Issue 
A&A
Volume 665, September 2022



Article Number  A24  
Number of page(s)  23  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202243074  
Published online  02 September 2022 
Linking a universal gas density profile to the coreexcised Xray luminosity in galaxy clusters up to z ∼ 1.1
^{1}
Université ParisSaclay, Université Paris Cité, CEA, CNRS, AIM de ParisSaclay, 91191 GifsurYvette, France
email: gabriel.pratt@cea.fr
^{2}
HH Wills Physics Laboratory, University of Bristol, Tyndall Ave, Bristol BS8 1TL, UK
^{3}
IRFU, CEA, Université ParisSaclay, 91191 GifsurYvette, France
Received:
10
January
2022
Accepted:
26
May
2022
We investigate the regularity of galaxy cluster gas density profiles and the link to the relation between coreexcised luminosity, L_{Xc}, and mass from the Y_{X} proxy, M_{YX}, for 93 objects selected through their SunyaevZeldovich effect (SZE) signal. The sample spans a mass range of M_{500} = [0.5−20]×10^{14} M_{⊙}, and lies at redshifts 0.05 < z < 1.13. To investigate differences in Xray and SZE selection, we compare to the local Xrayselected REXCESS sample. Using XMMNewton observations, we derive an average intracluster medium (ICM) density profile for the SZEselected systems and determine its scaling with mass and redshift. This average profile exhibits an evolution that is slightly stronger than selfsimilar (α_{z} = 2.09 ± 0.02), and a significant dependence on mass (α_{M} = 0.22 ± 0.01). Deviations from this average scaling with radius, which we quantify, indicate different evolution for the core regions as compared to the bulk. We measure the radial variation of the intrinsic scatter in scaled density profiles, finding a minimum of ∼20% at R ∼ [0.5−0.7] R_{500} and a value of ∼40% at R_{500}; moreover, the scatter evolves slightly with redshift. The average profile of the SZEselected systems adequately describes the Xrayselected systems and their intrinsic scatter at low redshift, except in the very central regions. We examine the evolution of the scaled core properties over time, which are positively skewed at later times, suggesting an increased incidence of centrally peaked objects at lower redshifts. The relation between coreexcised luminosity, L_{Xc}, and mass is extremely tight, with a measured logarithmic intrinsic scatter of σ_{lnLXcMYx} ∼ 0.13. Using extensive simulations, we investigate the impact of selection effects, intrinsic scatter, and covariance between quantities on this relation. The slope is insensitive to selection and intrinsic scatter between quantities; however, the scatter is very dependent on the covariance between L_{Xc} and Y_{X}. Accounting for our use of the Y_{X} proxy to determine the mass, for observationally motivated values of covariance we estimate an upper limit to the logarithmic intrinsic scatter with respect to the true mass of σ_{lnLXcM} ∼ 0.22. We explicitly illustrate the connection between the scatter in density profiles and that in the L_{Xc} − M relation. Our results are consistent with the overall conclusion that the ICM bulk evolves approximately selfsimilarly, with the core regions evolving separately. They indicate a systematic variation of the gas content with mass. They also suggest that the coreexcised Xray luminosity, L_{Xc}, has a tight and wellunderstood relation to the underlying mass.
Key words: Xrays: galaxies: clusters / galaxies: clusters: intracluster medium / largescale structure of Universe
© G. W. Pratt et al. 2022
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
In a Λ cold dark matter Universe, halo assembly is driven by the hierarchical gravitational collapse of the dominant dark matter component. To first order, this process is selfsimilar and scalefree because gravity has no characteristic scale. If the baryon content remains constant, powerlaw relations link the baryonic observable properties – such as Xray luminosity L_{X}, or SunyaevZeldovich effect (SZE) signal Y_{SZ}, or total optical richness Λ – to the cluster mass and redshift. Each of these observables is sensitive to a different underlying intrinsic physical characteristic (e.g., the distribution of gas or the number of red sequence galaxies above a given threshold). However, the detection of a baryonic observable also depends on the intrinsic properties of the signal itself. For example, while the SZE signal is proportional to the gas density, the Xray emission is proportional to the square of the gas density. This means that Xray measurements are very sensitive to the physical conditions in the core regions, while SZE measurements are much less so.
The Xray luminosity, L_{X}, is an attractive quantity because it can be measured from very few source counts once the redshift is known. For a virialised galaxy cluster where the intracluster medium (ICM) is in hydrostatic equilibrium, L_{X} depends only on the halo mass, M, redshift, z, and the distribution of the gas in the dark matter potential. Powerlaw L_{X} − M relations have indeed been observed (e.g., Maughan 2007; Rykoff et al. 2008; Zhang et al. 2008; Connor et al. 2014; Pratt et al. 2009; Vikhlinin et al. 2009; Schellenberger & Reiprich 2017; Lovisari et al. 2020). However, these relations exhibit a large intrinsic scatter (∼40 per cent), linked to the presence of cool cores and merging activity (e.g., Pratt et al. 2009). Exclusion of the core regions, by measuring L_{X} in an annulus excluding the cluster centre, significantly reduces the intrinsic scatter (Fabian et al. 1994; Maughan 2007; Pratt et al. 2009; Mantz et al. 2018).
Radial gas density profiles have been a key means of obtaining information about the ICM since the advent of Xray imaging. The high spatial resolution observations afforded by XMMNewton and Chandra observations have revealed the complexity of both the core regions, which are strongly affected by nongravitational processes (Croston et al. 2008; Pratt et al. 2010), and the outskirts, where the gas distribution becomes progressively more inhomogeneous (e.g., Eckert et al. 2015). Key open questions are how the ICM evolves over time in the dark matter potential, and how this connects to the formation and evolution of cool cores.
The advent of SZE surveys has resulted in the detection of large numbers of clusters at z > 0.5 (Hasselfield et al. 2013; Bleem et al. 2015; Planck Collaboration XXVII 2016; Hilton et al. 2021), extending the redshift leverage for studies of how the population changes over time. The suggestion that Xrayselected and SZEselected samples may not have the same distribution of dynamical states (e.g., Planck Collaboration IX 2011; Rossetti et al. 2016; AndradeSantos et al. 2017; Lovisari et al. 2017) has prompted examination of the relationship between the baryon signatures and the true underlying cluster population. Indeed, the dynamical state may well be as fundamental a characteristic as the mass or the redshift (Bartalucci et al. 2019). At the same time, McDonald et al. (2017) found that the new SZEselected samples suggest that cool cores are in place very early in the history of a cluster, and have not changed in size, density, or total mass up to the present. These authors further suggest that much of what was thought to be cool core growth over time is in fact due to the selfsimilar evolution of the cluster bulk around this static core.
Here we investigate the properties of the gas density profiles and the coreexcised Xray luminosity, L_{Xc}, of 31 Xrayselected clusters at z < 0.2 and 93 SZEselected clusters at z < 1.13. We describe a universal gas density profile for the SZEselected objects and quantify its variation with redshift and mass. We quantify the radial variation in scaled profiles with respect to the bestfitting evolution and the bestfitting mass dependence. Outside the cores, the median scaled gas density profiles are remarkably similar, showing no dependence on selection. We obtain the radial variation of the intrinsic scatter in scaled profiles and investigate the evolution of this scatter with redshift. We find that L_{Xc}, measured in the [0.15−1] R_{500} region^{1}, is an extremely wellbehaved mass proxy that does not depend on cluster selection, and shows little evolution beyond selfsimilar in the broad redshift range of the sample.
Throughout the paper we assume a flat ΛCDM cosmology with Ω_{m} = 0.3, Ω_{Λ} = 0.7, and H_{0} = 70 km s^{−1} Mpc^{−1}. The SunyaevZeldovich flux in units of square arcminutes is denoted Y_{SZ}; the quantity , in units of square megaparsecs, is the spherically integrated Compton parameter within R_{500}, where D_{A} is the angular diameter distance of the cluster. Unless stated otherwise, logarithmic quantities, including scatter, are given to base e, and uncertainties are quoted at the 68% confidence level.
2. Data and analysis
2.1. Dataset
The dataset consists of 31 Xrayselected clusters at 0.05 < z < 0.2 and 93 SZEselected objects at 0.08 < z < 1.13, with six systems in common. The local Xrayselected dataset is REXCESS (Böhringer et al. 2007; Croston et al. 2008; Pratt et al. 2009). The SZEselected systems consist of a local sample comprising a subset of 44 objects at z < 0.5 from the Planck Early SZ sample (ESZ; Planck Collaboration VIII 2011), which Bartalucci et al. (2019) show is representative of the full ESZ; these were complemented by a further 49 distant objects observed by XMMNewton in a series of three large programmes obtained as part of the M2C project. These cover the redshift ranges 0.5 < z < 0.7 (LP1, ID 069366, 072378), 0.7 < z < 0.9 (LP2, ID 078388), and z > 0.9 (LP3, ID 074440). The LP1 sample was selected from objects detected at a signaltonoise S/R > 4 in the Planck SZ catalogue, and confirmed by Autumn 2011 to be at z > 0.5. The LP2 sample consists of clusters with estimated masses M_{500} > 5 × 10^{14} M_{⊙} at 0.7 < z < 0.9 in the PSZ2 catalogue. The LP3 sample is derived from the five highestmass objects at z > 0.9 from the combined Planck and SPT catalogues (Bartalucci et al. 2017, 2018). Full observation details for the sample can be found in Table A.1.
The lefthand panel of Fig. 1 shows the distribution of the 118 clusters in the redshiftmass plane. Here and in the following, we group the SZEselected systems into three subsamples in the redshift ranges z < 0.3 (blue), 0.3 ≤ z < 0.6 (light green), and z ≥ 0.6 (dark green), containing 37, 27, and 29 objects, respectively. The righthand panel of Fig. 1 shows the stacked mass histogram. This plot makes clear that the REXCESS sample (light blue) has a lower median mass (M_{500} = 2.7 × 10^{14} M_{⊙}) than any of the SZEselected subsamples (M_{500} = 6.4, 7.9, and 5.0 × 10^{14} M_{⊙}, respectively). The LP2 subsample is subject to significant Eddington bias in the Planck signal, leading in the most extreme case to an estimated mass of only 6.5 × 10^{13} M_{⊙} for PSZ2 G208.57−44.31. We show below that this has a negligible effect on our results.
Fig. 1. Sample properties. Left: redshiftmass distribution of the clusters used in this paper. The SZEselected clusters comprise a subset of 44 systems from the Planck Early SZ sample (Planck Collaboration VIII 2011) at z < 0.5 and a further 49 clusters at z > 0.5. REXCESS (Böhringer et al. 2007) is an Xrayselected sample of 31 objects at z < 0.25. Right: stacked histogram of the mass distribution. The REXCESS sample has a lower median mass than the SZEselected samples. 
2.2. Analysis
