11institutetext:  1 Université Paris-Saclay, Université Paris Cité, CEA, CNRS,
AIM de Paris-Saclay, 91191 Gif-sur-Yvette, France
11email<EMAIL_ADDRESS>
2 HH Wills Physics Laboratory, University of Bristol, Tyndall Ave, Bristol,
BS8 1TL, UK
3 IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France

# Linking a universal gas density profile to the core-excised X-ray luminosity
in galaxy clusters up to $z\sim 1.1$

G.W. Pratt 11 M. Arnaud 11 B.J. Maughan 22 J.-B. Melin 33

(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 core-excised luminosity, $L_{\rm Xc}$, and mass
from the $Y_{\rm X}$ proxy, $M_{\rm Y_{X}}$, for 93 objects selected through
their Sunyaev-Zeldovich effect (SZE) signal. The sample spans a mass range of
$M_{\rm 500}=[0.5-20]\times 10^{14}$ M⊙, and lies at redshifts $0.05<z<1.13$.
To investigate differences in X-ray and SZE selection, we compare to the local
X-ray-selected REXCESS sample. Using XMM-Newton observations, we derive an
average intra-cluster medium (ICM) density profile for the SZE-selected
systems and determine its scaling with mass and redshift. This average profile
exhibits an evolution that is slightly stronger than self-similar
($\alpha_{\rm z}=2.09\pm 0.02$), and a significant dependence on mass
($\alpha_{\rm M}=0.22\pm 0.01$). 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 $\sim 20\%$ at
$R\sim[0.5-0.7]\,R_{500}$ and a value of $\sim 40\%$ at $R_{500}$; moreover,
the scatter evolves slightly with redshift. The average profile of the SZE-
selected systems adequately describes the X-ray-selected 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 core-excised
luminosity, $L_{\rm Xc}$, and mass is extremely tight, with a measured
logarithmic intrinsic scatter of $\sigma_{\ln{L_{\rm Xc}|M_{\rm Yx}}}\sim
0.13$. 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
$L_{\rm Xc}$ and $Y_{\rm X}$. Accounting for our use of the $Y_{\rm X}$ 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 $\sigma_{\ln{L_{\rm Xc}|M}}\sim 0.22$. We explicitly
illustrate the connection between the scatter in density profiles and that in
the $L_{\rm Xc}$$-M$ relation. Our results are consistent with the overall
conclusion that the ICM bulk evolves approximately self-similarly, with the
core regions evolving separately. They indicate a systematic variation of the
gas content with mass. They also suggest that the core-excised X-ray
luminosity, $L_{\rm Xc}$, has a tight and well-understood relation to the
underlying mass.

###### Key Words.:

galaxies: clusters: general – galaxies: clusters: intracluster medium –
X-rays: galaxies: clusters

##  1 Introduction

In a $\Lambda$ 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 self-similar and scale-free because gravity has
no characteristic scale. If the baryon content remains constant, power-law
relations link the baryonic observable properties – such as X-ray luminosity
$L_{\rm X}$, or Sunyaev-Zeldovich effect (SZE) signal $Y_{\rm SZ}$, or total
optical richness $\Lambda$ – 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 X-ray
emission is proportional to the square of the gas density. This means that
X-ray measurements are very sensitive to the physical conditions in the core
regions, while SZE measurements are much less so.

The X-ray luminosity, $L_{\rm X}$, 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 intra-cluster medium (ICM) is in
hydrostatic equilibrium, $L_{\rm X}$ depends only on the halo mass, $M$,
redshift, $z$, and the distribution of the gas in the dark matter potential.
Power-law $L_{\rm X}-M$ 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 ($\sim 40$
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 $L_{\rm X}$ 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).

Figure 1: Sample properties. Left: Redshift-mass distribution of the clusters
used in this paper. The SZE-selected 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
X-ray-selected 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
SZE-selected samples.

Radial gas density profiles have been a key means of obtaining information
about the ICM since the advent of X-ray imaging. The high spatial resolution
observations afforded by XMM-Newton and Chandra observations have revealed the
complexity of both the core regions, which are strongly affected by non-
gravitational 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
X-ray-selected and SZE-selected samples may not have the same distribution of
dynamical states (e.g. Planck Collaboration IX 2011; Rossetti et al. 2016;
Andrade-Santos 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 SZE-selected 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 self-similar evolution of the cluster bulk
around this static core.

Here we investigate the properties of the gas density profiles and the core-
excised X-ray luminosity, $L_{\rm Xc}$, of 31 X-ray-selected clusters at
$z<0.2$ and 93 SZE-selected clusters at $z<1.13$. We describe a universal gas
density profile for the SZE-selected objects and quantify its variation with
redshift and mass. We quantify the radial variation in scaled profiles with
respect to the best-fitting evolution and the best-fitting 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 $L_{\rm Xc}$, measured in the
$[0.15-1]\,R_{500}$ region111$R_{\Delta}$ is the radius within which the mean
density is $\Delta$ times the critical density at the redshift of the object.,
is an extremely well-behaved mass proxy that does not depend on cluster
selection, and shows little evolution beyond self-similar in the broad
redshift range of the sample.

Throughout the paper we assume a flat $\Lambda$CDM cosmology with $\Omega_{\rm
m}=0.3$, $\Omega_{\Lambda}=0.7$, and $H_{0}=70$ km s-1 Mpc-1. The Sunyaev-
Zeldovich flux in units of square arcminutes is denoted $Y_{\rm SZ}$; the
quantity $D_{\rm A}^{2}\,Y_{\rm SZ}$, in units of square megaparsecs, is the
spherically integrated Compton parameter within $R_{500}$, where $D_{\rm A}$
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 X-ray-selected clusters at $0.05<z<0.2$ and 93 SZE-
selected objects at $0.08<z<1.13$, with six systems in common. The local
X-ray-selected dataset is REXCESS (Böhringer et al. 2007; Croston et al. 2008;
Pratt et al. 2009). The SZE-selected 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 XMM-Newton 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 signal-to-noise ${\rm S/N}>4$ 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 $M_{500}>5\times 10^{14}\,{\rm M_{\odot}}$ at $0.7<z<0.9$ in
the PSZ2 catalogue. The LP3 sample is derived from the five highest-mass
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
Tables A, A, A, and A.

