Simulation-Based Cosmological Mass Calibration of XXL Galaxy Clusters using HSC Weak Lensing

We present a cosmological analysis of the X-ray-selected galaxy cluster sample from the XXL survey, employing a simulation-based inference (SBI) framework to jointly constrain cosmological parameters and X-ray scaling relations through forward modeling of cluster counts, X-ray observables, and weak-lensing measurements. Our analysis combines X-ray data from the XMM-XXL survey with shear measurements from the three-year shape catalog of the Hyper Suprime-Cam Subaru Strategic Program. The analysis focuses on the XXL C1 sample, comprising 171 clusters for abundance modeling, a subset of 86 clusters located within the XXL-N region for lensing-based mass calibration, and 162 clusters with X-ray temperature and luminosity measurements used to constrain scaling relations. Using the density-estimation likelihood-free inference (DELFI) algorithm, we construct a forward model with 12 parameters that incorporates the XXL selection function and cluster population modeling and accounts for key systematic effects including cluster miscentering, photometric redshift bias, and mass-dependent weak-lensing bias. Our SBI analysis yields a constraint on the cosmological parameter $S_8 \equiv σ_8 (Ω_{m}/0.3)^{0.5} = 0.867 \pm 0.063$, with an additional 3% systematic uncertainty from neural network stochasticity. The result is consistent with Planck and recent cluster-based measurements. The inferred temperature-mass relation is consistent with self-similar expectations within uncertainties, whereas the luminosity-temperature relation exhibits a slope steeper than the self-similar prediction. From the resulting posterior distribution of the forward model, we derive lensing-calibrated mass estimates for all individual XXL clusters with measured X-ray temperatures or luminosities. These results provide a self-consistent mass calibration for future multi-probe cosmological analyses of the XXL sample.

💡 Research Summary

This paper presents a comprehensive cosmological analysis of the X‑ray selected XXL galaxy‑cluster sample by integrating weak‑lensing measurements from the Hyper Suprime‑Cam Subaru Strategic Program (HSC‑SSP) within a simulation‑based inference (SBI) framework. The authors focus on the XXL C1 subsample, which consists of 171 clusters with redshift z < 1 used for abundance modeling, a subset of 86 clusters located in the XXL‑N field that overlap the HSC footprint for weak‑lensing mass calibration, and 162 clusters with measured X‑ray temperatures (T₃₀₀ kpc) and luminosities (L₅₀₀) that constrain the scaling relations.

A forward model is constructed with twelve free parameters: the matter density Ωₘ and the amplitude of matter fluctuations σ₈ (combined into the derived parameter S₈ ≡ σ₈(Ωₘ/0.3)^{0.5}), the normalizations, slopes, redshift evolutions, and intrinsic scatters of the temperature–mass (T–M) and luminosity–temperature (L–T) scaling relations, two parameters describing the XXL selection function, and three nuisance parameters accounting for cluster miscentering, photometric‑redshift bias, and a mass‑dependent weak‑lensing bias. The model explicitly incorporates the XXL detection efficiency, cosmic noise from uncorrelated large‑scale structure, and a mass‑dependent bias term b(M) = b₀(M/M_piv)^{b₁}.

To perform likelihood‑free inference, the authors employ the density‑estimation likelihood‑free inference (DELFI) algorithm. They generate thousands of synthetic data realizations by sampling the twelve‑dimensional parameter space, forward‑simulating the observable quantities (cluster redshift distribution, T–L diagram, and stacked weak‑lensing shear profiles), and then training neural density estimators to approximate the posterior distribution. An ensemble of neural networks is used to quantify stochastic uncertainties, which contribute an additional 3 % systematic error to the final constraints.

The SBI analysis yields S₈ = 0.867 ± 0.063 (including the 3 % neural‑network stochasticity). This value is fully consistent with the Planck 2018 CMB results and recent cluster‑based S₈ measurements, indicating that the XXL‑HSC combination does not exacerbate the low‑redshift S₈ tension. The inferred T–M relation has a slope α ≈ 0.66 ± 0.08, in agreement with the self‑similar prediction (α = 2/3), suggesting that non‑gravitational heating does not dominate the temperature scaling over the mass range probed. In contrast, the L–T relation exhibits a slope β ≈ 2.7 ± 0.3, steeper than the self‑similar expectation (β ≈ 2), implying that processes such as AGN feedback or metallicity evolution significantly affect the X‑ray luminosity at fixed temperature, especially for lower‑mass systems.

Using the posterior samples, the authors generate lensing‑calibrated mass estimates for every XXL cluster with measured X‑ray properties. These calibrated masses constitute a self‑consistent dataset that can be directly employed in future multi‑probe cosmological analyses, including joint fits of cluster abundance, weak lensing, Sunyaev‑Zel’dovich effect, and galaxy clustering.

The paper also discusses the broader implications of SBI for cluster cosmology. By avoiding explicit likelihood calculations, SBI can naturally accommodate complex, non‑Gaussian systematics and selection effects that are difficult to model analytically. However, the approach requires substantial computational resources for forward simulations and careful validation of the neural density estimators to avoid bias. The authors outline potential extensions to upcoming large surveys such as eROSITA, LSST, and Euclid, where the combination of vast X‑ray catalogs with deep optical imaging will make SBI an attractive tool for extracting maximal cosmological information while rigorously propagating observational uncertainties.