Estimating the heterogeneity of pressure profiles within a complete sample of 55 galaxy clusters: A Bayesian hierarchical model

¹ INAF-Osservatorio Astronomico di Brera, via Brera 28, 20121 Milano, Italy
² Insubria University, Department of Pure and Applied Sciences, Computer Science Division, via Ottorino Rossi, 21100 Varese, Italy
³ INAF-Osservatorio Astronomico di Brera, via Emilio Bianchi 46, 23807 Merate, Italy

^⋆ Corresponding author: fabio.castagna@inaf.it

Received: 15 May 2025
Accepted: 7 August 2025

Abstract

Galaxy clusters exhibit heterogeneity in their pressure profiles, even after rescaling, highlighting the need for adequately sized samples to accurately capture variations across the cluster population. We present a Bayesian hierarchical model that simultaneously fits individual cluster parameters and the underlying population distribution, providing estimates of the population-averaged pressure profile and the intrinsic scatter, as well as accurate pressure estimates for individual objects. We introduce a highly flexible, low-covariance, and interpretable parametrization of the pressure profile based on restricted cubic splines. We model the scatter properly accounting for outliers, and we incorporate corrections for beam and transfer function, as is required for Sunyaev-Zel’dovich (SZ) data. Our model is applied to the largest non-stacked sample of individual cluster radial profiles, extracted from SPT+Planck Compton-y maps. This is a complete sample of 55 clusters, with 0.05 < z < 0.30 and M₅₀₀ > 4 × 10¹⁴ M_⊙, enabling subdivision into sizable morphological classes based on eROSITA data. Fitting is computationally feasible within a few days on a modern (2024) personal computer. The shape of the population-averaged pressure profile, at our 250 kpc full width at half maximum resolution, closely resembles the universal pressure profile, despite the flexibility of our model for accommodating alternative shapes, with a ∼12% lower normalization, similar to what is needed to alleviate the tension between cosmological parameters derived from the cosmic microwave background and Planck SZ cluster counts. Beyond r₅₀₀, our pressure profile is steeper than previous determinations. The intrinsic scatter is consistent with or lower than previous estimates, despite the broader diversity expected from our SZ selection. Our flexible pressure modelization identifies a few clusters with nonstandard concavity in their radial profiles but no outliers in amplitude. When dividing the sample by morphology, we find remarkably similar pressure profiles across classes, though regular clusters show evidence of lower scatter and a more centrally peaked profile compared to disturbed ones.