The left-hand panel of Figure 1 shows the distribution of the 118 clusters in
the redshift-mass plane. Here and in the following, we group the SZE-selected
systems into three sub-samples in the redshift ranges $z<0.3$ (blue), $0.3\leq
z<0.6$ (light green), and $z\geq 0.6$ (dark green), containing $37,27$, and
$29$ objects, respectively. The right-hand panel of Figure 1 shows the stacked
mass histogram. This plot makes clear that the REXCESS sample (light blue) has
a lower median mass (${\rm M_{500}}=2.7\times 10^{14}\,{\rm M}_{\odot}$) than
any of the SZE-selected sub-samples (${\rm M_{500}}=6.4$, $7.9$, and
$5.0\times 10^{14}\,{\rm M}_{\odot}$, respectively). The LP2 sub-sample is
subject to significant Eddington bias in the Planck signal, leading in the
most extreme case to an estimated mass of only $6.5\times 10^{13}\,{\rm
M}_{\odot}$ for PSZ2 G208.57$-$44.31. We show below that this has a negligible
effect on our results.

###  2.2 Analysis

As our aim was to compare the SZE-selected clusters to the low-redshift X-ray-
selected REXCESS systems, we followed the X-ray data reduction and analysis
procedures described in Croston et al. (2008), Pratt et al. (2009), and Pratt
et al. (2010). Event files were reprocessed with the XMM-Newton 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.

Vignetting-corrected, background-subtracted [0.3-2] keV surface brightness
profiles were extracted in annular bins centred on the X-ray 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
X-ray peak, with a binning of $R_{\rm out}/R_{\rm in}=1.33=1.5$ depending on
data quality. After subtraction of the FWC spectra, all spectra were grouped
to a minimum of 25 counts per bin. The FWC-subtracted 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 $\Gamma=1.4$. The
spectra were fitted in the $[0.5-10]$ keV range using $\chi^{2}$ 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 best-fitting 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
$Z=0.3\,Z_{\odot}$ when its relative uncertainty exceeded $30\%$.

####  2.2.1 Luminosity

The core-excised X-ray luminosity $L_{\rm Xc}$ 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
$M_{500}-Y_{\rm X}$ 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 core-excised X-ray
luminosity $L_{\rm Xc}$ 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 best-fitting 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 $L_{\rm Xc}$ 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 PSF-corrected count rates in each aperture (see below).

####  2.2.2 Density profiles

Figure 2: Deprojected, PSF-corrected density profiles for 118 galaxy clusters,
normalised by the critical density $\rho_{\rm crit}$ and $R_{500}$. Fully
self-similar clusters would trace the same locus in this plot. The profiles
are colour-coded by mass $M_{500}$ in the left-hand panel, and by redshift $z$
in the right-hand panel. There are clear trends with respect to both
quantities.

The vignetting-corrected background-subtracted [0.3-2] keV surface brightness
profiles were used to obtain the deprojected, PSF-corrected density profiles
using the regularised, non-parametric 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 $\Lambda(\theta)$ 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 EPIC-MCT-TN-011222http://www.iasf-
milano.inaf.it/~simona/pub/EPIC-MCT/EPIC-MCT-TN-012.pdf and EPIC-MCT-
TN-012333http://www.iasf-milano.inaf.it/~simona/pub/EPIC-MCT/EPIC-MCT-
TN-012.pdf. In Bartalucci et al. (2017), the deprojected density profiles from
XMM-Newton 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 $\sim 5$ arcseconds (Fig. 6 of Bartalucci et al.
2017). The gas density $\rho_{\rm gas}=n_{\rm e}\times(\mu_{\rm e}\,m_{\rm
p})$, where $n_{\rm e}$ is the electron density measured in X-rays, $m_{\rm
p}$ is the proton mass, and $\mu_{\rm e}=1.148$ is the mean molecular weight
per free electron:

$\rho_{\rm gas}=1.92\times 10^{-24}\left(\frac{n_{\rm e}}{\rm
cm^{-3}}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm g}\ {\rm
cm^{-3}}.$ (1)

##  3 Gas density profiles

###  3.1 Model

In the self-similar model, a cluster can be completely defined by only two
parameters: its mass, $M_{\rm V}$, and its redshift, $z$. A fundamental
property of this model is the cluster overdensity, $\Delta$, with respect to
the reference density of the Universe $\rho_{\rm Uni}(z)$, from which the
virial mass $M_{\rm V}=\Delta\rho_{\rm Uni}(z)\,(4\pi/3)\,R_{\rm V}^{3}$, and
radius $R_{\rm V}$, can thereafter be defined. Cluster profiles then exhibit a
universal form when the radii are scaled to $R_{\rm V}$.

The original, and simplest, self-similar model concerns top-hat spherical
collapse in the Standard CDM ($\Omega=1$) cosmology. Here a cluster at
redshift $z$ is represented by a spherical perturbation that has just
collapsed, with $\rho_{\rm Univ}(z)$ being the critical density of the
Universe, $\rho_{\rm c}=3\,H^{2}(z)/(8\,\pi\,G)$, and $\Delta=178$. Of course,
the hierarchical formation of structure in a $\Lambda$CDM Universe is a very
complex dynamical process: objects continuously accrete matter along large-
scale 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
$\rho_{\rm Univ}(z)$, 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 self-similar when scaling by fixed overdensities with respect to the
critical density $\rho_{\rm c}(z)$. The zone in question corresponds to
overdensities of $\Delta\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}200$. As a scaling radius, we therefore
chose an $R_{500}$ corresponding to $\Delta=500$, the radius within which the
mean matter density is $\rho_{\rm 500}=500\,\rho_{\rm c}(z)$. The
corresponding total mass within this radius, $M_{\rm 500}$, is

