CHEX-MATE: Turbulence in the intra-cluster medium from X-ray surface brightness fluctuations

The intra-cluster medium is prone to turbulent motion that will contribute to the non-thermal heating of the gas, complicating the use of galaxy clusters as cosmological probes. Indirect approaches can estimate the intensity and structure of turbulent motions by studying the associated fluctuations in gas density and X-ray surface brightness. In this work, we aim to constrain the gas density fluctuations occurring in the CHEX-MATE sample to obtain a detailed view of their properties in a large population of clusters. To do so, we use a simulation-based approach to constrain the parameters of the power spectrum of density fluctuations, assuming a Kolmogorov-like spectrum and including the stochastic nature of the fluctuation-related observables in the error budget. Using a machine-learning approach, we learn an approximate likelihood for each cluster. This method requires clusters not to be too disturbed, as fluctuations can originate from dynamic processes such as merging. Accordingly, we removed the less relaxed clusters (centroid shift w > 0.02) from our sample, resulting in a sample of 64 clusters. We defined different subsets of CHEX-MATE to determine properties of density fluctuations as a function of dynamical state, mass, and redshift, and we investigated the correlation with the presence or not of a radio halo. We found a positive correlation between the dynamical state and density fluctuation variance, a non-trivial behaviour with mass, and no specific trend with redshift or the presence of a radio halo. The injection scale is mostly constrained by the core region. The slope in the inertial range is consistent with the Kolmogorov theory. When interpreted as originating from turbulent motion, the density fluctuations in R-500 yield an average Mach number of M-3D similar or equal to 0.4 +/- 0.2, an associated non-thermal pressure support of P-turb/P-tot similar or equal to (9 +/- 6)%, or a hydrostatic mass bias b(turb) similar or equal to 0.09 +/- 0.06. These findings align with expectations from existing literature.