CHEX-MATE: A LOFAR pilot X-ray – radio study on five radio halo clusters

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Introduction
Galaxy clusters are excellent laboratories for studying nonthermal phenomena, such as particle acceleration processes on large scales.In the last decades, a growing number of studies evidenced the presence of radio synchrotron emission in many of these objects (e.g.van Weeren et al. 2019, for a review).In particular, the most puzzling cases of radio emission are the extended radio sources (e.g.radio relics, mini-halos, giant halos).Giant radio halos are among the most extended cases of such emission.They are diffuse sources characterised by a steep spectral index (α < −1, where S ν ∝ ν α ) most frequently found in massive merging clusters.Historically, two main scenarios have been proposed to explain radio halo origin: turbulent re-acceleration and hadronic models.In the former, relativistic electrons are re-accelerated via a Fermi-II like process by the turbulence injected in the Intra-Cluster Medium (ICM) by cluster-cluster merger events (Brunetti et al. 2001;Petrosian 2001).In the latter case, instead, synchrotron emitting electrons are produced as secondary particles, i.e. as a decay product of heavy nuclei/particle collisions.These secondary models exploit cosmic ray protons (CRp) long lifetime over which they can diffuse on ∼Mpc scales and then, through collisions with ICM heavy nuclei, produce Cosmic Ray electrons (CRe) throughout all the cluster environment (e.g.Dennison 1980;Blasi & Colafrancesco 1999;Pfrommer & Enßlin 2004).Such CRp can be produced mainly by AGN activity but also by accretion shocks and galactic outflows (Brunetti & Jones 2014).The non-detection of gamma rays from π 0 decay from clusters (Ackermann et al. 2016;Brunetti et al. 2017;Adam et al. 2021), the connection of radio halos with mergers (Cassano et al. 2010(Cassano et al. , 2023;;Cuciti et al. 2021) and the spectral properties of halos (e.g.Blasi 2001;Brunetti et al. 2008;Macario et al. 2013;Bruno et al. 2021) suggest that the role of hadronic collisions is subdominant.Therefore, turbulence acceleration should be the driving mechanism for halo production and the required energy to obtain this (re-)acceleration is expected to be injected via merger events.Indeed clusters form by accretion of sub-clusters and groups through mergers, which release up to ∼ 10 64 ergs into the ICM in a cluster crossing time (∼ Gyr, Tormen et al. 2004).Such energy is mainly dissipated as gas heating through shocks, enhancing the ICM thermal bremsstrahlung emission in the X-rays.However, a small fraction of this energy can be channelled into turbulence and re-accelerates relativistic electrons triggering synchrotron radio emission (Brunetti & Jones 2014).As the clusters' mass sets the initial energy budget, it is expected that more massive systems host the most powerful radio halos which emit at higher energies, up to GHz frequencies.Less energetic events, due to the merger of less massive systems, will give rise to less luminous halos, emitting only at low radio frequencies: the Ultra Steep Spectrum Radio Halos, a crucial class of objects to understand the radio halo formation mechanism and for which low frequency observations are providing a growing evidence of their existence.(e.g.Wilber et al. 2018;Duchesne et al. 2021;Di Gennaro et al. 2021).
However, the details of the re-acceleration process are yet to be understood.Based on which turbulence mechanism re-accelerates relativistic particles (transit time damping; Alfven resonant scattering, etc.) the interplay between the various microphysical players (amplification of magnetic fields, heating of the plasma, etc) might lead to very different and complex scenarios (e.g.Brunetti & Lazarian 2007, 2011;Miniati & Beresnyak 2015;Brunetti & Lazarian 2016).Spatially resolved studies of radio and X-ray derived quantities are becoming possible for a large sample of objects in these years.They have the great potential to put constraints on the microphysics of the ICM and, in turn, on the particle acceleration mechanisms (e.g.Govoni et al. 2001;Rajpurohit et al. 2021b).Recent works exploited the correlation among radio and Xray surface brightness in the centre and in the periphery of clusters to derive constraints on the non-trivial distribution of the synchrotron emitting electrons (e.g.Biava et al. 2021).Similarly, from radio spectral index profiles it is now possible to constrain some of the crucial parameters of the model, such as the typical re-acceleration time (t acc , e.g.Rajpurohit et al. 2023).Specifically, LOFAR statistical studies on large samples follow the predictions of the simplest homogeneous models, with parameters that do not vary throughout the cluster volume (e.g.Cassano et al. 2023).However when radio spectral index profiles are available that allow to constrain model parameters such as t acc this might not be the case (e.g.Rajpurohit et al. 2023;Bonafede et al. 2022).In this work, we found further evidence of that, with the model that poorly reproduces the observed quantities (Sec.5.4).So far, spatially resolved studies have been performed using different radio and X-ray facilities, with different sensitivities and at different frequencies.All these effects can deeply impact the outcomes and make it difficult to compare results from different works.Therefore, a systematic radio and X-ray spatially resolved analysis is required to confirm or disprove the found relations and pose new tests for radio halos theoretical models.With the present pilot work, which allows us to verify our analyses for future wider studies, we want to start the first systematic study of radio halo resolved brightness properties and their possible correlations with thermal ICM emission.In particular, we aim to study a representative sample of objects, for which we have uniform and homogeneous coverage in both radio and X-ray bands.This is possible by matching the Cluster HEritage project with XMM-Newton -Mass Assembly and Thermodynamics at the Endpoint of structure formation (CHEX-MATE, CHEX-MATE Collaboration et al. 2021) and the LOFAR Two-meter Sky Survey Data Release 2 (LoTSS DR2, Shimwell et al. 2022) datasets, which provide the necessary data in the two requested bands and for a large sample of PSZ2 objects (Botteon et al. 2022a), both in mass and in redshift.We analyze the thermal -non-thermal connection in a first subsample of five radio halo clusters, selected from a wider sample of 18 CHEX-MATE hosting radio halos among the 40 objects observed by LoTSS DR2 (the ongoing survey has already observed the totality of the CHEX-MATE sample in the Northern sky, 77 out of 82 objects).The paper is organized as follows: in Section 2 we present observations and data reduction; in Section 3 we introduce the studied objects; in Section 4 we describe the data analysis; in Section 5 we comment on our results.In Section 6 we draw our conclusions.As for the other papers of the CHEX-MATE collaboration, in this work, we assume a flat, ΛCDM Universe cosmology with H 0 = 70 km/s/Mpc and Ω m,0 = 0.3 (CHEX-MATE Collaboration et al. 2021).