As our aim was to compare the SZEselected clusters to the lowredshift Xrayselected REXCESS systems, we followed the Xray data reduction and analysis procedures described in Croston et al. (2008), Pratt et al. (2009, 2010). Event files were reprocessed with the XMMNewton Science analysis System v15 and associated calibration files. Standard filtering for clean events (PATTERN < 4 and < 13 for MOS1/2 and pn detectors, respectively, and FLAG = 0) and soft proton flares was applied. The instrumental and particle background was obtained from custom stacked, recast data files derived from observations obtained with the filter wheel in the CLOSED position (FWC), renormalised using the count rate in a high energy band free of cluster emission.
Vignettingcorrected, backgroundsubtracted [0.3–2] keV surface brightness profiles were extracted in annular bins centred on the Xray peak. Temperature profiles were produced using the procedures described in Pratt et al. (2010). These were extracted in logarithmically spaced annular bins centred on the Xray peak, with a binning of R_{out}/R)_{in} = 1.33−1.5 depending on data quality. After subtraction of the FWC spectra, all spectra were grouped to a minimum of 25 counts per bin. The FWCsubtracted spectrum of the region external to the cluster was fitted with a model consisting of two MEKAL components plus an absorbed power law with a fixed slope of Γ = 1.4. The spectra were fitted in the [0.5−10] keV range using χ^{2} statistics, excluding the [1.4−1.6] keV band (due to the Al line in all three detectors), and, in the pn, the [7.45−9.0] keV band (due to the strong Cu line complex). In these fits the MEKAL models were unabsorbed and have solar abundances, and the temperature and normalisations are free parameters; the powerlaw component is absorbed by the Galactic absorption. Since it has a fixed slope, only its normalisation is an additional free parameter in the fit. This bestfitting model was added as an extra component to the annular spectral fits, with its normalisation rescaled to the ratio of the areas of the extraction regions (corrected for bad pixels, chip gaps, etc). In the annular spectral fits, the temperature and metallicity of the cluster component were left free, and the absorption was fixed to the HI value (Kalberla et al. 2005). The metallicity was fixed to a value of Z = 0.3 Z_{⊙} when its relative uncertainty exceeded 30%.
2.2.1. Luminosity
The coreexcised Xray luminosity L_{Xc} was measured in the [0.15−1] R_{500} region for all objects. Here the only change with respect to the analysis in Pratt et al. (2009) was the use of an updated M_{500} − Y_{X} relation from Arnaud et al. (2010) to estimate the relevant masses M_{500} and scaled apertures R_{500}. We show below that this change has a negligible impact on the results. The coreexcised Xray luminosity L_{Xc} was calculated both in the bolometric ([0.01−100] keV) and soft ([0.5−2] keV) bands for comparison to previous work. As in Pratt et al. (2009), the luminosities were calculated from the [0.3–2] keV band surface brightness profile count rates, using the bestfitting spectral model estimated in the [0.15−1] R_{500} aperture to convert from count rates to luminosity. In cases where the surface brightness profile did not extend to R_{500} (seven systems), we extrapolated using a power law with a slope measured from the data at large radius. Errors on L_{Xc} take into account the uncertainties in the spectral model, the count rates, and the value of R_{500}, and were estimated from Monte Carlo realisations in which the luminosity calculation was derived for 100 surface brightness profiles, the profiles and R_{500} values each being randomised according to the observed uncertainties. Once obtained, the luminosities were further corrected for point spread function (PSF) effects by calculating the ratio of the observed to PSFcorrected count rates in each aperture (see below). Table B.1 lists the resulting masses and luminosities for the present sample, in addition to the name, redshift, RA, and Dec.
2.2.2. Density profiles
The vignettingcorrected backgroundsubtracted [0.3–2] keV surface brightness profiles were used to obtain the deprojected, PSFcorrected density profiles using the regularised, nonparametric technique described in Croston et al. (2006), and applied to the REXCESS sample in Croston et al. (2008). The surface brightness profiles were converted to gas density by calculating an emissivity profile Λ(θ) in XSPEC, taking into account the absorption and instrumental response, and using a parameterised model of the projected temperature and abundance profiles (see e.g., Pratt & Arnaud 2003). The Croston et al. (2006) method uses the parametric PSF model of Ghizzardi (2001) as a function of the energy and angular offsets, the parameters of which can be found in EPICMCTTN011^{2} and EPICMCTTN012^{3}. In Bartalucci et al. (2017), the deprojected density profiles from XMMNewton observations of a number of clusters obtained using this method were compared to Chandra observations, for which the PSF can be neglected. It was shown that the results obtained with the Croston et al. (2006) method reproduced the deprojected Chandra density profiles accurately down to an effective resolution limit of ∼5 arcseconds (Fig. 6 of Bartalucci et al. 2017). The gas density ρ_{gas} = n_{e} × (μ_{e} m_{p}), where n_{e} is the electron density measured in Xrays, m_{p} is the proton mass, and μ_{e} = 1.148 is the mean molecular weight per free electron:
3. Gas density profiles
3.1. Model
In the selfsimilar model, a cluster can be completely defined by only two parameters: its mass, M_{V}, and its redshift, z. A fundamental property of this model is the cluster overdensity, Δ, with respect to the reference density of the Universe ρ_{Uni}(z), from which the virial mass , and radius R_{V}, can thereafter be defined. Cluster profiles then exhibit a universal form when the radii are scaled to R_{V}.
The original, and simplest, selfsimilar model concerns tophat spherical collapse in the Standard CDM (Ω = 1) cosmology. Here a cluster at redshift z is represented by a spherical perturbation that has just collapsed, with ρ_{Univ}(z) being the critical density of the Universe, ρ_{c} = 3 H^{2}(z)/(8 π G), and Δ = 178. Of course, the hierarchical formation of structure in a ΛCDM Universe is a very complex dynamical process: objects continuously accrete matter along largescale filaments, and there is no strict boundary that would separate a virialised region from the infall zone. The definitions of the mass and the corresponding overdensity are therefore ambiguous, as is the choice of ρ_{Univ}(z), as one can use either the critical density or the mean density (see Voit 2005, for a review).
Using numerical simulations, Lau et al. (2015) showed that the structure of the inner part of clusters that is typically covered by X–ray observations is more selfsimilar when scaling by fixed overdensities with respect to the critical density ρ_{c}(z). The zone in question corresponds to overdensities of Δ ≳ 200. As a scaling radius, we therefore chose an R_{500} corresponding to Δ = 500, the radius within which the mean matter density is ρ_{500} = 500 ρ_{c}(z). The corresponding total mass within this radius, M_{500}, is
with
and where E(z) = [Ω_{m}(1 + z)^{3} + Ω_{Λ}]^{1/2} is the evolution of the Hubble parameter with redshift in a flat cosmology. The scaled gas density profile expressed as a function of scaled radius x is then
In the selfsimilar model, ρ(x) follows a universal shape and its normalisation is independent of mass and redshift. In such a case ⟨ρ_{gas}⟩∝ρ_{500}, where the angle brackets denote the average within R_{500}, as expected for gas evolution purely driven by gravitation. Figure 2 shows the scaled density profiles of all 118 systems. If the clusters were perfectly selfsimilar, they would trace the same locus in this plot. This is clearly not the case. The colourcoding by mass and by redshift highlights that at large radius, the scaled profiles of the highermass, higherredshift systems lie systematically above those of lowermass, lower redshiftobjects. These trends suggest a dependence of the scaling on M_{500} and/or redshift.
Fig. 2. Deprojected, PSFcorrected density profiles for 118 galaxy clusters, normalised by the critical density ρ_{crit} and R_{500}. Fully selfsimilar clusters would trace the same locus in this plot. The profiles are colourcoded by mass M_{500} in the lefthand panel, and by redshift z in the righthand panel. There are clear trends with respect to both quantities. 
To better understand this dependence, we fitted the observed scaled profiles with a model consisting of a median analytical profile, the normalisation of which is allowed to vary with z and M_{500}, with a radially varying intrinsic scatter. The median profile was expressed as
where f(x) is the function describing the profile shape. Here we adopted a generalised NavarroFrenkWhite (GNFW) model (Nagai et al. 2007):
where x_{s} is the scaling radius, and the parameters (α, γ, 3β) are the central (x ≪ x_{s}), intermediate (x ∼ x_{s}), and outer (x ≫ x_{s}) slopes, respectively. The case α = 0 and γ = 2 corresponds to the standard β model (Cavaliere & FuscoFemiano 1976), while the case γ = 2 corresponds to the AB model introduced by Pratt & Arnaud (2002). The latter was used to model the median density profile of the REXCESS sample (Piffaretti et al. 2011).
The normalisation is given by the product f_{0} A(z, M_{500}), where A(z, M_{500}) describes the departure from standard selfsimilarity in terms of a possible mass and/or redshift dependence of the scaled gas density. For this we assumed a powerlaw dependence on M_{500} and E(z):
The standard selfsimilar model corresponds to α_{M} = 0, α_{z} = 0. We expect α_{M} > 0, as it is well established that the gas mass fraction of local clusters decreases with decreasing mass due to nongravitational effects (e.g., Pratt et al. 2010). The model above allows us to disentangle mass dependence and possible evolution.
Equations (6) and (7) translate into a gas mass fraction within R_{500}, f_{g, 500}, which varies with mass and redshift as a function of α_{M} and α_{z}:
where M_{gas}(< R_{500}) is the gas mass within R_{500} and we have used Eqs. (2) and (4). The quantity I(x_{s}, α, γ, 3β) is the three dimensional integral value for f_{0} = 1, which depends solely on the shape parameters.
We introduced a radially varying intrinsic scatter term around the model profile, assuming a lognormal distribution at each radius. Taking into account measurement errors, the probability of measuring a given scaled gas density ρ at given scaled radius x for a cluster of mass M_{500} at redshift z is then
where 𝒩 is the lognormal distribution. The variance term, σ^{2}(x), is the quadratic sum of the statistical error, σ_{stat} on the measured log ρ and of the intrinsic scatter on log ρ_{m} at radius x, σ_{int}(x).
We expect the intrinsic scatter to increase towards the centre, as observed in Fig. 2, due to the increasing effect of nongravitational physics on the density profiles. Ghirardini et al. (2019) studied the intrinsic scatter of massive local clusters in the XCOP sample, modelling the radially varying scatter σ_{int}(x) with a logparabola function. However, we found that such an analytical form significantly overestimates the scatter in the inner core, which was not covered by their data. To allow for more freedom we used a nonanalytical form for the intrinsic scatter, where σ_{int}(x) is defined at n equally spaced points in log(x) in the typical observed radial range, [x_{min}–x_{max}]. The scatter, σ_{int}(x) at other radii is computed by spline interpolation. We used n = 7 between x_{min} = 0.01 and x_{max} = 1.
The likelihood of a set of scaled density profiles measured for a sample of i = 1, N_{c} clusters of mass M_{500, i} and redshift z_{i} is:
where N_{R}[i] is the number of points of the profile of cluster i, and the quantity ρ_{i, j} = ρ_{gas}[i, j]/ρ_{500}(z_{i}) is the scaled density measured at each scaled radius x_{i, j} = r[i, j]/R_{500}(z_{i}, M_{500, i}), with ρ_{gas}[i, j] and r[i, j] being the physical gas density and radius. The statistical error on log ρ_{i, j} is σ_{stat, i, j}.
