CHEX-MATE: Scaling relations of radio halo profiles for clusters in the LoTSS DR2 area

The thermal and non-thermal components in galaxy clusters have properties that, although shaped from different physical phenomena, can share some similarities, mainly driven by their halo mass and the accretion processes. Scaling relations have been proven to exist for both components and studied in X-ray (thermal) and radio (non-thermal) bands. On the X-ray side, both integrated and spatially resolved profiles have shown a predictable and correlated behaviour. At the radio wavelength, such investigations are so far limited to the integrated quantities (e.g. total power and mass). We aimed to investigate the scaling relations between the mass of a galaxy cluster and its radio emission at low frequencies, treating both the integrated and the spatially resolved quantities for a sample of well-selected objects in a self-consistent analysis. We crossmatched LoTSS DR2 and CHEX-MATE datasets in order to get the deepest and most homogeneous radio data of a representative sample of objects. Among the 40 CHEX-MATE objects in the LOFAR DR2 area, we investigated the 18 objects showing radio halo emission, which span a broad mass range, by extracting and analysing their radio emission profiles. We analytically derived the expected relation between the radio power (Pν) and radio surface brightness profile (IR(r)), and performed a comparison with observational results. We obtained that properly accounting for the mass and redshift dependence in the radio profile can reduce the overall scatter by a factor of ∼4, with an evident residual dependence on the cluster dynamical status. We show that assuming the halo size RH ∼ R500 did not allow us to reconcile the expected (from our analytical derivations) and observed mass profile scaling. Instead, accounting for no RH - M relation, allowed us to reconcile the observed radio profile mass scaling and the one predicted starting from the Pν - M relation. We discuss the implications of a lack of RH - M relation, assessing possible systematics and biases in the analyses, and interpreting it as a natural consequence of the structure formation process, where the halo size depends on both the cluster dynamical status, related to the strength of the merger, and mass. Finally, we also considered the role of the magnetic field in the Pν - M relation, putting constraints on its dependence upon the cluster mass and finding consistent results with expectations from our radio power mass scaling.