CHEX-MATE
The CHEX-MATE project (CHEX-MATE Collaboration et al. 2021) is a three mega-second XMM-Newton Multi-Year Heritage Programme to obtain X-ray observations of a minimallybiased, signal-to-noise limited sample of 118 galaxy clusters detected by Planck through the Sunyaev-Zel'dovich effect (Planck Collaboration et al. 2016).The program aims to study the ultimate products of structure formation in time and mass, using a census of the most recent objects to have formed (Tier-1: 0.05 < z < 0.2 ; M 500 1 > 2 × 10 14 M ⊙ ), together with a sample of the highest-mass objects in the Universe (Tier-2: z < 0.6; M 500 > 7.25 × 10 14 M ⊙ ).The project acquired X-ray exposures of uniform depth that ensure a detailed mapping of the thermodynamic properties in the cluster volume where the non-thermal plasma is present.Therefore, the CHEX-MATE cluster sample is the best choice for a systematic and statistical analysis of cluster thermodynamic properties, allowing studies in a wide range of mass and redshift.

X-ray data reduction
The X-ray data used in this work have been reduced using the CHEX-MATE pipeline described in Bartalucci et al. (2023) and here we report only the main steps.The clusters were observed with the European Photon Imaging Camera (EPIC, Turner et al. 2001 andStrüder et al. 2001).Datasets were reprocessed using the Extended-Science Analysis System (E-SAS 2 , Snowden et al. 2008) embedded in SAS version 16.1.Flare events have been removed by using the tools 1 M 500 = 500 4 3 ρ c R 3 500 with ρ c the critical density and R 500 the radius within which the average cluster density is 500 ρ c 2 cosmos.esa.int/web/xmm-newton
mos-filter and pn-filter analyzing the light curves in the [2.5-8.5]keV energy range.Time intervals with count rates exceeding 3σ times the mean count rate have been excised.Point sources were filtered from the analysis following the scheme detailed in Section 2.2.3 of Ghirardini et al. (2019) and further described in Bartalucci et al. (2023).
To produce the scientific images used for the analysis, we proceeded as follows.We extracted the photon-count images in the [0.7-1.2]keV band (i.e. the band that maximizes the signal-tonoise of the cluster thermal emission Ettori et al. 2010) for each camera and produced the exposure maps with the tool eexpmap.
The background affecting X-ray observations is due to a sky component, composed of the local Galactic emission and the Cosmic X-ray background (Kuntz & Snowden 2000), and an instrumental component, due to the interaction of high-energy particles with the detector.We followed the strategy described in Ghirardini et al. (2019) to remove the latter one by producing background images accounting for the particle background and the residual soft protons, whereas we used a constant component in the profile to characterize the sky component as described in Bartalucci et al. (2023) The images, exposure, and background maps of the 3 cameras are merged to maximise the statistics following the procedure described in Bartalucci et al. (2023).

LoTSS
The LoTSS (Shimwell et al. 2017) is a deep, 120 − 168 MHz radio survey, that produces high resolution (∼ 6") and high sensitivity (∼ 100 µJy beam −1 ) images of the Northern sky.It had its first data release (LoTSS-DR1) in 2019 (Shimwell et al. 2019, released area ∼ 400 deg 2 ) and the second release in 2022 (Shimwell et al. 2022) providing images and radio catalogs for ∼ 5, 700 deg 2 of the Northern sky.One of the main goals of this survey is to find new diffuse radio sources inside galaxy clusters, such as giant radio halos, to determine their origin and to test theoretical and numerical models.Thanks to its high sensitivity to diffuse sources at low frequencies, the LoTSS allows us to perform detailed studies of radio halos at low frequencies, where these objects show brighter emissions.
For this work, we will use the public data of the PSZ2 galaxy clusters covered in the LoTSS-DR2, available online 3 and that have been described in Botteon et al. (2022a).

Radio data reduction
In the following, we report an overview of the calibration and imaging procedures applied to the radio data.Briggs (1995) weighting scheme with robust=-0.5 and applying Gaussian uv tapers in arcsec equivalent approximately to 25, 50, and 100 kpc at the cluster redshift.
Finally, to better study the diffuse emission, discrete source components have been subtracted from visibility data and then new source subtracted images have been obtained with the same taper values.
Since we are particularly interested in studying the diffuse halo emission, in this work we use images with the further excision of compact source emission and with the lowest resolution.Additionally, prior to exploiting the final images, we perform a visual inspection checking for contaminating sources, both compact and extended.In particular, we search for residual source emission as a consequence of poor subtraction, such as tailed AGN, and for very low (high) surface brightness regions clearly not associated with the radio halo emission (e.g.diffuse emission clearly detached from the central halo, revived fossil plasma or cluster's sub-components).Hence, we manually mask these regions in both X-ray and radio images (see Fig. 1 for the excluded regions).at the moment of the start of the project for these objects the X-ray data analysis was completed down to the spectral analysis.These objects (see Tab. 1) span a broad range in redshift, mass, and merging status, as probed by the morphological parameter M (Campitiello et al. 2022).The latter quantity combines four different morphological indicators: light concentration (c), centroid shift (w) and the power ratios P 20 and P 30 .M is computed by summing the deviations of each parameter from the mean of its distribution (computed over the whole CHEX-MATE sample) in units of standard deviation.In this way, you can obtain a unique indicator of the cluster morphological status, where relaxed (perturbed) clusters have low (high) M values (see Campitiello et al. 2022, for further details on this parameter).By performing a Kolmogorov-Smirnov (KS) test we found a high probability (≳ 95%) for our five targets to be representative of the 18 clusters redshift and M distributions.The KS test on the mass 4 distribution returns a lower value (70%) than in the z and M case, but still high enough to consider the sub-sample representative of the full sample.Therefore, these objects represent an optimal explorative sample to test and tune our analyses for future wider studies.General information about the five studied targets is found in Table 1, while radio and X-ray images are presented in Fig. 1.

Abell 2069
Abell 2069 (A2069) is a merging galaxy cluster with a M 500 = 5.10 × 10 14 M ⊙ located at redshift z=0.115 which is part of the "A2069-supercluster" (Einasto et al. 1997).It consists of two merging components (A2069-A and A2069-B), which are well detected by X-ray images.In the main component A2069-A, two bright elliptical galaxies are present, separated by a projected distance of about 55 kpc (Gioia et al. 1982).In X-rays it shows an elongated thermal emission which seems to be connected with the X-ray emission of A2069-B.Faint and extended non-thermal emission has been detected by Farnsworth et al. (2013) who classified it as a radio halo.This result was also confirmed by Drabent et al. (2015).Drabent et al. (2015) detected diffuse radio emission coming from A2069-B, where Chandra observations showed also the presence of a cold front (Owers et al. 2009).The radio diffuse emission from both A2069-A and A2069-B is detected also in the 144 MHz LOFAR images, as well as an emitting region among the two components.In this work, we show how this connecting region may be separated by the A2069-A halo emission through the radio-X-ray analysis (see also Sec.5.1).The detailed analysis of the LOFAR observations of this target will be presented by Drabent et al. (in preparation).