$\displaystyle M_{\rm 500}=(4\pi/3)\,R_{500}^{3}\,\rho_{\rm 500},$ (2)

with

$\displaystyle\rho_{\rm 500}=500\,\rho_{\rm c}(z)=4.603\times
10^{-27}\,E(z)^{2}\ {\rm g\ cm^{-3}},$ (3)

and where $E(z)=[\Omega_{\rm m}(1+z)^{3}+\Omega_{\Lambda}]^{1/2}$ 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

$\displaystyle\rho(x)=\frac{\rho_{\rm gas}(R)}{\rho_{\rm
500}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm where}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x=\frac{R}{R_{500}}.$ (4)

In the self-similar model, $\rho(x)$ follows a universal shape and its
normalisation is independent of mass and redshift. In such a case
$\langle\rho_{\rm gas}\rangle\propto\rho_{\rm 500}$, 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 self-similar, they would trace the
same locus in this plot. This is clearly not the case. The colour-coding by
mass and by redshift highlights that at large radius, the scaled profiles of
the higher-mass, higher-redshift systems lie systematically above those of
lower-mass, lower redshift-objects. These trends suggest a dependence of the
scaling on $M_{\rm 500}$ and/or redshift.

Figure 3: Marginalised posterior likelihood for the parameters of the best-
fitting density profile model detailed in Sect. 3.1.  Figure 4: The universal
cluster ICM density profile. Left: Scaled density profiles of the SZE-selected
clusters (grey points), overplotted with the best-fitting GNFW model with free
evolution and mass dependence: $\rho_{\rm gas}/\rho_{500}(R/R_{500})\propto
E(z)^{\alpha_{\rm z}}\,M^{\alpha_{\rm M}}$ with $\alpha_{\rm z}=2.09\pm 0.02$
and $\alpha_{\rm M}=0.22\pm 0.01$ (orange line). The model includes a radially
varying intrinsic scatter term (orange envelope). Right: Comparison of the
best-fitting model, defined on the SZE-selected sample, to the best-fitting
model for the X-ray-selected REXCESS sample (light blue). Here, the points
with error bars are the REXCESS sample.

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 $M_{\rm 500}$, with a radially varying
intrinsic scatter. The median profile was expressed as

$\rho_{\rm m}(x,z,M_{\rm 500})=A(z,M_{\rm 500})f(x),$ (5)

where $f(x)$ is the function describing the profile shape. Here we adopted a
generalised Navarro-Frenk-White (GNFW) model (Nagai et al. 2007):

$f(x)=\frac{f_{0}}{(x/x_{\rm s})^{\alpha}\left[1+(x/x_{\rm
s})^{\gamma}\right]^{(3\beta-\alpha)/\gamma}},$ (6)

where $x_{\rm s}$ is the scaling radius, and the parameters
$(\alpha,\gamma,3\beta)$ are the central ($x\ll x_{\rm s}$), intermediate
($x\sim x_{\rm s}$), and outer ($x\gg x_{\rm s}$) slopes, respectively. The
case $\alpha=0$ and $\gamma=2$ corresponds to the standard $\beta$ model
(Cavaliere & Fusco-Femiano 1976), while the case $\gamma=2$ 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 $f_{0}\,A(z,M_{\rm 500})$, where
$A(z,M_{\rm 500})$ describes the departure from standard self-similarity in
terms of a possible mass and/or redshift dependence of the scaled gas density.
For this we assumed a power-law dependence on $M_{\rm 500}$ and $E(z)$:

$A(z,M_{\rm 500})=E(z)^{\alpha_{\rm z}}\,\left[\frac{M_{\rm 500}}{5\times
10^{14}\,{\rm M_{\odot}}}\right]^{\alpha_{\rm M}}.$ (7)

The standard self-similar model corresponds to $(\alpha_{\rm M}=0,\alpha_{\rm
z}=0)$. We expect $\alpha_{\rm M}>0$, as it is well established that the gas
mass fraction of local clusters decreases with decreasing mass due to non-
gravitational 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 $R_{500}$, $f_{\rm
gas,500}$, which varies with mass and redshift as a function of $\alpha_{\rm
M}$ and $\alpha_{\rm z}$:

$\displaystyle f_{\rm gas,500}=\frac{M_{\rm gas}(<R_{500})}{M_{\rm 500}}$
$\displaystyle=$ $\displaystyle\frac{1}{3}\int_{0}^{1}{\rho(x)x^{2}dx}$ (8)
$\displaystyle=$ $\displaystyle f_{0}\,A(z,M_{\rm 500})\,I(x_{\rm
s},\alpha,\gamma,3\beta),$ (9)

where $M_{\rm gas}(<R_{500})$ is the gas mass within $R_{500}$ and we have
used Eq. 2 and Eq. 4. The quantity $I(x_{\rm s},\alpha,\gamma,3\beta)$ 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 log-normal distribution at each radius. Taking into
account measurement errors, the probability of measuring a given scaled gas
density $\rho$ at given scaled radius $x$ for a cluster of mass $M_{\rm 500}$
at redshift $z$ is then

$\displaystyle p\,(\rho|x,z,M_{\rm 500})=\mathcal{N}\,[\log{\rho_{\rm
m}\,(x,z,M_{\rm 500})},\sigma^{2}(x)]$ (10)
$\displaystyle\sigma^{2}(x)=\sigma^{2}_{\rm int}(x)+\sigma^{2}_{\rm stat},$
(11)

where $\mathcal{N}$ is the log-normal distribution.The variance term,
$\sigma^{2}(x)$, is the quadratic sum of the statistical error, $\sigma_{\rm
stat}$ on the measured $\log{\rho}$ and of the intrinsic scatter on
$\log{\rho_{\rm m}}$ at radius $x$, $\sigma_{\rm int}(x)$.

We expect the intrinsic scatter to increase towards the centre, as observed in
Fig. 2, due to the increasing effect of non-gravitational 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 $\sigma_{\rm int}(x)$ with a log-parabola 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 non-analytical form for the intrinsic scatter, where $\sigma_{\rm
int}(x)$ is defined at $n$ equally spaced points in $\log(x)$ in the typical
observed radial range, $[x_{\rm{min}}$–$x_{\rm{}_{max}}]$. The scatter,
$\sigma_{\rm int}(x)$ at other radii is computed by spline interpolation. We
used $n=7$ between $x_{\rm min}=0.01$ and $x_{\rm max}=1$.