We fitted the data (i.e. the set of observed ρ_{i, j}) using Bayesian maximum likelihood estimation with Markov chain Monte Carlo (MCMC) sampling. Using the emcee package developed by ForemanMackey et al. (2013), we maximised the log of the likelihood, which reads (up to an additive constant)
The fit marginalises over a total of fourteen parameters: four describing the shape of the median profile (x_{s}, α, γ, 3β), a global normalisation, f_{0}, the slopes α_{M} and α_{z} that describe the nonstandard mass and evolution dependences, and seven additional parameters describing the intrinsic scatter profile. We used flat priors on all parameters.
3.2. Results
To establish a baseline, we fitted the model described above to the 93 SZEselected systems. The resulting bestfitting model is
with
and
where
Figure 3 shows the marginalised posterior likelihood for the parameters of the bestfitting density profile model detailed in Sect. 3.1. All parameters are well constrained: in particular, we note that the mass and evolution parameters α_{M} and α_{z} do not show any degeneracies, implying that we clearly separate the mass and redshift effects.
Fig. 3. Marginalised posterior likelihood for the parameters of the bestfitting density profile model detailed in Sect. 3.1. 
The lefthand panel of Fig. 4 shows the density profiles of the SZEselected clusters together with the bestfitting model. The intrinsic scatter term is represented by the orange envelope; the numerical values for this term are given in Table 1. The righthand panel of Fig. 4 shows the bestfitting model for the SZEselected systems compared to the profiles from the Xrayselected REXCESS sample. The agreement is excellent beyond the core; in the inner regions, there is a hint that the Xrayselected systems may show more dispersion. We will return to this point below in Sect. 5.1.5.
Fig. 4. The universal cluster ICM density profile. Left: scaled density profiles of the SZEselected clusters (grey points), overplotted with the bestfitting GNFW model with free evolution and mass dependence: ρ_{gas}/ρ_{500}(R/R_{500})∝E(z)^{αz} M^{αM} with α_{z} = 2.09 ± 0.02 and α_{M} = 0.22 ± 0.01 (orange line). The model includes a radially varying intrinsic scatter term (orange envelope). Right: comparison of the bestfitting model, defined on the SZEselected sample, to the bestfitting model for the Xrayselected REXCESS sample (light blue). Here, the points with error bars are the REXCESS sample. 
Numerical values for the bestfitting intrinsic scatter term, measured at seven equally spaced points in log(R/R_{500}), in the range [0.01−1] R_{500}.
4. Luminosity scaling relations
We now turn to the scaling relation between L_{Xc} and the mass M_{500}. The bolometric Xray luminosity of a cluster can be written (Arnaud & Evrard 1999)
where f_{gas} = M_{gas}/M is the gas mass fraction, and Λ(T) is the cooling function. The quantity is a dimensionless structure factor that depends on the spatial distribution of the gas density (e.g., clumpiness at small scale, shape at large scale, etc.). Further assuming (i) virial equilibrium of the gas in the dark matter potential [M ∝ T^{3/2}]; (ii) simple Bremsstrahlung emission [Λ(T)∝T^{1/2}]; (iii) similar internal structure []; (iv) a constant gas mass fraction [], the standard selfsimilar relation between bolometric Xray luminosity and mass, L_{X} ∝ M^{4/3}, can be obtained. Similar arguments can be used to obtain the softXray luminositymass relation of L_{X} ∝ M.
4.1. Fitting method
We fitted the data with a powerlaw relation of the form
where L_{0} = 1 × 10^{44} erg s^{−1} and 5 × 10^{44} erg s^{−1} for the soft and bolometric bands, respectively, and X_{0} = 5 × 10^{14} M_{⊙} keV and 4 × 10^{14} M_{⊙} for Y_{X} and M, respectively. Fitting was undertaken using linear regression in the loglog plane, taking uncertainties in both variables into account, and including the intrinsic scatter. We fitted the data using a Bayesian maximum likelihood estimation approach with Markov chain Monte Carlo (MCMC) sampling. We write the likelihood as defined by Robotham & Obreschkow (2015)
with
and the intrinsic scatter, σ^{2}, as a free parameter. MCMC sampling was undertaken using the emcee package developed by ForemanMackey et al. (2013), with flat priors in the ranges [ − 2.0, 3.0] and [0.0, 1.0] for B_{L} and σ, respectively. The results, reported in Table 2, were compared to those obtained with the LINMIX (Kelly 2007) Bayesian regression package: these were indistinguishable and so are not reported here.
Fits to the coreexcised Xray luminosity L_{Xc} − M relation.
4.2. Results
4.2.1. L_{Xc} − Y_{X}
We first fitted the relation between the bolometric L_{Xc} and the mass proxy Y_{X} for the SZEselected systems only. With the evolution term left free, the bestfitting relation, shown in Fig. 5, is
Fig. 5. Relation between L_{Xc} and Y_{X}. The blue envelope is the bestfitting relation given in Eq. (23), and the results from Maughan (2007) are also shown for comparison. 
with an intrinsic scatter of σ_{LXcYX} = 0.09 ± 0.01. This result yields evolution and mass dependences that are in excellent agreement with the selfsimilar predictions of −9/5 and 1.0, respectively. It is also in excellent agreement with the REXCESS only results of Pratt et al. (2009) and that of Maughan (2007), the latter of which was estimated from Chandra data and is overplotted on the figure.
4.2.2. L_{Xc} − M: Low redshift with fixed evolution
We initially fixed the evolution factor, n, to the selfsimilar values of −2 and −7/3 for the soft and bolometric bands, respectively. We first fitted the bolometric L_{Xc}–M for the REXCESS data only, using a mass pivot of 2 × 10^{14} M_{⊙}, as used by Pratt et al. (2009). The resulting normalisation, A_{L} = 1.06 ± 0.04, slope B_{L} = 1.73 ± 0.05, and intrinsic scatter σ_{lnLXc} = 0.16 ± 0.03, are in excellent agreement with those found by Pratt et al. (2009) using orthogonal Bivariate Correlated Errors and intrinsic Scatter (BCES; Akritas & Bershady 1996) fitting. Similarly good agreement was found for the softband L_{Xc}–M relation, showing that the scaling relation parameters that are obtained for the Xrayselected are robust to the change in underlying M − Y_{X} relation used to estimate the mass, and also to differences in fitting method.
We then fitted the SZEselected clusters at z < 0.3 (37 systems). The results are given in Table 2 and show that the normalisation and slope for this subsample are in agreement within 1σ with those found for REXCESS. This indicates that there is no difference in the scaling relation between the local Xray and SZEselected samples, once the core region has been excised. There is a slight hint that the intrinsic scatter of the SZEselected sample about the bestfitting relation is lower than that for REXCESS, although this is only a ∼2σ effect for the bolometric luminosity, and is less significant for the softband. The data and bestfitting relations, including the 1σ scatter envelopes, are shown in the lefthand panel of Fig. 6.
Fig. 6. Relation between the coreexcised Xray luminosity L_{Xc} and mass estimated from the Y_{X} proxy, for the bolometric and softband luminosities of 118 systems. Left: data points with the best fitting relation to the Xrayselected REXCESS sample with the evolution factor fixed to the selfsimilar value of n = −2 (grey envelope). The dark grey envelope shows the best fitting relation to the 37 SZEselected systems at z < 0.3 with n = −2. Right: histogram of the log space residuals from the best fitting relation to the SZEselected objects at z < 0.3. 
4.2.3. L_{Xc} − M: Free evolution
The righthand panel of Fig. 6 shows the histogram of the residuals of the full SZEselected sample (93 systems, 0.08 < z < 1.13) with respect to the bestfitting relation to the systems at z < 0.3. The peak is offset by ΔlnL ≲ 0.1, indicating that some evolution beyond selfsimilar is in fact needed.
We then fitted the full SZEselected sample with a powerlaw relation, including a free evolution factor, n. The results are given in Table 2 and the bestfitting relations are shown in the lefthand panel of Fig. 7; the righthand panel of Fig. 7 shows the residual histograms. The latter are wellcentred on zero. The bestfitting evolution terms, n = −2.23 ± 0.09 for the softband and n = −2.50 ± 0.10 for the bolometric luminosity, suggest that stronger than selfsimilar evolution is significant at the ∼2σ level.
Fig. 7. Relation between the coreexcised Xray luminosity L_{Xc} and mass estimated from the Y_{X} proxy, for the bolometric and softband luminosities of 118 systems. Left: best fitting relation (grey envelope) to the full sample (data points) with the evolution factor n left free to vary. The bestfitting values of n are given in Table 2. Right: histogram of the log space residuals from the best fitting relation. Solid lines show the bestfitting Gaussian distributions with σ corresponding to the bestfitting intrinsic scatter in log space (Table 2). 
5. Discussion
5.1. Gas density profiles
5.1.1. Comparison with previous work
Pioneering work on parametric models of scaled density profiles (Neumann & Arnaud 1999) obtained from ROSAT allowed the dispersion in radial slopes to be constrained. Croston et al. (2008) studied the scaled density profiles of the REXCESS sample, obtaining for the first time constraints on the radial dependence of the intrinsic scatter. The scaled density profile of the XCOP sample, obtained assuming a selfsimilar evolution factor E(z)^{2}, was presented in Ghirardini et al. (2019). Figure 8 compares the scaled density profiles and the bestfitting model from our SZEselected sample to their median scaled density profile and 68% dispersion. The agreement is good out to the maximum XCOP radius of ∼2 R_{500}, although with subtle differences in the inner regions (R < 0.1 R_{500}). Their profile is less peaked, likely due to their not having corrected the density profiles for PSF effects, and has a smaller dispersion than our sample, which may be linked to their more limited mass coverage. The righthand panel compares the intrinsic scatter measurements, which also agree quite well, although the scatter of the present sample is better constrained. The bestfitting intrinsic scatter obtained from our sample when the evolution factor is forced to the selfsimilar value of E(z)^{2} is also shown in grey.
Fig. 8. Scaled density profiles. Left: scaled density profiles (points) and bestfitting model (orange envelope) for SZEselected systems in our sample compared to the median and 68% dispersion from the XCOP sample (Ghirardini et al. 2019, magenta envelope). Right: comparison of best fitting intrinsic scatter model (blue) with that found by Ghirardini et al. (2019, magenta). The bestfitting intrinsic scatter obtained from our sample when the evolution factor is forced to the selfsimilar value of E(z)^{2} is also shown in grey. 
We can also compare to the results obtained by Mantz et al. (2016), who modelled the evolution and mass dependence of the scaled density profiles of a morphologically relaxed cluster sample of 40 systems at 0.08 < z < 1.06. While their evolution dependence of 2.0 ± 0.2 is in agreement with our results, they find a mass dependence that is consistent with zero (0.03 ± 0.06). The difference with respect to our results may be due simply to cluster selection. They studied dynamically relaxed, hot systems, for which the mass leverage is more limited. Once scaled, they found a scatter in scaled density of ≲20% at R_{2500}. For the typical mass of the present sample, R_{2500} ≈ 0.45 R_{500}, where our intrinsic scatter measurements are in good agreement with theirs.