Abell 2244
Abell 2244 (A2244) is galaxy cluster located at z=0.095 and its SZ derived mass is M 500 = 4.34 × 10 14 M ⊙ .X-ray morphological indicators classify A2244 as a relaxed object, indeed the concentration parameter (c) and centroid shifts (w) measured by Zhang et al. (2023) would assign this object to the quadrant of relaxed objects as defined in Cassano et al. (2010).
A similar classification was provided by Botteon et al. (2022a) and Campitiello et al. (2022).Despite the regular and centrally peaked X-ray morphology, Donahue et al. (2005) reported the lack of a central temperature gradient and a central entropy 4 All the derived masses reported in this work have been extracted from the MMF3 Planck catalogue (Planck Collaboration et al. 2016) intermediate between the ones of cool cores and non-cool cores, all signs of a dynamical disturbance.The likely candidate for such a disturbance is a group of galaxies that interacted with the main cluster leaving a consistent trail of gas in the South.
Given the mass of the object as derived from the SZ signal of M 500 = 4.3 × 10 14 M ⊙ and the mass of the group to be in the range M 500 = 2 − 5 × 10 13 M ⊙ as typical for a 1 keV system (the central temperature measured in the group from our X-ray analysis) we have a relatively off-axis merger with a mass ratio of 10-20.High sensitivity LOFAR 144 MHz observations of A2244 revealed Mpc-scale diffuse emission, not associated with compact sources, likely caused by the turbulence generated by the group passage.
PSZ2G066.41+27.03PSZ2G066.41+27.03, is the highest redshift cluster of our sample (and the 2nd highest of all CHEX-MATE), z=0.575, and it is also quite massive, M 500 = 7.69 × 10 14 M ⊙ .No studies have been performed on this target so far.PSZ2G066.41+27.03shows powerful X-ray diffuse emission with a complex and elongated structure.The morphological parameters of this object would classify it as clearly disturbed, both estimating them within 500 kpc and within R 500 (Botteon et al. 2022a;Campitiello et al. 2022;Zhang et al. 2023).
The radio halo emission detected by LOFAR in PSZ2G066.41+27.03was claimed by Botteon et al. (2022a) for the first time.At these frequencies, the target shows a bright and extended diffuse radio emission, with few compact sources in the field.

Abell 2409
Abell 2409 (A2409) is a low mass galaxy cluster with M 500 = 4.99 × 10 14 M ⊙ and located at z=0.147.As for PSZ2G066.41+27.03,no dedicated studies have been performed on this target.Its morphological parameters classification is not so straightforward, with c = 0.4 ± 0.07 and w = (4.6 ± 1) × 10 −3 that made Campitiello et al. (2022) classify this object as "mixed" (in agreement with the classification made by Botteon et al. 2022a which puts A2409 close to the disturbed quadrant of Cassano et al. 2010).From the CHEX-MATE X-ray images, the overall shape is fairly roundish, however, some inhomogeneities at the very centre of the emission are observed.The most evident one is a small elongated emission in the North-South direction and a probable surface brightness discontinuity in the South-West region.
LoTSS observations show relatively limited diffuse emission coming from A2409, which Botteon et al. (2022a) classified as radio halo emission.However, the authors noted that it was not possible to reconstruct a radio halo profile for this object due to its low surface brightness.In the LoTSS image, an elongated structure is located in the West direction, apparently not associated with radio galaxies or other compact sources.

Abell 1758
Abell 1758 (A1758) is a cluster pair located at z=0.278 and where both the two components (North and South) are undergoing major merger events (e.g.Schellenberger et al. 2019).Be-Fig.1: Radio (left) and X-ray (right) images of A2069, A2244, PSZ2G066.41+27.03,A2409 and A1758.Radio and X-ray contours are present in both images (radio contours in black and magenta; X-ray contours in blue and white).The radio contours are plotted at 2, 4, 8×σ RMS , while the X-ray ones start at a level of 0.2 cts and are spaced with a factor of 2. In cyan are highlighted the excluded regions (see Sec. 4.1).The radio image resolutions are, respectively: 58, 69, 20, 45 and 30 arcsec.For A2069 we also show a larger yellow region used to mask the entire subcomponent emission, as presented in Sec.5.1.
cause of its higher mass (M 500 = 7.80 × 10 14 M ⊙5 ) A1758-N is the most studied cluster of the system.Lensing studies found a bimodal mass distribution of two subclusters, A1758-N East and A1758-N West (Monteiro-Oliveira et al. 2017).In the radio band, the radio halo emission in A1758-N was investigated from 1.4 GHz to 54 MHz (Kempner & Sarazin 2001;Giovannini et al. 2009;Venturi et al. 2013;Botteon et al. 2018Botteon et al. , 2020b)).Instead, A1758-S shows a faint radio halo emission discovered by Botteon et al. (2018).This diffuse emission is also difficult to disentangle from the inter-cluster emission present in this system, which is prominent in the 144 MHz images (Botteon et al. 2020b).A1758-NW shows evident diffuse halo emission.In the East region, instead, there are two emission blobs apparently not associated with any optical galaxy and that might be regions where the plasma might has been somehow locally compressed or reaccelerated.Given their complex nature, it is non-trivial to determine what has originated them and if and how they are associated with larger scale emission (e.g.halo substructures, AGN fossil plasma; see Botteon et al. 2018, for more information on the diffuse sources in A1758-NE.).Therefore, we exclude these cluster regions from our analysis, focusing our study only on A1758-NW.The difference between the A1758-NW and A1758-NE components radio emission emerges also in this work, where we find a better agreement between radio and X-ray surface brightness when excluding the NE part and A1758-S.