The likelihood of a set of scaled density profiles measured for a sample of
$i=1,N_{\rm c}$ clusters of mass $M_{\rm 500,i}$ and redshift $z_{\rm i}$ is:

$\displaystyle\mathcal{L}=\prod_{i=0}^{N_{\rm c}}\prod_{j=0}^{N_{\rm
R}[i]}p\,(\rho_{\rm i,j}|x_{\rm i,j},z_{\rm i},M_{\rm 500,i}),$ (12)

where $N_{\rm R}[i]$ is the number of points of the profile of cluster $i$,
and the quantity $\rho_{\rm i,j}=\rho_{\rm gas}[i,j]/\rho_{\rm 500}(z_{\rm
i})$ is the scaled density measured at each scaled radius $x_{\rm
i,j}=r[i,j]/R_{500}(z_{\rm i},M_{\rm 500,i})$, with $\rho_{\rm gas}[i,j]$ and
$r[i,j]$ being the physical gas density and radius. The statistical error on
$\log{\rho_{\rm i,j}}$ is $\sigma_{\rm stat,i,j}$.

We fitted the data (i.e. the set of observed $\rho_{\rm i,j}$) using Bayesian
maximum likelihood estimation with Markov Chain Monte Carlo (MCMC) sampling.
Using the emcee package developed by Foreman-Mackey et al. (2013), we
maximised the log of the likelihood, which reads (up to an additive constant)

$\displaystyle\ln{\mathcal{L}}=-0.5\sum_{i,j}\left[\ln{\sigma^{2}_{\rm
i,j}}+\frac{\left(\log{\rho_{\rm i,j}}-\log{\rho_{\rm
m,i,j}}\right)^{2}}{\sigma^{2}_{\rm i,j}}\right]$ (13) $\displaystyle\rho_{\rm
m,i,j}=\rho_{\rm m}(x_{\rm i,j},z_{\rm i},M_{\rm 500,i})$ (14)
$\displaystyle\sigma^{2}_{\rm i,j}=\sigma^{2}_{\rm int}(x_{\rm
i,j})+\sigma^{2}_{\rm stat,i,j}.$ (15)

The fit marginalises over a total of fourteen parameters: four describing the
shape of the median profile $(x_{\rm s},\alpha,\gamma,3\beta)$, a global
normalisation, $f_{0}$, the slopes $\alpha_{\rm M}$ and $\alpha_{\rm z}$ that
describe the non-standard 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 SZE-
selected systems. The resulting best-fitting model is

$\displaystyle\rho_{\rm m}(x,z,M_{\rm 500})=A(z,M_{\rm 500})f(x),$ (16)

with

$\displaystyle A(z,M_{\rm 500})=E(z)^{2.09\pm 0.02}\times\left[\frac{M_{\rm
500}}{5\times 10^{14}\,h_{70}^{-1}\,{\rm M}_{\odot}}\right]^{0.22\pm 0.01}$
(17)

and

$\displaystyle f(x)=\frac{f_{0}}{(x/x_{\rm s})^{\alpha}\left[1+(x/x_{\rm
s})^{\gamma}\right]^{(3\beta-\alpha)/\gamma}},$ (18)

where

$\displaystyle f_{0}=1.20\pm 0.15,$ $\displaystyle x_{\rm s}=0.28\pm 0.01,$
$\displaystyle\alpha=0.42\pm 0.06,$ $\displaystyle\beta=0.78\pm 0.03,\ {\rm
and}$ $\displaystyle\gamma=1.52\pm 0.16.$

Figure 3 shows the marginalised posterior likelihood for the parameters of the
best-fitting density profile model detailed in Sect. 3.1. All parameters are
well constrained: in particular, we note that the mass and evolution
parameters $\alpha_{\rm M}$ and $\alpha_{\rm z}$ do not show any degeneracies,
implying that we clearly separate the mass and redshift effects.

The left-hand panel of Fig. 4 shows the density profiles of the SZE-selected
clusters together with the best-fitting model. The intrinsic scatter term is
represented by the orange envelope; the numerical values for this term are
given in Table 1. The right-hand panel of Fig. 4 shows the best-fitting model
for the SZE-selected systems compared to the profiles from the X-ray-selected
REXCESS sample. The agreement is excellent beyond the core; in the inner
regions, there is a hint that the X-ray-selected systems may show more
dispersion. We will return to this point below in Sect. 5.1.5.

Table 1: Numerical values for the best-fitting intrinsic scatter term, measured at seven equally spaced points in $\log(R/R_{500})$, in the range $[0.01-1]\,R_{500}$.  Radius | $\sigma_{\rm int}$
---|---
0.010 | $0.98\pm 0.06$
0.021 | $0.75\pm 0.03$
0.046 | $0.60\pm 0.02$
0.100 | $0.38\pm 0.01$
0.215 | $0.19\pm 0.01$
0.464 | $0.12\pm 0.01$
1.000 | $0.19\pm 0.01$

##  4 Luminosity scaling relations

We now turn to the scaling relation between $L_{\rm Xc}$ and the mass $M_{\rm
500}$. The bolometric X-ray luminosity of a cluster can be written (Arnaud &
Evrard 1999)

$\displaystyle L(T)=f_{\rm gas}^{2}(T)[M(T)\Lambda(T)]\hat{Q}(T),$ (19)

where $f_{\rm gas}=M_{\rm gas}/M$ is the gas mass fraction, and $\Lambda(T)$
is the cooling function. The quantity $\hat{Q}(T)=\langle\rho_{\rm
gas}^{2}\rangle/\langle\rho_{\rm gas}\rangle^{2}$ 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 [$M\propto
T^{3/2}$]; (ii) simple Bremsstrahlung emission [$\Lambda(T)\propto T^{1/2}$];
(iii) similar internal structure [$\hat{Q}(T)={\rm const.}$]; (iv) a constant
gas mass fraction [$f_{\rm gas}^{2}(T)={\rm const.}$], the standard self-
similar relation between bolometric X-ray luminosity and mass, $L_{\rm
X}\propto M^{4/3}$, can be obtained. Similar arguments can be used to obtain
the soft-X-ray luminosity-mass relation of $L_{\rm X}\propto M$.