5.1.2. Radial dependence of the scaling
Once the best fitting model was obtained, we quantified how well the model represents the data by calculating the variation of scaled density at different scaled radii. To better disentangle redshift and mass evolution, we extracted two subsamples: one covering a large redshift range at nearly constant mass, and another covering a large mass range at nearly constant redshift. These subsamples are illustrated in the z − M plane by the green and blue regions in the lefthand panel of Fig. 9. We defined ten radial bins in terms of x = r/R_{500} and measured the gas density in each bin. We then scaled the density by the bestfitting model and fitted a function of the form ρ_{gas}(x)/ρ_{crit, 0} ∝ E(z)^{αz} M^{αM}. For a selfsimilarly evolving population, the gas density scales with the critical density ρ_{crit} and there is no mass dependence, and so α_{z} = 2 and α_{M} = 0.
Fig. 9. Deviations from the average scaling with radius. Left: redshiftmass distribution of the SZEselected sample used in this work. The shaded regions indicate cuts for two subsamples: a large redshift range at nearly fixed mass, and a large mass range at nearly fixed redshift. Middle: degree to which the radial ICM density profile evolves as a function of redshift at nearly fixed mass. The dotted line shows the selfsimilar expectation (α_{z} = 2). The dashed line shows the bestfitting evolution, which varies from slower than selfsimilar in the centre (α_{z} ∼ 0.3) to faster than selfsimilar around R_{500} (α_{z} ∼ 2.4). Envelopes show the 1 and 2σ uncertainties. Right: degree to which the radial ICM density profile scales with mass at nearly fixed redshift. The dotted line shows the selfsimilar expectation (α_{z} = 0). The dashed line shows the bestfitting mass dependence of α_{z} = 0.22. The scaled density at nearly fixed redshift does not depend on radius. 
The middle and righthand panels of Fig. 9 show the degree to which the two subsamples vary from selfsimilar scaling and from the bestfitting model scaling, as a function of scaled radius. Uncertainties are large because of the reduced number of data points and the scatter in the data. At nearly constant mass the overall variation with redshift is slightly greater than selfsimilar. However, the density evolves differently in the core and in the outer regions. The density evolution is consistent with zero in the core: α_{z} = 0.28 ± 1.10 at x = 0.01, a value that is in good agreement the result found by McDonald et al. (2017) although with large uncertainties. However, the density in the outer regions appears to evolve more strongly than selfsimilar: α_{z} = 2.42 ± 0.22 at x = 1.20, a result that is significant at slightly more than 1σ. At nearly fixed redshift, the density varies with mass in agreement with the α_{M} = 0.22 ± 0.01 scaling established above. At R_{500}, this mass dependence is significant at ∼2σ, but does not depend on radius.
5.1.3. Median and scatter of scaled profiles
We now turn to the ensemble properties of the scaled density profiles. The lefthand panel of Fig. 10 shows the median scaled profile, obtained in the loglog plane, and 68% dispersion for REXCESS Xrayselected sample and the SZEselected sample split into three redshift bins. It is clear that once scaled, the four subsamples are remarkably similar beyond 0.2 R_{500}. In the core region, the median central density decreases progressively with redshift. The median scaled central densities of the REXCESS and of the SZEselected sample at z < 0.3 are virtually indistinguishable. We will return to the central regions in Sect. 5.1.5 below.
Fig. 10. Scaled profiles and scatter. Left: median scaled profiles (solid lines) and 68% scatter (envelopes) for REXCESS and the SZEselected sample split into three redshift bins. Beyond ∼0.2 R_{500} the scaled profiles are almost indistinguishable. Right: radial profile of the intrinsic scatter for the various subsamples. The bestfitting intrinsic scatter model obtained from the SZEselected sample is also shown. Intrinsic scatter is less than 20% between 0.2 ≲ R_{500} ≲ 1.0. The gold line shows the model intrinsic scatter profile corrected for the covariance between M_{gas} and R_{500} (see Sect. 5.1.4), which results in a suppression of the scatter by a factor of about two at R_{500}. 
The radial variation of the intrinsic scatter about these median profiles is quantified in the righthand panel of Fig. 10, together with that of the bestfitting intrinsic scatter model obtained above (from the SZEselected clusters only). The intrinsic scatter of this model falls below 20% in the radial range 0.2 ≲ R_{500} ≲ 1.0. There is excellent agreement between the bestfitting intrinsic scatter model and the observed profiles, which all follow broadly the same trend with scaled radius: a steep decrease with a minimum at ∼0.5−0.7 R_{500}, followed by an increase towards larger radii.
Within R ≲ 0.2 R_{500} the intrinsic scatter in the scaled gas density profiles climbs steeply towards the centre. This increase is intimately linked to the complex physics of the core regions, dynamical activity, and to the presence or absence of cool core systems in the various samples. In this connection, the sample with the largest intrinsic scatter in the central regions is REXCESS, reflecting the presence of cool core systems in this dataset.
Beyond ∼0.2 R_{500} the relative dispersion of all samples dips below 20%, and at R_{500} the dispersion in profiles is ∼15%. In the SZEselected subsamples, there is a clear evolution, in the sense that the lowredshift systems exhibit the lowest intrinsic scatter values while the highredshift systems show higher values. The scatter will be related to intrinsic clustertocluster variations linked to inhomogeneities that will depend on the mass accretion rate and associated dynamical state, together with a component due to uncertainties in the total mass. It is possible that both of these effects conspire to produce higher intrinsic scatter values for higherredshift systems: one expects an increase in dynamical activity with redshift, while uncertainties in the cluster mass measurement will also increase in the same sense.
5.1.4. Suppression of scatter due to covariance between M_{gas} and R_{500}
The observed scatter in the density profiles may be suppressed by the use of M_{gas} in the computation of R_{500} when scaling the radial coordinate. All other things being equal, a cluster with a higher than average ρ_{gas} (for its mass) at some radius, will have a higher than average M_{gas} and hence Y_{X} relative to its mass. Since Y_{X} is then used to estimate R_{500} assuming a mean scaling relation, one would then overestimate R_{500} for this cluster. The radial scaling for this cluster would then be too large, which would move its density profile back towards the mean profile, reducing the apparent scatter. The reduction in scatter will depend on the slope of the density profile (i.e. the reduction is larger where the profile is steeper), so will be radially dependent.
The possible magnitude of this effect was estimated by generating synthetic cluster density profiles with a known amount of scatter, and then scaling them in radius following the method used for the observed clusters in order to test how much the scatter was changed. In more detail, the bestfitting median profile presented in Sect. 3.2 was normalised to match a 6 keV cluster at z = 0.15 (i.e. when integrated to R_{500}, the gas mass and Y_{X} were consistent with the scaling relations used for the observed clusters). For these reference values, the ‘true’ R_{500} is 1210 kpc.
A large number of realisations of this median profile were then generated by resampling the normalisation from a lognormal distribution with a standard deviation of 0.2 (approximating a constant 20% scatter in ρ_{gas} at all radii). For each realisation, R_{500} was then computed in the same manner as for the observed clusters; the profile was integrated to compute Y_{X} (assuming a fixed temperature of 6 keV) and hence R_{500}, with the process performed iteratively until R_{500} converged. The profile was then scaled in radius by R_{500} and the process was repeated for each realisation of the density profile.
When the distribution of densities in the realisations was measured at the ‘true’ value of R_{500} = 1210 kpc, the input scatter of 20% was recovered. However, when the profiles were each scaled in radius by the value of R_{500} estimated for each realisation from the M − Y_{X} relation, the scatter at a scaled radius of unity was found to be 10%; that is, the scatter is suppressed by a factor of two at around R_{500} due to the dependence of R_{500} on M_{gas} in our analysis. The same factor of two suppression was found for different values of the input scatter. We calculated the reduction in scatter at different scaled radii, obtaining a radial profile of the suppression factor. The intrinsic scatter profile corrected for this suppression factor is plotted in gold in Fig. 10. This method makes a number of simplifying assumptions (e.g., there is no scatter in T, the profile is fixed at the median form), but based on this analysis, we estimate that the scatter measured for the observed clusters is likely to be underestimated by a factor of approximately two at R_{500}.
5.1.5. Change of central regions over time
McDonald et al. (2017) showed that while the ICM outside the core regions of their SZEselected sample, covering the redshift range 0.25 < z < 1.2, evolved selfsimilarly with redshift, the central absolute median density (i.e. expressed in units of cm^{−3}) did not. They interpreted this result as being due to an unevolving core component embedded in a selfsimilarly evolving bulk.
Our sample is of comparable size to that of McDonald et al. (2017), but the evolution with mass and redshift has been decoupled and quantified (Sect. 3.1). The effective 5″ resolution of XMMNewton after PSF correction (Bartalucci et al. 2017) allows us to measure the density of the SZEselected sample across all redshifts down to a scaled radius of R ∼ 0.05 R_{500}. The lefthand panel of Fig. 11 shows a histogram of the resulting values^{4} scaled according to the bestfitting model (Eq. (17)) established in Sect. 3.2. The SZEselected sample was further divided into three redshift bins to better visualise how the sample changes over time. This histogram is characterised by a strong peak centred on a scaled central density of −0.2 (in log space), which is clearly visible, and coincident, in all three SZEselected subsamples. While the histogram of the z > 0.6 subsample has no detectable skewness, the histogram of the z < 0.3 subsample exhibits a distinct tail to higher scaled central densities, which is characterised by a moderately large positive skewness of G_{1} = 0.86 that is significant at > 90% (Doane & Seward 2011). This may indicate the gradual appearance of objects with more peaked scaled central densities towards lower redshifts.
Fig. 11. Central density. Left: histogram of central densities for the SZEselected systems, scaled according to the bestfitting model (Eq. (17)) derived in Sect. 3.2, measured at 0.05 R_{500}. The solid line is a kernel density plot with a smoothing width of 0.15. Middle: histogram of scaled central densities for the z < 0.3 SZEselected systems compared to the Xrayselected sample, measured at 0.015 R_{500}. The solid line is a kernel density plot with a smoothing width of 0.15. Right: histogram of central densities for the SZEselected sample at 40 kpc. 