Point-to-point analysis
The pioneering work by Govoni et al. (2001) showed the importance of the study of the correlation between the X-ray and radio brightness and posed the basis of its use to derive information on the mechanism responsible for radio emission.This approach has been developed in many subsequent studies up to the recent developments exploiting the strength of low frequency data (e.g.Rajpurohit et al. 2021a).Past works have shown that perturbed galaxy clusters tend to present a positive correlation between radio and X-ray emission which, in general, is found to be sub-linear.(e.g.Giacintucci et al. 2005;Cova et al. 2019;Xie et al. 2020;Hoang et al. 2021).The majority of the results point toward a weaker radial decline of the non-thermal component with respect to the thermal one.However, differences in the instrument, affecting the observing frequency and resolution, and the adopted analysis technique can impact the final results, making it difficult to draw general conclusions.(e.g.Shimwell et al. 2014;Botteon et al. 2020a;Rajpurohit et al. 2021b;Bonafede et al. 2022;Sikhosana et al. 2023) Thanks to LoTSS and CHEX-MATE high sensitivity images in radio and X-ray bands, we can now perform a homogeneous thermal-non-thermal analysis on a spatially resolved basis for a sample of clusters.This also allows us to compare the results for different clusters and perform a statistical study on cluster radio -X-ray relations.
To perform such a study on our sample, we extract the average surface brightness from radio (I R ) and X-ray (I X ) images by constructing a grid (following Govoni et al. 2001) that covers the whole radio halo emission, which, in general, appears less extended than the thermal one.Each box of the grid has an area equal to the radio beam resolution, which is typically larger than the XMM-Newton point spread function in the X-rays.Following Botteon et al. (2020a)  average radio surface brightness is above 2 × σ RMS .Prior to extracting the values from the grid, we took particular care in the excision of sources which are unrelated to the diffuse radio halo emission, as detailed in Sec.2.4.Once the grids are set, we proceed with the surface brightness estimation in each box from both radio and X-ray images.
In particular, we derive the cluster X-ray count rate from the background-subtracted and exposure-corrected images.Given the homogeneous coverage in both the X-ray and radio bands of the sample and the uniformity of the data, we aim to directly compare all the I R − I X correlations of all clusters.Since we want to use physical units for the X-ray images, we perform the conversion from detector units (count/s) to physical units (erg/cm 2 /s).We calculate the conversion from count-rates to fluxes using a thermal spectral model of the source, with the mean temperature as defined in Bartalucci et al. (2023), abundance fixed at Z = 0.3 (using Asplund et al. 2009 solar tables) modified by Galactic absorption.The response files used are the ones for the MOS2 camera on-axis as this is the reference camera for the merged image (see Bartalucci et al. 2023), where the exposure map is scaled according to the response of that camera.Finally, the brightness values we report are the unabsorbed ones and k-corrected to have a 0.7-1.2keV rest-frame flux (see Jones et al. 1998).For the radio band, instead, the k-correction is accounted for by dividing the extracted brightness for the term (1 + z) 1+α assuming a spectral index α = -1.3, as made by Botteon et al. (2022a).Finally, we also account in both bands for the cosmological dimming factor (1 + z) 4 .
We have investigated the thermal -non-thermal relation fitting I X and I R with a power law : where the slope A tells us whether the radio plasma components (CRe and magnetic fields) decline faster or slower with radius than the thermal ICM distribution.
Using the python package PyMC (Salvatier J 2016;Wiecki et al. 2022), we developed a Bayesian regression model that allows us to perform a linear fit of logI R on logI X accounting for x and y errors, selection effects (i.e.Malmquist bias, Malmquist 1922) and the intrinsic scatter in the regression relationship (see also Appendix A). the five objects.
We observe a strong correlation between X-ray and radio brightness.The strength of those correlations is shown also by high Spearman correlation coefficients (r S ), which are reported in Tab. 2. We also notice that the correlations appear stronger when excluding the contaminant regions from the images.Figure 2 also shows that, for each of the clusters, a positive I R −I X correlation exists.All correlations are sub-linear, implying a shallower radial decline of the non-thermal ICM component with respect to the thermal one.We report in Tab. 3 the best-fit values for the slope, intercept and intrinsic scatter of linear relation.In Fig. 2, we see the surface brightness differences of our objects.In particular, the two higher mass clusters (A1758-N and PSZ2G066.41+27.03)present brighter radio emission.The brightest X-ray, emitting regions are found in A2244 and A2409 which are also the two more relaxed objects of our sample (Campitiello et al. 2022) and then a higher central peak might be expected.We also note the absence of any correlations between mass, morphological parameters (M-parameter, c and w, from Campitiello et al. 2022) and the recovered I R − I X correlation slope.Due to the small size of our sample, we cannot make any general claim on this particular lack of correlation and we will further investigate this kind of connection with the full sample.

Radial analysis
Past works have shown how the radial analysis of thermal and non-thermal properties provides additional information about the ICM environment (e.g.interaction between different ICM components and on the properties of CRe and magnetic fields at large radii; Govoni et al. 2001;Pearce et al. 2017;Rajpurohit et al. 2021b;Bruno et al. 2021;Botteon et al. 2020a;Cuciti et al. 2022;Rajpurohit et al. 2023;Bruno et al. 2023).Therefore, we proceed to analyze the radial emission properties in both radio and X-ray bands.

Radial profiles
We extracted radial profiles in both X-ray and radio bands.Given the fact that the radio emission is generally less extended than the X-ray one, the radial profile extraction is guided by the halo emission and starts from the radio halo centre.
We fit the radio profiles using the Halo-Flux Density CAlculator (Halo-FDCA, Boxelaar et al. 2021).This code directly fits the two dimensional surface brightness profile with an exponential model (see also Sec. 5.2) and uses a Markov chain Monte Carlo method for uncertainties estimation.Since this code allows also us to choose the exponential model geometry (circle, ellipse, rotated ellipse and skewed), we exploit the information provided by Botteon et al. (2022a) on the halo profile to select the bestfitting model.The surface brightness model is given by: where I 0 is the central surface brightness and G(r) is a radial function that takes different forms depending on the selected model.For our cases of circle and rotated ellipse models we adopt, respectively: where: where R e is the characteristic e-folding radius and r 2 = x 2 + y 2 .
For the data preparation process we followed the procedure described by Botteon et al. (2022a).We have masked contaminant regions (see Sec: 4.1 and Fig. 1) and reduced the Field of View (FoV) of each radio image to approximately 1.5R 500 × 1.5R 500 before passing them as input to the code.From the Halo-FDCA output, we determine the centre of each halo.Then we extract the radio profile using linearly spaced annuli up to 2.5 R e (the outer limit where we expect to find halo emission, Bonafede et al. 2017).Following Cuciti et al. (2021), we choose the width of the annuli to be half of the Full-Width at Half Maximum of the image beam.We performed the profile extraction based on the shape of the halo, using circular or elliptical annuli depending on what has been used for the halo fitting.
The results of the fits are presented in Table 4.In Fig. 3 we report the observed 1-D radio and X-ray (extracted in the same annuli) radial profile of the five studied clusters.For each annulus, we compute the mean of the radio brightness and associated an error of δS = σ RMS / √ N beam (where N beam is the number of beam in the annulus).The solid red line represents the best-fit halo profile obtained with Halo-FDCA.We see that all our objects show a monotonically decreasing profile, in both radio and X-ray data and with roughly comparable trends in both bands.These 1-D plots provide a good representation of the cluster emission only for those with circular shapes.Therefore, for a highly elliptical halo as PSZ066.41+27.03,we do not report the best-fit line in the 1-D plot, instead, we show the residual image after the subtraction of the elliptical exponential model.We also note that the bump of radio emission at r ∼ 800 kpc in A1758-NW is probably due to residual contaminant emission from the masked regions.
Here we see how, in general, this simple exponential model manages to reproduce the overall radial trend of the halos in our sample (as well as for many of the halos in the past, which is one of the reasons why this simple model was chosen, Murgia et al. 2009).However, when observed in detail, it seems that there are some systematic discrepancies in reproducing simultaneously inner and outer halo regions with a single exponential profile (see also Cuciti et al. 2021;Botteon et al. 2022a).Although it does not affect the rest of our analysis, we will better discuss this point in Sec.5.2.