Figure 5: Relation between $L_{\rm Xc}$ and $Y_{\rm X}$. The blue envelope is
the best-fitting relation given in Eqn. 23, and the results from Maughan
(2007) are also shown for comparison. Figure 6: Relation between the core-
excised X-ray luminosity $L_{\rm Xc}$ and mass estimated from the $Y_{\rm X}$
proxy, for the bolometric and soft-band luminosities of 118 systems. Left:
Data points with the best fitting relation to the X-ray-selected REXCESS
sample with the evolution factor fixed to the self-similar value of $n=-2$
(grey envelope). The dark grey envelope shows the best fitting relation to the
37 SZE-selected systems at $z<0.3$ with $n=-2$. Right: Histogram of the log
space residuals from the best fitting relation to the SZE-selected objects at
$z<0.3$.  Figure 7: Relation between the core-excised X-ray luminosity $L_{\rm
Xc}$ and mass estimated from the $Y_{\rm X}$ proxy, for the bolometric and
soft-band 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 best-fitting values of $n$ are given in Table 4. Right:
Histogram of the log space residuals from the best fitting relation. Solid
lines show the best-fitting Gaussian distributions with $\sigma$ corresponding
to the best-fitting intrinsic scatter in log space (Table 4).  Table 2: Fits
to the core-excised X-ray luminosity $L_{\rm Xc}$$-M$ relation. $L_{\rm Xc}$
is measured in the $[0.15-1]\,R_{500}$ region.

Relation | Selection | Redshift | $n$ | $L_{0}$ | $A_{\rm L}$ | $B_{\rm L}$ | $\sigma_{\rm\ln{L}}$
---|---|---|---|---|---|---|---
|  |  |  |  $(10^{44}$ erg s${}^{-1})$ |  |  |
bolometric | REXCESS | $0.05<z<0.20$ | $-7/3$ | $5$ | $0.70\pm 0.03$ | $1.73\pm 0.05$ | $0.16\pm 0.03$
| SZ | $z<0.3$ | $-7/3$ | $5$ | $0.73\pm 0.02$ | $1.69\pm 0.05$ | $0.09\pm 0.02$
| all SZ | $z>0.05$ | $-2.50\pm 0.09$ | $5$ | $0.70\pm 0.02$ | $1.74\pm 0.02$ | $0.12\pm 0.01$
0.5-2 keV | REXCESS | $0.05<z<0.20$ | $-2$ | $1$ | $1.06\pm 0.04$ | $1.48\pm 0.05$ | $0.17\pm 0.03$
| SZ | $z<0.30$ | $-2$ | $1$ | $1.12\pm 0.04$ | $1.37\pm 0.06$ | $0.13\pm 0.02$
| all SZ | $z>0.05$ | $-2.23\pm 0.10$ | $1$ | $1.05\pm 0.03$ | $1.46\pm 0.03$ | $0.15\pm 0.02$

444The data were fitted in log-log space with a relation of the form
$E(z)^{n}\,(L/L_{0})=A_{\rm L}\,(M/4\times 10^{14}\,{\rm M}_{\odot})^{B_{\rm
L}}$.

###  4.1 Fitting method

We fitted the data with a power-law relation of the form

$E(z)^{n}\,L/L_{0}=A_{\rm L}\,(X/X_{0})^{B_{\rm L}},$ (20)

where $L_{0}=1\times 10^{44}$ erg s-1 and $5\times 10^{44}$ erg s-1 for the
soft and bolometric bands, respectively, and $X_{0}=5\times 10^{14}\,{\rm
M}_{\odot}\,{\rm keV}$ and $4\times 10^{14}\,{\rm M}_{\odot}$ for $Y_{\rm X}$
and $M$, respectively. Fitting was undertaken using linear regression in the
log-log 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)

$\displaystyle\ln{\mathcal{L}}=\sum_{i=1}^{N}\left[\ln{\frac{B_{\rm
L}^{2}+1}{\sigma_{i}^{2}}}-\frac{\left(\ln{(L_{i}/L_{0})}-\ln{A_{\rm
L}}-B_{\rm L}\ln{(X_{i}/X_{0})}\right)^{2}}{\sigma_{i}^{2}}\right],$ (21)

with

$\displaystyle\sigma_{i}^{2}=\sigma^{2}+B_{\rm
L}^{2}\sigma_{X,i}^{2}+\sigma_{L,i}^{2}$ (22)

and the intrinsic scatter, $\sigma^{2}$, as a free parameter. MCMC sampling
was undertaken using the emcee package developed by Foreman-Mackey et al.
(2013), with flat priors in the ranges $[-2.0,3.0]$ and $[0.0,1.0]$ for
$B_{\rm L}$ and $\sigma$, respectively. The results, reported in Table 4, were
compared to those obtained with the `LINMIX` (Kelly 2007) Bayesian regression
package: these were indistinguishable and so are not reported here.

Figure 8: Scaled density profiles. Left: Scaled density profiles (points) and
best-fitting model (orange envelope) for SZE-selected systems in our sample
compared to the median and $68\%$ dispersion from the X-COP 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
best-fitting intrinsic scatter obtained from our sample when the evolution
factor is forced to the self-similar value of $E(z)^{2}$ is also shown in
grey.

