between the thermal Sunyaev--Zel'dovich (tSZ) effect, galaxy clustering, and weak lensing. We perform a Fisher forecast incorporating realistic survey noise, foreground modeling for clustered CIB and radio sources, and full marginalization over cosmological and astrophysical nuisance parameters, including per-bin galaxy bias perturbations, photometric redshift shifts, intrinsic alignments, and baryonic feedback modeled with HMCode2020. With optimized tomographic binning (10 lens and 5 source bins for LSST; 6 lens and 5 source bins for CSST), we forecast marginalized constraints of $0.98\%$ for SO+LSST, $1.60\%$ for CMB-S4+LSST, and $2.40\%$ for CMB-S4+CSST. Tomography improves $b_{\mathrm{HSE}}$ precision by factors of approximately three relative to non-tomographic analyses, reflecting the role of redshift information in breaking the $b_{\mathrm{HSE}}$--${\sigma }_8$ degeneracy. Optical-only probes provide no direct constraint on $b_{\mathrm{HSE}}$, whereas inclusion of tSZ-containing spectra enables percent-level calibration under realistic systematic assumptions. The results demonstrate that multi-probe tomographic analyses with Stage-IV surveys can achieve robust control of hydrostatic mass bias, strengthening cluster-based constraints on structure growth.

20 pages, 7 figures, 3 tables. Published in The Astrophysical Journal, 1004, 103 (2026). We have added useful citations in this version

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