Radial slope's trend
Finally, we combine the two analyses presented so far to investigate if the correlation slope of the I R − I X relation changes with the radius (e.g.Bruno et al. 2023).Such studies provide information on how the ratio between X-ray and radio components varies across the cluster size.The value of the slope tells us how radio and X-ray components relate.Therefore, a constant slope throughout the whole cluster extension would indicate that their relation is constant, implying that the radial variations of the thermal and non-thermal components are the same.Instead, if we find fluctuations in the slope values, it means that their relation is changing and the value of the slope provides information on which component is increasing/decreasing with respect to the other.Here we investigate the radial variation of the I R − I X relation, both by deriving the correlation slope and studying directly the I R /I X ratio as already done in other works (Biava et al. 2024;Bonafede et al. 2022;Rajpurohit et al. 2023).Here we present and discuss only slope studies and report the results of the ratios in Appendix C. To reduce as much as possible the introduction of biases (as also made by Bruno et al. 2023), here we choose to study these possible slope changes computing it as: where k(r) is the correlation slope of the I R ∝ I k X scaling, ∆ ln I R (I X ) is the difference between the logarithm of the radio (X-ray) brightness in two consecutive annuli (the annuli are centred in the radio peak as for the brightness profile).Again, to reduce the dependence from the chosen separation of the annuli, we try three different widths for them, respectively 1, 1.5 and 2 times the beam width.As shown in Fig. 4, we obtain good agreement among the three different binning choices, finding that they reproduce similar features in the radial profile.The main difference lies in the fact that smaller bins retain larger errors and more scatter but are capable of resolving finer features than the larger ones (e.g. the case of A1758-NW).Figure 4 shows the evident complex trends in the presented halos, displaying how a uniform k is generally not representative of the whole cluster extension.

Comments on point-to-point analysis
As pointed out by Govoni et al. (2001), the point-to-point analysis can provide not only information on the spatial distribution of the thermal and non-thermal ICM but also on the mechanism responsible for halo emission.They showed how, under some reasonable assumptions on the magnetic field profile, the expected correlation between I X and I R in (re-)acceleration and hadronic models is different.In particular, the sublinear scaling often observed in radio halos is hardly reproduced by the hadronic scenarios, as they would require an unphysical amount of CRp in the cluster peripheries to justify such I R − I X relation (e.g.sub-linear I R − I X relations.It implies a flatter profile of the radio components with respect to the X-ray one, suggesting the distribution of CRe and/or the magnetic field to be rather flat compared to the thermal ICM.This potentially indicates a larger size of the radio emission than the X-ray one when not limited by sensitivity, as already observed in a few cases (Shweta et al. 2020;Botteon et al. 2020aBotteon et al. , 2022b;;Rajpurohit et al. 2021b;Bruno et al. 2023).Additionally, we note that if we mask a larger region in A2069, similarly to what is made for the sky regions around A1758-NW (the yellow region in the first panel of Fig. 1), the recovered slope becomes steeper (∼ 0.5).This may indicate that there is a projected diffuse emission between A2069-A and A2069-B.Here we do not speculate further on its origin since is beyond the scope of this work.However, we remark on the potential of these novel analyses in discovering new features thanks to the combination of radio and X-ray data.
Regime changes of the thermal-non-thermal relation throughout the cluster can be investigated through variations of the correlation slope.To test a possible change in the slope, indicating a departure from a single power law relation, we fit the I R − I X relation with a broken power law .As we show in Appendix A, it is worth investigating slope changes if the intrinsic scatter is ≲ 30%.All targets except PSZ2G066.41+27.03have an intrinsic scatter (as retrieved by the power law fit) lower than 30%.Therefore, for these objects, we can search for slope changes.
We use the same approach as described in Sec.4.1, but this time searching for a relationship like: where x c is the X-ray surface brightness value at which the change from the slope α 1 to α 2 occurs.With this analysis, we find that for three (A1758-NW, A2069 and A2244), out of five, objects we found that a broken power law is preferred rather than a simple power law, but with a low significance6 .We also note that all best-fit results, also with a broken power law, remain sublinear.

Discussion on radio profiles
As pointed out in Sec.4.2.1, there are discrepancies when fitting the radio halo profiles with a single exponential profile.The choice of this profile shape is motivated by two reasons: it is a simple function with only two free parameters and it provides a reasonable description of many radio halos (albeit not being physically motivated).Thanks to the new generation of radio interferometers, it is now possible to observe radio halos at a high signal-to-noise ratio with relatively high angular resolution.This allows to recover features in the emission that were invisible with past instruments, showing that the radio emitting structure is difficultly reproduced by such a simple profile.Indications of a departure from the single exponential profile were already present in Cuciti et al. (2021) and Botteon et al. (2022a).Recently, Botteon et al. (2023) pointed out the presence of substructure in radio halos using MeerKAT data.
Here we find similar results.We see a common departure from the exponential profile in the central regions for the most perturbed systems.Specifically, the model tends to overestimate the central emission while recovers optimally the most extended one (this is also true for PSZ2G066.41+27.03,for which in the model-subtracted image negative residuals are found in the central parts while close to zero in the outer ones).Instead, for the two more relaxed clusters, A2244 and A2409, the model seems to systematically underestimate the emission in the cluster centre.This suggests a centrally peaked distribution (Sec.4.1) of the non-thermal ICM component for both objects, alongside a more concentrated thermal component.Instead, the outer regions seem to have the opposite behaviour.A flattening of the radio profile is observed in both objects (even though without high significance for A2409, mainly due to its faint diffuse emission).Therefore, our results support the emerging idea that, despite the overall agreement between the exponential model and the radial halo profile (both on our sample and previous works), radio halos are more complex structures than the ones observed with past radio telescopes.Extreme cases of this have been shown in Cuciti et al. (2022) and Biava et al. (2024) where the authors clearly showed how the single exponential model fails to describe the entire halo profile.