At z < 0.3, the effective 5″ resolution of XMMNewton after PSF correction allows us to measure the density down to a scaled radius of R ∼ 0.015 R_{500}. The middle panel of Fig. 11 shows a histogram of the scaled central density of the SZEselected clusters at z < 0.3 compared to that of REXCESS. The positive skewness of the SZEselected systems is confirmed at greater significance (G_{1} = 1.06), while the histogram of the REXCESS sample exhibits two peaks in scaled central density: a main peak that is coincident with the peak of the SZEselected subsamples, and a secondary peak at a scaled density of 0.7 (in log space). The latter peak is due to cool core systems and may indicate that centrally peaked systems are overrepresented in Xrayselected samples, as has been argued by Rossetti et al. (2017) from their comparison of the image concentration parameter in Planck clusters to those for Xrayselected systems, and also by AndradeSantos et al. (2017).
The righthand panel of Fig. 11 shows the histogram of the central density of the SZEselected sample at 40 kpc, measured in physical units (cm^{−3}). There is a broad maximum at n_{e, 40 kpc} ∼ 0.01 cm^{−3}, and the histograms of the three subsamples coincide. A KolmogorovSmirnov test indicates that all three subsamples come from the same parent distribution. This result suggests, in agreement with McDonald et al. (2017), that the absolute central density remains constant over the redshift range probed by the current sample.
5.2. The L_{Xc} − M relations
5.2.1. Comparison with other work
Figure 12 shows the bestfitting bolometric L_{Xc} − M relation for the SZEselected clusters in the present sample compared to a number of results from the literature (Maughan 2007; Mantz et al. 2010; Bulbul et al. 2019; Lovisari et al. 2020). With the exception of those obtained by Mantz et al. (2010), these studies generally find slopes that are steeper than the selfsimilar expectation of 4/3, ranging from 1.63 to 1.92. Studies that put constraints on the evolution with redshift (Mantz et al. 2010; Bulbul et al. 2019; Lovisari et al. 2020) generally find good agreement with selfsimilar expectations (although with large uncertainties). However, any measurement of the dependence of a quantity on the mass will be affected strongly by the sample selection and on how the mass itself has been measured. Data fidelity and sample sizes are now such that systematic effects are starting to become dominant over measurement uncertainties.
Fig. 12. Comparison of the bolometric L_{Xc} − M relation to previous work (Maughan 2007; Mantz et al. 2010; Bulbul et al. 2019; Lovisari et al. 2020). 
Concerning the sample selection, the results in Sect. 4.2 show that the L_{Xc} − M relations of Xray and SZEselected systems are in good agreement, suggesting that once the core regions are excluded, effects due to detection methods relying on the ICM do not have any impact. Similarly, Lovisari et al. (2020) showed that their relaxed and disturbed samples had similar L_{Xc} − M relation slopes and normalisations. This suggests that the L_{Xc} − M relation may also be relatively robust to selection effects linked to cluster dynamical state, likely due to the small intrinsic scatter.
A more fundamental issue is the mass measurement itself (Pratt et al. 2019). In the present work we have used Y_{X} as a mass proxy; the works listed above use variously Y_{X} (Maughan 2007), the gas mass (Mantz et al. 2010), the SZE signaltonoise (Bulbul et al. 2019), and the hydrostatic mass (Lovisari et al. 2020), as proxies. In this context, the shallower slope of the Mantz et al. (2010) relation compared to the others can be fully explained by their assumption of a constant gas mass fraction in the mass calculation (e.g., Rozo et al. 2014b).
All of the above mass estimates are derived from ICM observables, and all except Lovisari et al. (2020) use scaling laws that have been calibrated on Xray hydrostatic mass estimates. Independent mass measurements, such as those available from lensing, galaxy velocity dispersions, or caustic measurements (e.g., Maughan et al. 2016), are critical to making progress on this issue. In this connection, weaklensing mass measurements for individual clusters have been carried out by several projects, such as the Local Cluster Substructure Survey (LoCuSS; Okabe et al. 2010, 2013; Okabe & Smith 2016), the Canadian Cluster Comparison Project (CCCP; Hoekstra et al. 2012, 2015), the Cluster Lensing And Supernova survey with Hubble (CLASH; Merten et al. 2015; Umetsu et al. 2014, 2016), Weighing the Giants (WtG; von der Linden et al. 2014; Kelly et al. 2014; Applegate et al. 2014), and CHEXMATE (CHEXMATE Collaboration 2021). The cluster community is undertaking a major ongoing effort to critically compare various mass estimates, obtained from mass proxies and from direct Xray, lensing, or velocity dispersion analyses (e.g., Rozo et al. 2014a; Sereno & Ettori 2015, 2017; Sereno 2015; Groener et al. 2016). Ultimately, this effort will help to better constrain the parameters of the scaling relations.
5.2.2. Link between density profile and L_{Xc}
As noted in for example Maughan et al. (2008), the similarity of the ICM density profiles outside the core implies a low scatter in L_{Xc}, as is indeed observed here. In order to explore how much of the scatter in L_{Xc} is due to the variation in density profiles, we computed a ‘pseudo luminosity’ for each cluster. For this calculation, we used the measured density profile for each cluster and assumed isothermality at the measured coreexcised temperature for the cluster. The integral was performed over a cylindrical volume from projected radii of 0.15 R_{500} to R_{500}. For 16 of 118 clusters, the density profiles did not reach R_{500}, so the integrals were truncated at the maximum observed radius. In all cases the profiles reached to ≈90% of R_{500}, and the contribution to the luminosity of the outer parts of the profile is very small, so the effect of this truncation is negligible. The scatter in then provides an estimate of the scatter in the bolometric L_{Xc} due only to the scatter in density profiles.
The intrinsic scatter about the best fitting relation to versus M_{500} was then measured (assuming that the fractional statistical error on is the same as that in L_{Xc} for each cluster), giving a value of 11%. This implies that most or all of the intrinsic scatter in the bolometric L_{Xc} at fixed mass can be explained by the variation in the ICM density profiles.
In principle, the use of M_{Yx} to determine R_{500} and hence the aperture within which L_{Xc} and are measured, could introduce additional scatter in the L_{Xc} − M_{500} relation. If M_{Yx} were scattered high relative to the true mass of a cluster, then R_{500} would be overestimated and the 0.15−1 R_{500} aperture would be shifted to larger radii. This shift would reduce L_{Xc} since more of the luminosity comes from the inner edge of the aperture than the outer edge. Hence, if M_{Yx} were scattered high, then L_{Xc} and would be scattered low, adding to the observed scatter in the relation. We examined the impact of this effect by adding 10% scatter to M_{Yx} when computing . This increased the measured scatter in by less than 1%. The dependence of the luminosity aperture on M_{Yx} leads to a negligible contribution to the scatter in the L_{Xc} − M_{Yx} relation.
We therefore conclude that the measured intrinsic scatter in the bolometric L_{Xc} at fixed mass is dominated almost entirely by the variation in the ICM density profiles. The results do not depend on the aperture for reasonable assumptions on the scatter between M_{Yx} and the true mass M_{500}. Any residual scatter will come from inhomogeneity and/or substructure in the density distribution, or from the effects of structure in the temperature or metallicity distribution.
5.2.3. Impact of selection bias and covariance
We found above that the bolometric L_{Xc} − M relation has a slope B_{L} ∼ 1.7, which is steeper than the selfsimilar value of 1.3, and has a small scatter of σ_{lnLXc} ∼ 0.15. A similar difference in slope of Δ B_{L} ∼ +0.4 is seen relative to the selfsimilar softband L_{Xc} − M relation. One might wonder whether and to what extent selection effects and covariance may impact these results. Firstly, in survey data the observable that is used for cluster detection, 𝒪_{det}, can be affected by socalled Malmquist bias. This happens because clusters that are scattered to higher values of 𝒪_{det}, whether by noise or by the intrinsic scatter between the observable and the mass, will be preferentially detected, leading to a positive bias in the average value of 𝒪_{det}. This is a particular concern near the detection threshold, and will affect the apparent slope of any scaling relation with 𝒪_{det}. Secondly, the intrinsic scatter of each quantity around the mean relation may well be correlated, leading to a reduction in the measured scatter with respect to the true underlying value.
Since we showed in Sect. 4.2 that the results for the Xrayselected sample are compatible within 1σ with those for the local SZEselected sample, we focus here on the SZE selection. As detailed in Sect. 2.1, the SZEselected sample is composed of four different subsamples and with different selections: the ESZ, LP1, LP2 and LP3. Although the ESZ subsample was generated from high signaltonoise ratio (S/N > 6) detections, AndradeSantos et al. (2021) showed that its selection is dominated by the instrumental and astrophysical scatter, and that the intrinsic scatter in the Y_{SZ}–M relation has a negligible effect on the recovered scaling parameters. The three other SZE subsamples (LP1, LP2, LP3) are selected at an S/N lower than that of the ESZ sample, and are thus even more affected by the instrumental and astrophysical scatter. We restricted in consequence our selection study to the ESZ. This will provide upper limits to the effects of selection on the results for the L_{Xc} − M of the SZEselected systems in the current study.
Appendix C describes in detail the simulations we used, which were similar to those produced for the AndradeSantos et al. (2021) study. Simulated clusters, modelled with the Arnaud et al. (2010) pressure profile and drawn from a Tinker mass function (Tinker et al. 2008), were injected into the Planck Early SZ maps in the ‘cosmological’ mask region. The MultiMatched Filter extraction algorithm (Melin et al. 2006) was then applied, to obtain SZ detections at S/N > 6, corresponding to the threshold for the ESZ sample. We then matched the injected and recovered clusters to produce a mock ESZ catalogue, doing this twenty times to generate 3188 detections in total. To investigate selection and covariance effects, we assumed a Gaussian lognormal correlated distribution for Y_{SZ}, Y_{X}, and L_{Xc} at fixed true mass M, with a covariance matrix to describe correlations between parameters.
A firstorder estimate of the expected scatter in the L_{Xc} − M_{Yx} relation can be obtained by assuming that selection effects are negligible. In this case,
For a measured intrinsic scatter σ_{lnLXcMYx} ∼ 0.13, a covariance between L_{Xc} and Y_{X} of t = 0.85 (Farahi et al. 2019), and assuming a scatter between Y_{X} and true mass M of σ_{lnYXM} ∼ 0.16 motivated by recent numerical simulations (Planelles et al. 2014; Le Brun et al. 2017; Truong et al. 2018), the resulting dispersion between coreexcised luminosity and true mass is σ_{lnLXcM} ∼ 0.22. This suggests that the measured scatter is underestimated by a factor of 0.22/0.13 ∼ 1.7 due to the use of Y_{X} as a mass proxy.
We generated a series of simulations using the above values for true scatter and covariance between quantities, and used them to investigate the effects of selection, intrinsic scatter, and covariance on the results. We found:

The Malmquist bias induced on the L_{Xc} − Y_{SZ} relation due to intrinsic scatter in the relation is completely negligible, with the slope changing by less than 0.5%. The resulting impact on the slope of the L_{Xc} − M relation is ΔB_{L} = −0.01. This is important, because it implies that selection effects due to intrinsic scatter in the Y_{SZ} observable cannot account for the observed steeper slope of the L_{Xc} − M relations seen here.

Use of Y_{X} as a mass proxy introduces additional intrinsic scatter with respect to the underlying mass. The net impact on the recovered slope of the L_{Xc} − M relation is minimal, with ΔB_{L} = −0.04.

Intrinsic scatter in the L_{Xc} − Y_{X} relation has the effect of redressing the slope towards its original value, with ΔB_{L} = +0.02.

Covariance between Y_{X} and L_{Xc} again changes the slope. For t = 0.85 (Farahi et al. 2019), the impact on the slope of the L_{Xc} − M relation is ΔB_{L} = +0.04.
We thus conclude that the slope of the L_{Xc} − M_{Yx} relation found here is robust to selection effects due to intrinsic scatter in the Y_{SZ} and Y_{X} proxies, and to covariance between quantities. The dispersion, however, is very sensitive to the covariance. We estimate that the measured dispersion in the relation between the coreexcised luminosity and true mass, L_{Xc} − M, is underestimated by a factor of ∼1.7 due to the use of Y_{X} as a mass proxy.
5.3. Link to f_{gas} − M
The L_{Xc} − M relations derived above have a steeper dependence than selfsimilar, which cannot be explained by selection effects, intrinsic scatter, or covariance. There is evidence for a dependence of the gas content with mass, which has the effect of suppressing the luminosity preferentially in lowermass systems, leading to the observed steepening of the relation (see discussion in Pratt et al. 2009). Interestingly, with the assumption of a standard dependence of temperature on mass (T ∝ M^{2/3}, e.g., Arnaud et al. 2005; Mantz et al. 2016), use of Eq. (19) with the observed bolometric L_{Xc} ∝ M^{1.7} relation yields f_{gas} ∝ M^{0.21}.
The gas density profile model detailed in Sect. 3.1 yields an alternative method to obtain the dependence of the gas mass fraction with mass, as the model includes a dependence of f_{gas} = M_{gas}/M on the total mass M, via Eq. (8). The bestfitting gas density model (Eq. (17)) suggests f_{gas} ∝ M^{0.22 ± 0.01}. This dependence of the gas content on total mass is therefore in good agreement with the expected relation given the observed L_{Xc} − M dependence, and with previous findings (e.g., Pratt et al. 2009; Lovisari et al. 2015; Ettori 2015) based primarily on Xrayselected clusters.
6. Conclusions
We have examined the gas density profiles and the relation between the core excised Xray luminosity L_{Xc} and the total mass derived from the Y_{X} mass proxy for 118 Xray and SZEselected objects covering a mass range of M_{500} = [0.5−20]×10^{14} M_{⊙} and extending in redshift up to z ∼ 1.13. We first examined the scaled density profiles:

The gas density profiles do not scale perfectly selfsimilarly, exhibiting subtle trends in mass and redshift.

Motivated by this finding, we fitted an analytic gas density model to the 93 SZEselected systems. The analytic model is based on a generalised NFW profile, and correctly reproduces the scaled gas density profile and the radial variation of its intrinsic dispersion. Combined with the empirical mass scaling of the profiles, this analytic model defines the gas density profile of SZEselected clusters as a function of mass and redshift. This model is given in Eqs. (16)–(18).

The intrinsic dispersion in scaled profiles is greatest in the central regions, declining to a minimum at ∼0.5−0.7 R_{500}, and increasing thereafter. The dispersion is similar for Xrayselected clusters and for local SZEselected clusters, except in the centre, where the Xrayselected systems have a higher dispersion. There is a hint for an evolution of the dispersion with redshift, which may be linked to an increase in perturbed clusters at higher redshifts.

We investigated the effect of covariance between M_{gas} and R_{500} due to the use of M_{YX} as a mass proxy, obtaining a radial profile of the scatter suppression factor. Taking into account this suppression factor, we estimated a scatter in scaled density profiles of approximately 40% at R_{500}.

We quantified deviations from the average scaling with radius. These show no variation with mass, but which show a significant variation with redshift, in the sense that the core regions clearly evolve differently as compared to the bulk.

We examined the scaled central density measured at R = 0.05 R_{500} for the SZEselected systems, finding that only the z < 0.3 sample is skewed. This skewness is positive, and may indicate the increased presence of centrally peaked systems at later times.

We measured the scaled central density at R = 0.015 R_{500} for the Xray and SZEselected systems at z < 0.3. The scaled central density of the local Xrayselected sample exhibits two peaks. The main peak, corresponding to noncool core systems in the Xrayselected sample, is slightly offset to higher scaled central density from that of the local SZEselected sample. The secondary peak in the Xrayselected sample, corresponding to the cool core systems, is not seen in the SZEselected sample, although the latter does exhibit a clear tail to higher scaled central density as confirmed by the strongly positively skewed distribution.

The absolute value of the central density in the SZEselected sample measured at 40 kpc does not appear to evolve with redshift, consistent with the findings of McDonald et al. (2017).
We then examined the relation between the core excised Xray luminosity L_{Xc} and the total mass derived from the Y_{X, 500} mass proxy, M_{Yx}.

This relation is extremely tight, with a logarithmic intrinsic scatter of σ_{lnLxcMYx} ≲ 0.15 depending on subsample and band in which the luminosity is measured. Importantly, at low redshift, the bestfitting parameters of this relation do not depend on whether the sample was selected in Xrays or through the SZE, suggesting that L_{Xc} is a selectionindependent quantity.

The slope of the bolometric relation fitted to the SZEselected clusters, B ∼ 1.74 ± 0.02, is significantly steeper than selfsimilar. When left free to vary, the evolution of n = −2.5 ± 0.09 is in agreement with the selfsimilar value of −7/3 within < 2σ.

We thoroughly examined the impact of selection bias and covariance on the relation. We found that the slope of the L_{Xc} − M_{Yx} relation is robust to selection effects due to intrinsic scatter in the Y_{SZ} and Y_{X} proxies, and to covariance between quantities. The dispersion, however, is very sensitive to the covariance. For reasonable values of covariance, we estimate that the measured dispersion in the L_{Xc} − M relation is underestimated by a factor of at most ∼1.7 due to the use of Y_{X} as a mass proxy, implying a true scatter of σ_{lnLXcM} ∼ 0.22.