###  4.2 Results

####  4.2.1 $L_{\rm Xc}$$-Y_{\rm X}$

We first fitted the relation between the bolometric $L_{\rm Xc}$ and the mass
proxy $Y_{\rm X}$ for the SZE-selected systems only. With the evolution term
left free, the best-fitting relation, shown in Fig. 5, is

$\displaystyle E(z)^{-1.79\pm 0.08}\,(L_{\rm Xc}/L_{0})=(1.49\pm
0.03)\,(Y_{\rm X}/Y_{0})^{0.97\pm 0.02},$ (23)

with an intrinsic scatter of $\sigma_{L_{\rm Xc}|Y_{\rm X}}=0.09\pm 0.01$.
This result yields evolution and mass dependences that are in excellent
agreement with the self-similar 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_{\rm Xc}$$-M$: Low redshift with fixed evolution

We initially fixed the evolution factor, $n$, to the self-similar values of
$-2$ and $-7/3$ for the soft and bolometric bands, respectively. We first
fitted the bolometric $L_{\rm Xc}$–$M$ for the REXCESS data only, using a mass
pivot of $2\times 10^{14}\,{\rm M_{\odot}}$, as used by Pratt et al. (2009).
The resulting normalisation, $A_{\rm L}=1.06\pm 0.04$, slope $B_{\rm
L}=1.73\pm 0.05$, and intrinsic scatter $\sigma_{\ln{L_{\rm Xc}}}=0.16\pm
0.03$, 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
soft-band $L_{\rm Xc}$–$M$ relation, showing that the scaling relation
parameters that are obtained for the X-ray-selected are robust to the change
in underlying $M-Y_{\rm X}$ relation used to estimate the mass, and also to
differences in fitting method.

We then fitted the SZE-selected clusters at $z<0.3$ (37 systems). The results
are given in Table 4 and show that the normalisation and slope for this sub-
sample are in agreement within $1\sigma$ with those found for REXCESS. This
indicates that there is no difference in the scaling relation between the
local X-ray and SZE-selected samples, once the core region has been excised.
There is a slight hint that the intrinsic scatter of the SZE-selected sample
about the best-fitting relation is lower than that for REXCESS, although this
is only a $\sim 2\sigma$ effect for the bolometric luminosity, and is less
significant for the soft-band. The data and best-fitting relations, including
the $1\sigma$ scatter envelopes, are shown in the left-hand panel of Fig. 6.

####  4.2.3 $L_{\rm Xc}$$-M$: Free evolution

The right-hand panel of Fig. 6 shows the histogram of the residuals of the
full SZE-selected sample (93 systems, $0.08<z<1.13$) with respect to the best-
fitting relation to the systems at $z<0.3$. The peak is offset by
$\Delta\ln{L}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}0.1$, indicating that some evolution
beyond self-similar is in fact needed.

We then fitted the full SZE-selected sample with a power-law relation,
including a free evolution factor, $n$. The results are given in Table 4 and
the best-fitting relations are shown in the left-hand panel of Fig. 7; the
right-hand panel of Fig. 7 shows the residual histograms. The latter are well-
centred on zero. The best-fitting evolution terms, $n=-2.23\pm 0.09$ for the
soft-band and $n=-2.50\pm 0.10$ for the bolometric luminosity, suggest that
stronger than self-similar evolution is significant at the $\sim 2\sigma$
level.

Figure 9: Deviations from the average scaling with radius. Left: Redshift-mass
distribution of the SZE-selected sample used in this work. The shaded regions
indicate cuts for two sub-samples: 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 self-similar expectation ($\alpha_{\rm
z}=2$). The dashed line shows the best-fitting evolution, which varies from
slower than self-similar in the centre ($\alpha_{\rm z}\sim 0.3$) to faster
than self-similar around $R_{500}$ ($\alpha_{\rm z}\sim 2.4$). Envelopes show
the $1$ and $2\sigma$ uncertainties. Right: Degree to which the radial ICM
density profile scales with mass at nearly fixed redshift. The dotted line
shows the self-similar expectation ($\alpha_{\rm z}=0$). The dashed line shows
the best-fitting mass dependence of $\alpha_{\rm z}=0.22$. The scaled density
at nearly fixed redshift does not depend on radius.

##  5 Discussion

Figure 10: Scaled profiles and scatter. Left: Median scaled profiles (solid
lines) and $68\%$ scatter (envelopes) for REXCESS and the SZE-selected sample
split into three redshift bins. Beyond $\sim 0.2\,R_{500}$ the scaled profiles
are almost indistinguishable. Right: Radial profile of the intrinsic scatter
for the various sub-samples. The best-fitting intrinsic scatter model obtained
from the SZE-selected sample is also shown. Intrinsic scatter is less than
$20\%$ between $0.2\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}R_{500}\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}1.0$. The gold line
shows the model intrinsic scatter profile corrected for the covariance between
$M_{\rm gas}$ and $R_{500}$ (see Sect. 5.1.4), which results in a suppression
of the scatter by a factor of about two at $R_{500}$.

###  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 X-COP
sample, obtained assuming a self-similar evolution factor $E(z)^{2}$, was
presented in Ghirardini et al. (2019). Figure 8 compares the scaled density
profiles and the best-fitting model from our SZE-selected sample to their
median scaled density profile and $68\%$ dispersion. The agreement is good out
to the maximum X-COP radius of $\sim 2\,R_{500}$, 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 right-hand panel compares the intrinsic
scatter measurements, which also agree quite well, although the scatter of the
present sample is better constrained. The best-fitting intrinsic scatter
obtained from our sample when the evolution factor is forced to the self-
similar 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 $2.0\pm 0.2$ is in agreement with our results,
they find a mass dependence that is consistent with zero ($0.03\pm 0.06$). 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
$\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}20\%$ at $R_{2500}$. For the typical mass
of the present sample, $R_{2500}\approx 0.45\,R_{500}$, 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 sub-samples: one covering a large redshift at nearly constant
mass, and another covering a large mass range at nearly constant redshift.
These sub-samples are illustrated in the $z-M$ plane by the orange and blue
regions in the left-hand 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 best-fitting model and fitted a function of the form $\rho_{\rm
gas}(x)/\rho_{\rm crit,0}\propto E(z)^{\alpha_{\rm z}}\,M^{\alpha_{\rm M}}$.
For a self-similarly evolving population, the gas density scales with the
critical density $\rho_{\rm crit}$ and there is no mass dependence, and so
$\alpha_{\rm z}=2$ and $\alpha_{\rm M}=0$.