Radial slope's trend
From Fig. 4 it is evident that a single value of k can not be representative for the I R ∝ I k X scaling throughout the whole halo extension.We also notice a different slope k found in the point-to-point analysis with respect to the radial analysis.This difference can be due to the fact that (i) the radial profile reaches a distance of 2 R e from the cluster centre, while, for some objects, the pointto-point analysis stops at smaller radii; (ii) the boxes used for the point-to-point analysis have a smaller area than the annuli used in the radial analysis, possibly retaining real physical fluctuations on smaller spatial scales; (iii) high signal to noise regions affect more the fitting process, i.e. the fitted slope in the point-topoint analysis is more representative of central regions than for the outer ones.In the following, we discuss our results separately for the more relaxed (A2244 and A2409) and more disturbed (A2069, PSZ2G066.41+27.03and A1758-NW) objects, starting with the former.In Fig. 4, we see that for A2244 and A2409 the slopes show a clear decreasing trend with r.This means that there is a flattening of the I R − I X relation moving in the outer regions.Therefore, the non-thermal component radial decline becomes weaker and weaker with respect to the thermal one as outer radii are considered Such a result indicates, for these two less disturbed objects, an increasingly important contribution of the non-thermal component with respect to the thermal one with the radius.The three more disturbed clusters, instead, display different behaviours.A first common feature is that they do not show a decreasing slope trend as the less disturbed systems.These three objects display an overall increasing value of k, indicating that the radio emission steepens with the radius.In particular, k reaches linear and even superlinear values, indicating that the non-thermal component is decreasing faster than the thermal one (in contrast to what was found for A2244 and A2409).7 Hence, the overall picture of these three more disturbed objects is similar to what has been recently observed in Coma by Bonafede et al. (2022).Their slope's profiles indicate a steeper trend in the outer regions than in the inner ones.
A&A proofs: manuscript no.aanda.R1.singleFor a direct comparison with previous literature results, we also show in Appendix C the I R /I X ratio.

Modeling the I R − I X radial trend
Following the work made by Bonafede et al. (2022), we now consider a simple model for the radio emissivity to be compared with our observational results.Since this is a novel analysis, we stress that here we are not performing a fit of the model on our profiles.We just compare a simple model for radio emission with observations to see if general properties can be reproduced and explained with rather basic assumptions.Here we use the model for radio emission adopted also in Cassano et al. (2023) which can reproduce halo statistical properties, such as the halo fraction as a function of the mass.To derive such a model we assume a magnetic field scaling with the thermal gas as B(r) ∝ B 0 n 0.5 e (according to the best constrained magnetic field profile made by Bonafede et al. 2010).In the context of turbulent re-acceleration scenarios, the radio emissivity is (e.g.Brunetti & Vazza 2020): where B IC ≈ 3.25(1 + z) 2 µG is the CMB equivalent magnetic, η e is the re-acceleration efficiency and F is the turbulent energy flux.The latter is expressed as: with σ v the turbulent velocity dispersion on a scale L and ρ is the gas density.If we consider an isotropic distribution of the electrons' momentum space ( f (p)), η e can be written as (Brunetti & Lazarian 2007): Here, U CRe is the energy density of re-accelerated electrons and D pp /p 2 , considering the case of second-order acceleration with super-Alfvenic solenoidal turbulence (Brunetti & Lazarian 2016), can be expressed as: where M t is the turbulent Mach number and v A is the Alfvén velocity.Finally, if we consider a constant temperature profile and a constant turbulent Mach number, the synchrotron emissivity is8 : where ϵ X is the X-ray emissivity and X(r) = U CRe U th is the ratio of CRe energy (U CRe ) to the thermal gas energy (U th ).We compute this emission for different values of the central magnetic field B 0 , in the range 3-10 µG and assuming a constant X.We derive ϵ X by fitting a β-model to our X-ray images using the PYPROFFIT package presented in Eckert et al. (2020).Using this model, we derive the expected correlation slope between the radio and X-ray surface brightness as defined in Eq. 6.Finally, we compare model predictions with observational results (i.e. the slopes' trends in Fig. 4).The results of this comparison are presented in Fig. 5 (we note how all the theoretical k profiles asymptotically tend to 1 by construction).From Fig. 5, we note a generally poor agreement between the data and the model.Only in A1758-NW, we see a marginal agreement of both the trend and the values of k.This indicates that more complicated models should be considered.For instance, a non-constant value of X and/or more complex magnetic field profiles should be taken into account.The observed differences, however, are not the same in every cluster and a separation can be made, again, based on their dynamical disturbance.In A2244 and A2409, the observed k decreases with the radius, implying a flattening of the radio emission, which is opposite to the slopes' trends predicted by the model.A possible explanation of this behaviour is a non-constant value of X(r), implying changes in the energy content of CRe with respect to the one of the thermal gas.In particular, a weaker decline of the radio emission as observed here, requires an increment of X(r) with the radius, i.e. the ratio between CRe and thermal energy density increases.There is no shortage of physical motivation for an increasing X(r).For instance, the occurrence of more energetic merger shocks in the outskirts originates a higher amount of CRe (e.g.Vazza et al. 2017), as well as the amount of turbulence can be more substantial in the external regions due to an off-axis geometry of the merger (which can be found in clusters where the central core has not been disrupted by merging cluster or group passage).We therefore proceed to model a variable X(r) by assuming the following functional form: where r c is the core radius of the best-fit beta model for the X-ray brightness.We tested different values of γ and searched for the one which could minimize the difference between the model and the observed trend of Fig. 5.For both A2244 and A2409, we find the best agreement for γ ∼ 3.9.In Fig. 6, we compare the observed trends and model expectations considering different values for γ and assuming a central magnetic field B 0 = 3µG.To check whether the value of γ = 3.9 alone could explain all the observed properties of the halo, we compute the radio emission originated by this new model.We assume the proposed scaling for X(r) and use Eq. 12 to derive the radio surface brightness profile, as done for observational data.We find that such a steep X(r) behaviour would originate a surface brightness profile which, at large (r ≳ 200 kpc) radii, is brighter than the observed one (lower panels in Fig. 6).These results indicate that (i) an increasing X(r) is indeed required to produce a decreasing I R − I X slope and (ii) that only a varying X(r) is not enough to explain the radial trend.
In fact, other factors (e.g.diverse behaviour of B(r) and, in turn, a different expression for the factor D pp ) likely play a role in these cases, causing the observed shallower decline of the radio emission with respect to the X-ray thermal emission.
In A2069 and PSZG066.41+27.03,k(r) does not display a monotonic trend, though we notice a global increase of it with the radius.Again, this could be related to the dynamical status of the clusters and a non-constant X(r) parameter.However, for these objects we do not perform the same analysis made for A2244 and A2409.The complex k(r) trend (likely due to their dynamical disturbance) cannot be associated with a monotonic variation of X(r).In addition, in these clusters the thermal gas distribution will be naturally less smooth than in the more relaxed systems, making the magnetic field profile more uncertain.Departures of the magnetic field profile from the assumed B(r) ∝ n 0.5 thermal will cause an increase in the model complexity (e.g.non-constant Alfvén velocity) that is beyond the scope of this work.
Fig. 6: Comparison between predicted and observed profiles for different values of γ, assuming a central magnetic field B 0 = 3µG.Upper panels: observed (black points) and predicted (dashed), radio-X-ray slopes as a function of the radius.Lower panels: predicted and observed (shaded) radio brightness profiles.