We show explicitly that the scatter in the L_{Xc} − M relation can be accounted for almost entirely by objecttoobject variations in gas density profiles.
With our study we have examined the mass and redshift dependence of the ICM gas density profile, and made quantitative comparisons between Xray and SZEselected samples. Our overall conclusion is consistent with the view that the ICM bulk evolves approximately selfsimilarly, with the core regions evolving separately due to cooling and feedback from the central active galactic nucleus. Indeed, it suggests potentially subtle differences in the core regions between Xray and SZEselected systems. It also supports a view where the ICM gas mass fraction depends on mass up to high redshift, with a dependence for the present sample. Further progress can be undoubtedly be made by bringing to bear fully independent mass estimates, such as those that can be obtained from weak lensing and/or galaxy velocity dispersions. Such studies are one of the goals of the CHEXMATE project (CHEXMATE Collaboration 2021).
Acknowledgments
G.W.P., M.A., and J.B.M. acknowledge funding from the European Research Council under the European Union’s Seventh Framework Programme (FP720072013) ERC grant agreement no. 340519, and from the French space agency, CNES. B.J.M. acknowledges support from the Science and Technology Facilities Council (grant number ST/V000454/1). The results reported in this article are based on data obtained from the XMMNewton observatory, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
References
 Akritas, M. G., & Bershady, M. A. 1996, ApJ, 470, 706 [Google Scholar]
 AndradeSantos, F., Jones, C., Forman, W. R., et al. 2017, ApJ, 843, 76 [Google Scholar]
 AndradeSantos, F., Pratt, G. W., Melin, J.B., et al. 2021, ApJ, 914, 58 [NASA ADS] [CrossRef] [Google Scholar]
 Applegate, D. E., von der Linden, A., Kelly, P. L., et al. 2014, MNRAS, 439, 48 [Google Scholar]
 Arnaud, M., & Evrard, A. E. 1999, MNRAS, 305, 631 [NASA ADS] [CrossRef] [Google Scholar]
 Arnaud, M., Pointecouteau, E., & Pratt, G. W. 2005, A&A, 441, 893 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Arnaud, M., Pratt, G. W., Piffaretti, R., et al. 2010, A&A, 517, A92 [CrossRef] [EDP Sciences] [Google Scholar]
 Bartalucci, I., Arnaud, M., Pratt, G. W., et al. 2017, A&A, 598, A61 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bartalucci, I., Arnaud, M., Pratt, G. W., & Le Brun, A. M. C. 2018, A&A, 617, A64 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bartalucci, I., Arnaud, M., Pratt, G. W., Démoclès, J., & Lovisari, L. 2019, A&A, 628, A86 [EDP Sciences] [Google Scholar]
 Bleem, L. E., Stalder, B., de Haan, T., et al. 2015, ApJS, 216, 27 [Google Scholar]
 Böhringer, H., Schuecker, P., Pratt, G. W., et al. 2007, A&A, 469, 363 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bulbul, E., Chiu, I. N., Mohr, J. J., et al. 2019, ApJ, 871, 50 [Google Scholar]
 Cavaliere, A., & FuscoFemiano, R. 1976, A&A, 500, 95 [NASA ADS] [Google Scholar]
 CHEXMATE Collaboration (Arnaud, M. et al.) 2021, A&A, 650, A104 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Connor, T., Donahue, M., Sun, M., et al. 2014, ApJ, 794, 48 [NASA ADS] [CrossRef] [Google Scholar]
 Croston, J. H., Arnaud, M., Pointecouteau, E., & Pratt, G. W. 2006, A&A, 459, 1007 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Croston, J. H., Pratt, G. W., Böhringer, H., et al. 2008, A&A, 487, 431 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Doane, D. P., & Seward, L. E. 2011, J. Stat. Educ., 19 [CrossRef] [Google Scholar]
 Eckert, D., Roncarelli, M., Ettori, S., et al. 2015, MNRAS, 447, 2198 [Google Scholar]
 Ettori, S. 2015, MNRAS, 446, 2629 [NASA ADS] [CrossRef] [Google Scholar]
 Fabian, A. C., Crawford, C. S., Edge, A. C., & Mushotzky, R. F. 1994, MNRAS, 267, 779 [NASA ADS] [CrossRef] [Google Scholar]
 Farahi, A., Mulroy, S. L., Evrard, A. E., et al. 2019, Nat. Commun., 10, 2504 [NASA ADS] [CrossRef] [Google Scholar]
 ForemanMackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 [Google Scholar]
 Ghirardini, V., Eckert, D., Ettori, S., et al. 2019, A&A, 621, A41 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ghizzardi, S. 2001, in Flight calibration of the PSF for the MOS1 and MOS2 cameras, XMMSOCCALTN0022 [Google Scholar]
 Groener, A. M., Goldberg, D. M., & Sereno, M. 2016, MNRAS, 455, 892 [NASA ADS] [CrossRef] [Google Scholar]
 Hasselfield, M., Hilton, M., Marriage, T. A., et al. 2013, JCAP, 7, 008 [CrossRef] [Google Scholar]
 Hilton, M., Sifón, C., Naess, S., et al. 2021, ApJS, 253, 3 [Google Scholar]
 Hoekstra, H., Mahdavi, A., Babul, A., & Bildfell, C. 2012, MNRAS, 427, 1298 [NASA ADS] [CrossRef] [Google Scholar]
 Hoekstra, H., Herbonnet, R., Muzzin, A., et al. 2015, MNRAS, 449, 685 [NASA ADS] [CrossRef] [Google Scholar]
 Kalberla, P. M. W., Burton, W. B., Hartmann, D., et al. 2005, A&A, 440, 775 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Kay, S. T., Peel, M. W., Short, C. J., et al. 2012, MNRAS, 422, 1999 [NASA ADS] [CrossRef] [Google Scholar]
 Kelly, B. C. 2007, ApJ, 665, 1489 [Google Scholar]
 Kelly, P. L., von der Linden, A., Applegate, D. E., et al. 2014, MNRAS, 439, 28 [NASA ADS] [CrossRef] [Google Scholar]
 Lau, E. T., Nagai, D., Avestruz, C., Nelson, K., & Vikhlinin, A. 2015, ApJ, 806, 68 [NASA ADS] [CrossRef] [Google Scholar]
 Le Brun, A. M. C., McCarthy, I. G., Schaye, J., & Ponman, T. J. 2017, MNRAS, 466, 4442 [Google Scholar]
 Lovisari, L., Reiprich, T. H., & Schellenberger, G. 2015, A&A, 573, A118 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Lovisari, L., Forman, W. R., Jones, C., et al. 2017, ApJ, 846, 51 [Google Scholar]
 Lovisari, L., Schellenberger, G., Sereno, M., et al. 2020, ApJ, 892, 102 [Google Scholar]
 Mantz, A., Allen, S. W., Rapetti, D., & Ebeling, H. 2010, MNRAS, 406, 1759 [NASA ADS] [Google Scholar]
 Mantz, A. B., Allen, S. W., & Morris, R. G. 2016, MNRAS, 462, 681 [CrossRef] [Google Scholar]
 Mantz, A. B., Allen, S. W., Morris, R. G., & von der Linden, A. 2018, MNRAS, 473, 3072 [NASA ADS] [CrossRef] [Google Scholar]
 Maughan, B. J. 2007, ApJ, 668, 772 [Google Scholar]
 Maughan, B. J., Jones, C., Forman, W., & Van Speybroeck, L. 2008, ApJS, 174, 117 [NASA ADS] [CrossRef] [Google Scholar]
 Maughan, B. J., Giles, P. A., Rines, K. J., et al. 2016, MNRAS, 461, 4182 [NASA ADS] [CrossRef] [Google Scholar]
 McDonald, M., Allen, S. W., Bayliss, M., et al. 2017, ApJ, 843, 28 [Google Scholar]
 Melin, J. B., Bartlett, J. G., & Delabrouille, J. 2006, A&A, 459, 341 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Merten, J., Meneghetti, M., Postman, M., et al. 2015, ApJ, 806, 4 [Google Scholar]
 Nagai, D., Kravtsov, A. V., & Vikhlinin, A. 2007, ApJ, 668, 1 [Google Scholar]
 Nagarajan, A., Pacaud, F., Sommer, M., et al. 2019, MNRAS, 488, 1728 [Google Scholar]
 Neumann, D. M., & Arnaud, M. 1999, A&A, 348, 711 [NASA ADS] [Google Scholar]
 Okabe, N., & Smith, G. P. 2016, MNRAS, 461, 3794 [Google Scholar]
 Okabe, N., Takada, M., Umetsu, K., Futamase, T., & Smith, G. P. 2010, PASJ, 62, 811 [NASA ADS] [Google Scholar]
 Okabe, N., Smith, G. P., Umetsu, K., Takada, M., & Futamase, T. 2013, ApJ, 769, L35 [NASA ADS] [CrossRef] [Google Scholar]
 Piffaretti, R., Arnaud, M., Pratt, G. W., Pointecouteau, E., & Melin, J.B. 2011, A&A, 534, A109 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration VIII. 2011, A&A, 536, A8 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration IX. 2011, A&A, 536, A9 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planck Collaboration XXVII. 2016, A&A, 594, A27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Planelles, S., Borgani, S., Fabjan, D., et al. 2014, MNRAS, 438, 195 [Google Scholar]
 Pratt, G. W., & Arnaud, M. 2002, A&A, 394, 375 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pratt, G. W., & Arnaud, M. 2003, A&A, 408, 1 [CrossRef] [EDP Sciences] [Google Scholar]
 Pratt, G. W., Croston, J. H., Arnaud, M., & Böhringer, H. 2009, A&A, 498, 361 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pratt, G. W., Arnaud, M., Piffaretti, R., et al. 2010, A&A, 511, A85 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25 [Google Scholar]
 Robotham, A. S. G., & Obreschkow, D. 2015, PASA, 32, e033 [Google Scholar]
 Rossetti, M., Gastaldello, F., Ferioli, G., et al. 2016, MNRAS, 457, 4515 [Google Scholar]
 Rossetti, M., Gastaldello, F., Eckert, D., et al. 2017, MNRAS, 468, 1917 [Google Scholar]
 Rozo, E., Rykoff, E. S., Bartlett, J. G., & Evrard, A. 2014a, MNRAS, 438, 49 [NASA ADS] [CrossRef] [Google Scholar]
 Rozo, E., Bartlett, J. G., Evrard, A. E., & Rykoff, E. S. 2014b, MNRAS, 438, 78 [CrossRef] [Google Scholar]
 Rykoff, E. S., Evrard, A. E., McKay, T. A., et al. 2008, MNRAS, 387, L28 [NASA ADS] [Google Scholar]
 Schellenberger, G., & Reiprich, T. H. 2017, MNRAS, 469, 3738 [CrossRef] [Google Scholar]
 Sereno, M. 2015, MNRAS, 450, 3665 [Google Scholar]
 Sereno, M., & Ettori, S. 2015, MNRAS, 450, 3633 [NASA ADS] [CrossRef] [Google Scholar]
 Sereno, M., & Ettori, S. 2017, MNRAS, 468, 3322 [CrossRef] [Google Scholar]
 Tinker, J., Kravtsov, A. V., Klypin, A., et al. 2008, ApJ, 688, 709 [Google Scholar]
 Truong, N., Rasia, E., Mazzotta, P., et al. 2018, MNRAS, 474, 4089 [NASA ADS] [CrossRef] [Google Scholar]
 Umetsu, K., Medezinski, E., Nonino, M., et al. 2014, ApJ, 795, 163 [NASA ADS] [CrossRef] [Google Scholar]
 Umetsu, K., Zitrin, A., Gruen, D., et al. 2016, ApJ, 821, 116 [Google Scholar]
 Vikhlinin, A., Kravtsov, A. V., Burenin, R. A., et al. 2009, ApJ, 692, 1060 [Google Scholar]
 Voit, G. M. 2005, Rev. Mod. Phys., 77, 207 [Google Scholar]
 von der Linden, A., Allen, M. T., Applegate, D. E., et al. 2014, MNRAS, 439, 2 [NASA ADS] [CrossRef] [Google Scholar]
 Zhang, Y. Y., Finoguenov, A., Böhringer, H., et al. 2008, A&A, 482, 451 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
Appendix A: Sample observation details
Table A.1 contains the sample observation details, including: cluster name(s), redshift, coordinates, column density, exposure time, and XMMNewton OBSID used for the analysis.
Sample observation details. Columns: (1) Planck name; (2) SPT name; (3) ACT name (4) alternative name (nonexhaustive), REXCESS clusters in bold; (5) redshift; (6) RA; (7) Dec.; (8) neutral hydrogen column density integrated along the line of sight determined from the LAB survey (Kalberla et al. 2005); (9) observation exposure time for MOS1 and pn detectors, in ks; (10) XMMNewton OBSID.
Appendix B: Sample data
Table B.1 contains the name, redshift, RA, Dec., mass from the Y_{X} proxy, and softband and bolometric coreexcised Xray luminosity for the sample.
Sample observational data.
Appendix C: Sample selection bias and covariance tests
We used similar simulations as those produced for the AndradeSantos et al. (2021) study. Simulated clusters, modelled with the Arnaud et al. (2010) pressure profile and drawn from a Tinker mass function (Tinker et al. 2008), were injected into the Planck Early SZ maps in the ‘cosmological’ mask region. The Y_{SZ} value for each object was drawn from the Y_{SZ} − M relation of Arnaud et al. (2010), with a bias between the Xray calibrated mass and the true mass of (1 − b) = 0.65, a value used to obtain the observed cluster number counts in the Planck cosmology. As the slope of the Y_{SZ} − Y_{X} relation is expected to be close to unity, and we are only interested in slope variations, we assume that the normalisation and slope of the Y_{X} − M and Y_{SZ} − M relations are the same.
We draw the Y_{SZ}, Y_{X} and L_{X} quantities associated to each simulated cluster following a correlated Gaussian distribution with covariance matrix. If 𝒬_{M} is the latent value of 𝒬 at a given mass, obtained if there were no scatter, one can write
where Y_{M} and L_{M} are the latent values of Y and L at a given mass (the values obtained if there were no scatter), and , Y_{X}, and L_{Xc} are the true values. For these simulations we assume B_{Y} = 1.79 (Arnaud et al. 2010) and B_{L} = 1.37 (Table 2). 𝒩 is a Gaussian lognormal correlated distribution at fixed mass, where
For the covariance matrix V_{σ}, we assume σ_{lnYSZ} = 0.12 (Kay et al. 2012; Le Brun et al. 2017); σ_{lnYX} = 0.16 (Planelles et al. 2014; Le Brun et al. 2017; Truong et al. 2018); r = 0.4 (Farahi et al. 2019; Nagarajan et al. 2019); s = 0.4 (Farahi et al. 2019); and t = 0.85 (Farahi et al. 2019). As detailed in Sect. 5.2.3, with these assumptions, Eqn. 24 gives a firstorder estimate of σ_{lnLXc} = 0.22 in the absence of selection effects.
The MultiMatched Filter extraction algorithm (Melin et al. 2006) was then applied, to obtain SZ detections at S/N > 6, corresponding to the threshold for the ESZ sample. We then matched the injected and recovered clusters to produce a mock ESZ catalogue, doing this twenty times, resulting in a total of 3188 detections. Measurement errors were estimated from the maps as described in Melin et al. (2006). Fits to the simulated data were performed as described in Sect. 4.2. Our baseline simulation in the following assumes r = s = t = 0 (i.e. zero covariance between quantities).
We first fitted the Y_{M} − L_{M} relation, which assumes that there is zero intrinsic scatter in either of the observables with respect to the mass. The scatter in the Yaxis seen in Fig. C.1 is then entirely due to the observational uncertainties in the SZE measurements. The resulting slope of 0.759 ± 0.001 implies a change in the L_{Xc} − M relation of Δ B_{L} = −0.01. Adding intrinsic scatter of σ_{lnYSZ} = 0.12 (Kay et al. 2012; Le Brun et al. 2017), the slope of the Y_{SZ} − L_{M} relation is 0.759 ± 0.001, again implying a negligible change of Δ B_{L} = −0.01 on the slope of the L_{Xc} − M relation. These results imply that Malmquist bias in the Y_{SZ} observable are negligible, and cannot account for the significantly steeper than selfsimilar slope we find for the L_{Xc} − M relation.
Fig. C.1. Simulated Y_{SZ} reextracted from PSZ2 maps, plotted as a function of L_{Xc}. The quantity 𝒬_{M} is the latent variable with respect to the mass M, obtained if there were no scatter in the relation. The Figure shows the effect of progressively adding scatter in Y_{SZ}, Y_{X}, and L_{Xc} with M. The blue points include covariance of t = 0.85 between Y_{X} and L_{Xc}. 
We next studied the robustness of the recovery of the L_{Xc} − M relation slope to intrinsic scatter in the Y_{X} proxy, due to Y_{X} being a scattered estimates of Y_{SZ}. An intrinsic scatter of σ_{lnYX} = 0.16 (Planelles et al. 2014; Le Brun et al. 2017; Truong et al. 2018) pushes the slope of the Y_{X} − L_{M} relation to 0.741 ± 0.002, in turn changing the L_{Xc} − M relation slope by Δ B_{L} = −0.04.
We then added an intrinsic scatter of σ_{lnLXc} = 0.22, finding that this redresses the slope to 0.775 ± 0.005, implying a change of Δ B_{L} = +0.02 in the slope of the L_{Xc} − M relation. Finally, we added a covariance of t = 0.85 between Y_{X} and L_{Xc} (Farahi et al. 2019). This pushes the slope to a slightly steeper value of 0.785 ± 0.003, changing the L_{Xc} − M relation slope by Δ B_{L} = +0.04. The covariance significantly reduces the dispersion in the Y_{X} − L_{Xc} relation, by a factor of two. This can clearly be seen in the difference in dispersion between the red and blue points in Fig. C.1.
In conclusion, we find that selection effects and intrinsic scatter have a negligible effect on the slope of the L_{Xc} − M relation, and cannot account for the significantly steeper than selfsimilar value that we find in this work. The dispersion of the L_{Xc} − M relation that we derive is significantly underestimated, most likely by a factor of ∼1.7, due to the covariance between L_{Xc} and the mass proxy, Y_{X}.
All Tables
Numerical values for the bestfitting intrinsic scatter term, measured at seven equally spaced points in log(R/R_{500}), in the range [0.01−1] R_{500}.
Sample observation details. Columns: (1) Planck name; (2) SPT name; (3) ACT name (4) alternative name (nonexhaustive), REXCESS clusters in bold; (5) redshift; (6) RA; (7) Dec.; (8) neutral hydrogen column density integrated along the line of sight determined from the LAB survey (Kalberla et al. 2005); (9) observation exposure time for MOS1 and pn detectors, in ks; (10) XMMNewton OBSID.
All Figures
Fig. 1. Sample properties. Left: redshiftmass distribution of the clusters used in this paper. The SZEselected clusters comprise a subset of 44 systems from the Planck Early SZ sample (Planck Collaboration VIII 2011) at z < 0.5 and a further 49 clusters at z > 0.5. REXCESS (Böhringer et al. 2007) is an Xrayselected sample of 31 objects at z < 0.25. Right: stacked histogram of the mass distribution. The REXCESS sample has a lower median mass than the SZEselected samples. 