The middle- and right-hand panels of Fig. 9 show the degree to which the two
sub-samples vary from self-similar scaling and from the best-fitting 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
self-similar. However, the density evolves differently in the core and in the
outer regions. The density evolution is consistent with zero in the core:
$\alpha_{\rm z}=0.28\pm 1.10$ 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
self-similar: $\alpha_{\rm z}=2.42\pm 0.22$ at $x=1.20$, a result that is
significant at slightly more than $1\sigma$. At nearly fixed redshift, the
density varies with mass in agreement with the $\alpha_{\rm M}=0.22\pm 0.01$
scaling established above. At $R_{500}$, this mass dependence is significant
at $\sim 2\sigma$, 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
left-hand panel of Figure 10 shows the median scaled profile, obtained in the
log-log plane, and $68\%$ dispersion for REXCESS X-ray-selected sample and the
SZE-selected sample split into three redshift bins. It is clear that once
scaled, the four sub-samples 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 SZE-
selected sample at $z<0.3$ are virtually indistinguishable. We will return to
the central regions in Sect. 5.1.5 below.

The radial variation of the intrinsic scatter about these median profiles is
quantified in the right-hand panel of Figure 10, together with that of the
best-fitting intrinsic scatter model obtained above (from the SZE-selected
clusters only). The intrinsic scatter of this model falls below $20\%$ in the
radial range $0.2\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}R_{500}\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}1.0$. There is
excellent agreement between the best-fitting intrinsic scatter model and the
observed profiles, which all follow broadly the same trend with scaled radius:
a steep decrease with a minimum at $\sim 0.5-0.7R_{500}$, followed by an
increase towards larger radii.

Figure 11: Central density. Left: Histogram of central densities for the SZE-
selected systems, scaled according to the best-fitting model (Eqn. 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 $z<0.3$ SZE-selected systems compared to the X-ray-selected
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
SZE-selected sample at 40 kpc.

Within $R\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}0.2R_{500}$ 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 $\sim 0.2R_{500}$ the relative dispersion of all samples dips below
$20\%$, and at $R_{500}$ the dispersion in profiles is $\sim 15\%$. In the
SZE-selected sub-samples, there is a clear evolution, in the sense that the
low-redshift systems exhibit the lowest intrinsic scatter values while the
high-redshift systems show higher values. The scatter will be related to
intrinsic cluster-to-cluster 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
higher-redshift 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 $M_{\rm gas}$ and
$R_{500}$

The observed scatter in the density profiles may be suppressed by the use of
$M_{\rm gas}$ in the computation of $R_{500}$ when scaling the radial
coordinate. All other things being equal, a cluster with a higher than average
$\rho_{\rm gas}$ (for its mass) at some radius, will have a higher than
average $M_{\rm gas}$ and hence $Y_{\text{X}}$ relative to its mass. Since
$Y_{\text{X}}$ 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 best-fitting median
profile presented in Sect. 3.2 was normalised to match a
$6\leavevmode\nobreak\ \rm keV$ cluster at $z=0.15$ (i.e. when integrated to
$R_{500}$, the gas mass and $Y_{\rm X}$ 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 $\rho_{\rm
gas}$ 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 $Y_{\rm X}$ (assuming a fixed temperature of $6\rm keV$) 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 $M-Y_{\rm X}$ 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 $M_{\rm gas}$ 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 SZE-selected sample, covering the redshift range $0.25<z<1.2$, evolved
self-similarly 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 un-evolving core component embedded in a self-similarly 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 $5^{\prime\prime}$ resolution of XMM-Newton after PSF
correction (Bartalucci et al. 2017) allows us to measure the density of the
SZE-selected sample across all redshifts down to a scaled radius of $R\sim
0.05\,R_{500}$. The left-hand panel of Fig. 11 shows a histogram of the
resulting values555The measurement requires a small extrapolation, by
$<1^{\prime\prime}$ in the log-log plane, for seven systems.  scaled according
to the best-fitting model (Eqn. 17) established in Sect. 3.2. The SZE-selected
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 SZE-selected sub-samples. While the
histogram of the $z>0.6$ sub-sample has no detectable skewness, the histogram
of the $z<0.3$ sub-sample 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.

At $z<0.3$, the effective $5^{\prime\prime}$ resolution of XMM-Newton after
PSF correction allows us to measure the density down to a scaled radius of
$R\sim 0.015\,R_{500}$. The middle panel of Fig. 11 shows a histogram of the
scaled central density of the SZE-selected clusters at $z<0.3$ compared to
that of REXCESS. The positive skewness of the SZE-selected 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 SZE-selected sub-samples, and a secondary
peak at a scaled density of $\leavevmode\nobreak\ 0.7$ (in log space). The
latter peak is due to cool core systems and may indicate that centrally peaked
systems are over-represented in X-ray-selected samples, as has been argued by
Rossetti et al. (2017) from their comparison of the image concentration
parameter in Planck clusters to those for X-ray-selected systems, and also by
Andrade-Santos et al. (2017).

The right-hand panel of Fig. 11 shows the histogram of the central density of
the SZE-selected sample at 40 kpc, measured in physical units (cm-3). There is
a broad maximum at $n_{\rm e,40\,kpc}\sim 0.01\ {\rm cm}^{-3}$, and the
histograms of the three sub-samples coincide. A Kolmogorov-Smirnov test
indicates that all three sub-samples 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_{\rm Xc}$$-M$ relations

####  5.2.1 Comparison with other work

Figure 12 shows the best-fitting bolometric $L_{\rm Xc}$$-M$ relation for the
SZE-selected 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 self-
similar 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 self-similar
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.

Concerning the sample selection, the results in Sect. 4.2 show that the
$L_{\rm Xc}$$-M$ relations of X-ray- and SZE-selected systems are in good
agreement, suggesting that once the core regions are excluded, effects due to
detection methods relying on the ICM do not have any impact. Similarly,
Lovisari et al. (2020) showed that their relaxed and disturbed samples had
similar $L_{\rm Xc}$$-M$ relation slopes and normalisations. This suggests
that the $L_{\rm 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 $Y_{\rm X}$ as a mass proxy; the works listed
above use variously $Y_{\rm X}$ (Maughan 2007), the gas mass (Mantz et al.
2010), the SZE signal-to-noise (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. 2014).