Comparison with mini halos results
We studied the thermal -non-thermal connection for a homogeneously covered sample of five clusters.In particular, we focused our attention on dynamically disturbed clusters which show radio halo emission detected at low frequencies.
A similar analysis has been carried out on mini-halos by Ignesti et al. (2020), even though at ∼ 1 GHz, and by Biava et al. (2024) on a sample of 12 cool-core clusters with LOFAR.Mini-halos, are diffuse radio sources on hundreds of kpc, with steep spectra (α > 1), located at the centre of massive, relaxed cluster (e.g van Weeren et al. 2019).Their emission is often confined in the cool cores of the clusters, which, as for giant halos, suggests a connection between non-thermal and thermal plasma (e.g.Gitti et al. 2004Gitti et al. , 2007;;Mazzotta & Giacintucci 2008;Giacintucci et al. 2024).The origin of synchrotron emitting electrons in mini-halos is yet unclear.In fact, as for giant radio halos, leptonic and hadronic models have been proposed to explain such emission.In particular, the hadronic scenario for mini-halo production is not ruled out (Ignesti et al. 2020).
The new-generation radio telescopes are highlighting the underlying complexity of these systems, with an increasing number of multi-component mini-halo findings (e.g.Venturi et al. 2017;Savini et al. 2018;Biava et al. 2021;Riseley et al. 2022b,a).In the context of the thermal-non-thermal connection, the pointto-point analysis on mini-halos has shown a recurrent linear or super-linear relation between radio and X-ray surface brightness (e.g.Ignesti et al. 2020;Biava et al. 2021;Riseley et al. 2022b).This is in general in antithesis to what is observed for radio halo sources (see Sec 4.1).It suggests, as other results have shown, a physical difference in the origin of these diffuse sources.
Here, we remark such a difference providing the first consistent comparison of radio halos I X − I R trend.All our halos show sub-linear slopes, while the mini-halos presented in Biava et al. (2021) and Riseley et al. (2022b) show linear or super-linear slopes at LOFAR HBA frequencies.This evidences how the distribution of plasma is different in these two classes of sources.In radio halo clusters, the non-thermal plasma is more extended and less tightly connected to the thermal plasma.Instead, in coolcore clusters, the radio emission seems to be more tightly related to the thermal emission and, in some cases, even more peaked in the cluster centre.The same results are found if we compare our sample with the one of Ignesti et al. (2020), with cool-core clusters which show a linear I X − I R behaviour.We however note that this latter comparison can be only qualitative since the observing frequencies are different.
It is worth mentioning that for the two more relaxed clusters of our sample, a close-to-linear behaviour is observed in the cluster centre (especially for A2409).This might be an indication of a significant contribution of hadronic processes in the core.However, the point-to-point relations of both objects are sub-linear and they are classified as perturbed clusters following their c and w parameters (Botteon et al. 2022a;Campitiello et al. 2022;Zhang et al. 2023).This indicates that, even though in the central region they may look like mini-halo objects, they are different sources originated by different re-acceleration processes.Finally, we compare our results of Fig. C.1 with the recent work of Biava et al. (2024).The authors compute the ratio between radio and X-ray surface brightness as a function of the radius for their four cool-core clusters which present diffuse radio emission in the central regions at 144 MHz.The main analogy with our work comes from the fact that also in the central cluster regions of their objects, there are clear changes in the correlation slopes, as found for our five radio halos.This indicates that, even for the mini-halo emission, there is a non-trivial connection between the thermal and non-thermal components.In particular, in their work, the authors find that objects that show slope changes also have strong departures from a single exponential profile in the radio emission.This indicates the presence of more than a single non-thermal component embedded in the studied regions, which, might cause also slope changes.However, in their sample of four objects, Biava et al. (2024) find both increasing and decreasing I X /I R ratios, while we mainly find increasing ones which are more evident in outer regions.Even though the samples studied so far are small, this might be an indication of a more complex environment in central regions (e.g.due to the stronger impact of radio galaxies on thermal and non-thermal components) than in outer ones, where turbulence is the main responsible for radio emission.In our future work (Balboni et al. in preparation), we plan to shed light on these connections by using the complete sample of CHEX-MATE clusters covered by LoTSS-DR2.

