Estimating the heterogeneity of pressure profiles within a complete sample of 55 galaxy clusters: a Bayesian Hierarchical Model

TL;DR

We address the heterogeneity of galaxy-cluster pressure profiles by developing a forward-modeling Bayesian hierarchical approach that jointly infers the population-averaged pressure profile and the intrinsic scatter while delivering cluster-specific profiles. The method uses a restricted cubic spline representation in log-log space, a Student's t-distributed scatter to handle outliers, and SZ data corrections via Abel transforms and beam transfer functions, applied to a complete sample of 55 SZ-detected clusters from SPT+Planck. The results show the population profile is close to the universal pressure profile with about a 12% normalization deficit within r500 and a steeper outer decline, with intrinsic scatter peaking at intermediate radii and modest differences across morphologies. These findings help reconcile cosmological tensions and provide a robust, nonparametric framework for analyzing cluster thermodynamics across diverse samples and future multi-wavelength studies.

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 parameterization 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 required for 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 $M_{500}>4\times 10^{14}M_\odot$, enabling subdivision into sizable morphological classes based on eROSITA data. The shape of the population-averaged profile, at our 250 kpc FWHM resolution, closely resembles the universal pressure profile, despite the flexibility of our model to accommodate alternative shapes, with a ~12% lower normalization, similar to what is needed to alleviate the tension between cosmological parameters derived from the CMB and Planck SZ cluster counts. Beyond $r_{500}$, our 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 non-standard concavity in their radial profiles but no outliers in amplitude. When dividing the sample by morphology, we find remarkably similar pressure profiles across classes.

Figures (20)

• Figure 1: Studied sample (colored points) and entire SPT catalog Bocquet2019. Gray points within the redshift-mass selection area did not meet the S/N or footprint constraints. Crosses indicate galaxy clusters that were subsequently removed from our sample.
• Figure 2: Derivation of $r_{500}$ for two galaxy clusters, SPT-CLJ0051-4834 (left panel) and SPT-CLJ0328-5541 (right panel), shown as examples. Integrating the model ensures regularity in the case of noisy data (left panel) and does not affect high-quality data (right panel).
• Figure 3: Comparison of our estimates of $r_{500}$ (y-axis) with Planck for the 38 clusters in common (left panel) and with SPT for the entire sample (right panel). Our $r_{500}$ values are strongly correlated with the Planck ones, while a larger scatter is observed when compared with SPT. No morphology-dependent trend is observed.
• Figure 4: Distribution of the Reduced Chi-Square $\chi^2_{\nu=6}$ of the 58 individual fits and the population fit of 55 clusters. The red outlier with $\chi^2_{\nu}>7$ is SPT-CLJ0405-4916, which is characterized by a highly oscillating surface brightness distribution in its outermost region, while our background model, the pedestal, is a constant. For this reason, we excluded it from the population analysis.
• Figure 5: Distribution of the number of resolution elements, $\theta_{500}/$HWHM, across the initial sample of 58 objects. Because of our careful sample selection in mass and redshift, all clusters have $\theta_{500}/$HWHM > 5, i.e., are well-resolved. The two clusters with $\theta_{500}/$HWHM $> 30$ are excluded from later analyses because of the high computational time required to include them in the hierarchical model.