In the text 
Fig. 2. Deprojected, PSFcorrected density profiles for 118 galaxy clusters, normalised by the critical density ρ_{crit} and R_{500}. Fully selfsimilar clusters would trace the same locus in this plot. The profiles are colourcoded by mass M_{500} in the lefthand panel, and by redshift z in the righthand panel. There are clear trends with respect to both quantities. 

In the text 
Fig. 3. Marginalised posterior likelihood for the parameters of the bestfitting density profile model detailed in Sect. 3.1. 

In the text 
Fig. 4. The universal cluster ICM density profile. Left: scaled density profiles of the SZEselected clusters (grey points), overplotted with the bestfitting GNFW model with free evolution and mass dependence: ρ_{gas}/ρ_{500}(R/R_{500})∝E(z)^{αz} M^{αM} with α_{z} = 2.09 ± 0.02 and α_{M} = 0.22 ± 0.01 (orange line). The model includes a radially varying intrinsic scatter term (orange envelope). Right: comparison of the bestfitting model, defined on the SZEselected sample, to the bestfitting model for the Xrayselected REXCESS sample (light blue). Here, the points with error bars are the REXCESS sample. 

In the text 
Fig. 5. Relation between L_{Xc} and Y_{X}. The blue envelope is the bestfitting relation given in Eq. (23), and the results from Maughan (2007) are also shown for comparison. 

In the text 
Fig. 6. Relation between the coreexcised Xray luminosity L_{Xc} and mass estimated from the Y_{X} proxy, for the bolometric and softband luminosities of 118 systems. Left: data points with the best fitting relation to the Xrayselected REXCESS sample with the evolution factor fixed to the selfsimilar value of n = −2 (grey envelope). The dark grey envelope shows the best fitting relation to the 37 SZEselected systems at z < 0.3 with n = −2. Right: histogram of the log space residuals from the best fitting relation to the SZEselected objects at z < 0.3. 

In the text 
Fig. 7. Relation between the coreexcised Xray luminosity L_{Xc} and mass estimated from the Y_{X} proxy, for the bolometric and softband luminosities of 118 systems. Left: best fitting relation (grey envelope) to the full sample (data points) with the evolution factor n left free to vary. The bestfitting values of n are given in Table 2. Right: histogram of the log space residuals from the best fitting relation. Solid lines show the bestfitting Gaussian distributions with σ corresponding to the bestfitting intrinsic scatter in log space (Table 2). 

In the text 
Fig. 8. Scaled density profiles. Left: scaled density profiles (points) and bestfitting model (orange envelope) for SZEselected systems in our sample compared to the median and 68% dispersion from the XCOP sample (Ghirardini et al. 2019, magenta envelope). Right: comparison of best fitting intrinsic scatter model (blue) with that found by Ghirardini et al. (2019, magenta). The bestfitting intrinsic scatter obtained from our sample when the evolution factor is forced to the selfsimilar value of E(z)^{2} is also shown in grey. 

In the text 
Fig. 9. Deviations from the average scaling with radius. Left: redshiftmass distribution of the SZEselected sample used in this work. The shaded regions indicate cuts for two subsamples: a large redshift range at nearly fixed mass, and a large mass range at nearly fixed redshift. Middle: degree to which the radial ICM density profile evolves as a function of redshift at nearly fixed mass. The dotted line shows the selfsimilar expectation (α_{z} = 2). The dashed line shows the bestfitting evolution, which varies from slower than selfsimilar in the centre (α_{z} ∼ 0.3) to faster than selfsimilar around R_{500} (α_{z} ∼ 2.4). Envelopes show the 1 and 2σ uncertainties. Right: degree to which the radial ICM density profile scales with mass at nearly fixed redshift. The dotted line shows the selfsimilar expectation (α_{z} = 0). The dashed line shows the bestfitting mass dependence of α_{z} = 0.22. The scaled density at nearly fixed redshift does not depend on radius. 

In the text 
Fig. 10. Scaled profiles and scatter. Left: median scaled profiles (solid lines) and 68% scatter (envelopes) for REXCESS and the SZEselected sample split into three redshift bins. Beyond ∼0.2 R_{500} the scaled profiles are almost indistinguishable. Right: radial profile of the intrinsic scatter for the various subsamples. The bestfitting intrinsic scatter model obtained from the SZEselected sample is also shown. Intrinsic scatter is less than 20% between 0.2 ≲ R_{500} ≲ 1.0. The gold line shows the model intrinsic scatter profile corrected for the covariance between M_{gas} and R_{500} (see Sect. 5.1.4), which results in a suppression of the scatter by a factor of about two at R_{500}. 

In the text 
Fig. 11. Central density. Left: histogram of central densities for the SZEselected systems, scaled according to the bestfitting model (Eq. (17)) derived in Sect. 3.2, measured at 0.05 R_{500}. The solid line is a kernel density plot with a smoothing width of 0.15. Middle: histogram of scaled central densities for the z < 0.3 SZEselected systems compared to the Xrayselected sample, measured at 0.015 R_{500}. The solid line is a kernel density plot with a smoothing width of 0.15. Right: histogram of central densities for the SZEselected sample at 40 kpc. 

In the text 
Fig. 12. Comparison of the bolometric L_{Xc} − M relation to previous work (Maughan 2007; Mantz et al. 2010; Bulbul et al. 2019; Lovisari et al. 2020). 

In the text 
Fig. C.1. Simulated Y_{SZ} reextracted from PSZ2 maps, plotted as a function of L_{Xc}. The quantity 𝒬_{M} is the latent variable with respect to the mass M, obtained if there were no scatter in the relation. The Figure shows the effect of progressively adding scatter in Y_{SZ}, Y_{X}, and L_{Xc} with M. The blue points include covariance of t = 0.85 between Y_{X} and L_{Xc}. 

In the text 
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