Summary
In this work, we performed a pilot study investigating the radio and X-ray emission of five galaxy clusters from the CHEX-MATE sample.These objects have been selected to be representative of the 18 CHEX-MATE, radio halo clusters observed with LOFAR in LoTSS DR2.
The considered targets, according to the classification made by Campitiello et al. (2022), Botteon et al. (2022a) and Zhang et al. (2023), show signs of disturbance in the ICM.Thanks to the combination of high-quality data provided by the CHEX-MATE and LoTSS DR2 datasets we have been able to perform the first consistent comparison of spatially resolved properties in X-ray and radio bands.Our results can be summarized as follows: -We found a strong connection between I X and I R in all clusters (confirmed by a high Spearman ranking).All the analyzed objects presented sub-linear slopes in the I R − I X plot, suggesting a slower decline of the non-thermal (CRe + B) component with respect to the thermal one.-The five objects we have analysed do not show any particular trend of the best-fit slope with other parameters like mass, dynamical disturbance, temperature, total power, etc.Instead, we report the possible presence of a projected intercluster emission among A2069-A and A2069-B, remarking how this novel analysis can be used to discern among clusters' sub-components.-We found that for three clusters of the sample, a broken power law might be preferred, rather than a simple power law, to describe the I X − I R relation.We aim to further investigate these features for a wider sample, searching for general behaviours.
-Through the radial analysis, we found a rather smooth trend of the radio halo profile and the X-ray emission in all objects.
In agreement with Botteon et al. (2022aBotteon et al. ( , 2023)), we find that, when studied in detail, the single exponential model struggles to reproduce the whole radio halo profile.Interestingly, the two more relaxed objects, A2244 and A2409, also display a peaked radio profile, which is similar to what is observed in the X-ray, suggesting a tighter connection between the thermal and non-thermal components in central regions.-Investigating the radial behaviour of the I R − I X correlation slope we found departures from a constant value indicating radial shifts in the thermal and non-thermal contribution in all targets.However, we found differences between more and less perturbed objects: less disturbed clusters show a decreasing slope profile, indicating, indicating a flatter I R ∝ I k X scaling in external cluster regions, whereas clusters with a higher degree of disturbance show a more complex k radial behaviour and a general increasing slope which was also found in the Coma cluster (Bonafede et al. 2022) -We compared our findings with the expectations from turbulent re-acceleration models, under simplified assumptions (e.g.homogeneous conditions, a constant temperature, a constant turbulent Mach number, and a constant ratio of the energy density of CRe to thermal gas).Although the model is simplified, it has been proven successful in reproducing statistical properties of radio halos (e.g.Cassano et al. 2023).
We find that this model poorly reproduces almost every observed k profile, suggesting that more complicated models should be considered.-We showed how the model cannot reproduce the increasing amount of the non-thermal component with respect to the thermal one observed in more relaxed objects.This supports an increment in the amount of CRe energy density over the thermal one, for which different physical motivations can be presented.However, we showed that an increasing ratio of the energy densities is not sufficient to fully reproduce the observed properties and other model parameters (e.g.different B(r) or a non-constant acceleration efficiency) need to be taken into account.-Among the disturbed systems, A2069 and PSZ066.41+27.03display complex k(r) profiles.In these cases, we did not search for a best-fit X(r) since their high dynamical disturbance will further increase the uncertainties of the analysis results (e.g.uncertain B(r) profile).A1758-NW, instead is the only one marginally consistent with the model, possibly suggesting smaller variations of X(r).We also note that, to the best of our knowledge, PSZ066.41+27.03 is the highest redshift cluster for which a radio-X-ray spatially resolved study has been performed.-By comparing our homogeneous study with the ones made on samples of mini-halos, we remarked the difference between the sub-linear I R − I X scalings of radio halos and the linear or super-linear ones found in mini-halos.This indicates a tighter relation between thermal and non-thermal components in the latter.However, the comparison of the radial trend of the I R /I X ratio showed that for both mini-halos and radio halos a uniform trend is not evident.
Our work supports the idea of a non-thermal component that has a more uniform distribution with respect to the thermal one, despite not being completely decoupled from it and with a connection (I X − I R ) that changes throughout the cluster extension.In a forthcoming publication, we will investigate these results on a larger sample as may have important implications on radio halo origins.
Fig. 2: I R − I X relations of all the clusters in a single plot.The label reports the cluster names together with the best-fit slopes and their errors.The best-fit lines are the colored solid lines.

Fig. 3 :
Fig.3: Radial profiles for the five studied objects (in the cases of elliptical annuli, we consider the major axis for the radial distance).The radio emission with the best-fit model found from Halo-FDCA and errors (shaded) is shown in red; the X-ray emission extracted in the same regions is shown in blue.The red dotted line is the 1σ threshold of the radio images.For PSZ2G066.41+27we report the 1D profile and the residual image after the subtraction of the elliptical exponential model, with the masked regions in red.

Fig. 4 :
Fig.4: I R − I X slope as a function of the radius (i.e.distance from the halo centre) for the five presented targets.The k values are computed using Eq.6 among the consequent annuli.To reduce possible dependencies on the annulus width, we report the results for three different sizes of the annuli (1, 1.5, 2× beam size).The red dotted horizontal line indicates the best-fit slope found by the point-to-point analysis (with its error) up to the radius considered for that analysis.

Fig. 5 :
Fig. 5: The same k value of in Fig. 4 with overlayed model's predictions for different values of B 0 ranging between 3 -10 µG.

Fig. A. 1 :
Fig. A.1: Fitted slope change as a function of the threshold set on data.From upper left to lower right, the real data slope is 1.2, 1., 0.8, 0.6.Points are slightly offset on the x-axis for visualization purposes.

Fig. A. 2 :
Fig. A.2: Plots of the ∆elpd loo of the two models (broken and power law ) as a function of both scatter (x-axis) and threshold set (y-axis) in terms of percentage of excluded data.Different figures represent different cases of break point assumption; from upper left to bottom right panel x c = 2, 4, 6, 8.The orange and white contours trace a ∆elpd loo of 10 and 4 respectively.The horizontal dashed line is placed at 40%, while the two vertical lines are at one scatter of 0.20 and 0.30.

Fig
Fig. B.1: k-corrected I R − I X relations for the five studied clusters.The points are color coded based on their distance from the radio halo emission peak.The best-fit line is shown in black and the red dashed line is the σ RMS of the image.

Fig. C. 1 :
Fig. C.1: I R /I X ratio as a function of the radius (i.e.distance from the halo centre) for the five presented targets.For A2069, the red dotted vertical line defines the radius beyond which the average signal in the annulus falls below 1 σ RMS .The blue thinner lines are at R e and 2R e .

Table 1 :
Sample of the selected clusters present in LoTSS DR2 and CHEX-MATE observations, with their redshift, mass and Y S Z parameter within R 500 (from the Planck cluster catalogue PSZ2 Planck Collaboration et al. 2016), radio power at 150 MHz as found by

Table 2 :
, we consider only the cells where the Spearman correlation coefficients for the I R −I X relations of each cluster with and without considering "contaminant" regions.

Table 3 :
Correlation slopes, intercepts and recovered intrinsic scatter of the I X − I R relation for the five studied targets.

Table 4 :
Brunetti & Jones 2014).Here we provide a consistent comparison among five radio halos whose results confirm the general view that radio halos show Results of the radio exponential fit with Halo-FDCA.