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
X-ray 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, weak-lensing 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 (von der Linden et al. 2014; Kelly et al.
2014; Applegate et al. 2014, WtG;), and CHEX-MATE (CHEX-MATE Collaboration
2021). The cluster community is undertaking a major ongoing effort to
critically compare various mass estimates, obtained from mass proxies and from
direct X-ray, lensing, or velocity dispersion analyses (e.g. Rozo et al. 2014;
Sereno & Ettori 2015; Sereno 2015; Groener et al. 2016; Sereno & Ettori 2017).
Ultimately, this effort will help to better constrain the parameters of the
scaling relations.

Figure 12: Comparison of the bolometric $L_{\rm Xc}$$-M$ relation to previous
work (Maughan 2007; Mantz et al. 2010; Bulbul et al. 2019; Lovisari et al.
2020).

####  5.2.2 Link between density profile and $L_{\rm Xc}$

As noted in for example Maughan et al. (2008), the similarity of the ICM
density profiles outside the core implies a low scatter in $L_{\rm Xc}$, as is
indeed observed here. In order to explore how much of the scatter in $L_{\rm
Xc}$ is due to the variation in density profiles, we computed a ‘pseudo
luminosity’ $L_{\rm Xc}^{\prime}=\int\rho_{\rm gas}^{2}T_{\rm
X}^{1/2}\operatorname{d\\!}V$ for each cluster. For this calculation, we used
the measured density profile for each cluster and assumed isothermality at the
measured core-excised temperature for the cluster. The integral was performed
over a cylindrical volume from projected radii of $0.15R_{500}$ 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 $\approx 90\%$ 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 $L_{\rm Xc}^{\prime}$ then
provides an estimate of the scatter in the bolometric $L_{\rm Xc}$ due only to
the scatter in density profiles.

The intrinsic scatter about the best fitting relation to $L_{\rm
Xc}^{\prime}\,E(z)^{-2.5}$ versus $M_{\rm 500}$ was then measured (assuming
that the fractional statistical error on $L_{\rm Xc}^{\prime}$ is the same as
that in $L_{\rm Xc}$ for each cluster), giving a value of $11\%$. This implies
that most or all of the intrinsic scatter in the bolometric $L_{\rm Xc}$ at
fixed mass can be explained by the variation in the ICM density profiles.

In principle, the use of $M_{\rm Yx}$ to determine $R_{500}$ and hence the
aperture within which $L_{\rm Xc}$ and $L_{\rm Xc}^{\prime}$ are measured,
could introduce additional scatter in the $L_{\rm Xc}$$-M_{\rm 500}$ relation.
If $M_{\rm Yx}$ were scattered high relative to the true mass of a cluster,
then $R_{500}$ would be overestimated and the $0.15-1R_{500}$ aperture would
be shifted to larger radii. This shift would reduce $L_{\rm Xc}$ since more of
the luminosity comes from the inner edge of the aperture than the outer edge.
Hence, if $M_{\rm Yx}$ were scattered high, then $L_{\rm Xc}$ and $L_{\rm
Xc}^{\prime}$ would be scattered low, adding to the observed scatter in the
$L_{\rm Xc}^{\prime}-M_{\rm 500}$ relation. We examined the impact of this
effect by adding $10\%$ scatter to $M_{\rm Yx}$ when computing $L_{\rm
Xc}^{\prime}$. This increased the measured scatter in $L_{\rm Xc}^{\prime}$ by
less than $1\%$. The dependence of the luminosity aperture on $M_{\rm Yx}$
leads to a negligible contribution to the scatter in the $L_{\rm Xc}$$-M_{\rm
Yx}$ relation.

We therefore conclude that the measured intrinsic scatter in the bolometric
$L_{\rm Xc}$ 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 $M_{\rm Yx}$ and the true mass
$M_{\rm 500}$. 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_{\rm Xc}$$-M$ relation has a slope
$B_{\rm L}\sim 1.7$, which is steeper than the self-similar value of 1.3, and
has a small scatter of $\sigma_{\ln{L_{\rm Xc}}}\sim 0.15$. A similar
difference in slope of $\Delta\,B_{\rm L}\sim+0.4$ is seen relative to the
self-similar soft-band $L_{\rm Xc}$$-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,
$\mathcal{O}_{\rm det}$, can be affected by so-called Malmquist bias. This
happens because clusters that are scattered to higher values of
$\mathcal{O}_{\rm det}$, 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 $\mathcal{O}_{\rm det}$. This is a
particular concern near the detection threshold, and will affect the apparent
slope of any scaling relation with $\mathcal{O}_{\rm det}$. 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 X-ray-selected sample
are compatible within $1\sigma$ with those for the local SZE-selected sample,
we focus here on the SZE selection. As detailed in Sect. 2.1, the SZE-selected
sample is composed of four different sub-samples and with different
selections: the ESZ, LP1, LP2 and LP3. Although the ESZ sub-sample was
generated from high signal-to-noise ratio (${\rm S/N}>6$) detections, Andrade-
Santos et al. (2021) showed that its selection is dominated by the
instrumental and astrophysical scatter, and that the intrinsic scatter in the
$Y_{\rm SZ}$-$M$ relation has a negligible effect on the recovered scaling
parameters. The three other SZE sub-samples (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_{\rm Xc}$$-M$ of the SZE-selected systems
in the current study.

Appendix B describes in detail the simulations we used, which were similar to
those produced for the Andrade-Santos 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 Multi-Matched Filter extraction
algorithm (Melin et al. 2006) was then applied, to obtain SZ detections at
${\rm S/N}>6$, 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 $Y_{\rm SZ}$, $Y_{\rm X}$, and $L_{\rm Xc}$ at fixed true
mass $M$, with a covariance matrix to describe correlations between
parameters.

A first-order estimate of the expected scatter in the $L_{\rm Xc}$$-M_{\rm
Yx}$ relation can be obtained by assuming that selection effects are
negligible. In this case,

$\displaystyle\sigma^{2}_{\ln{L_{\rm Xc}|M_{\rm Yx}}}=\sigma^{2}_{\ln{L_{\rm
Xc}|M}}+\left({B_{\rm L}^{2}/B_{\rm Y}^{2}}\right)\,\sigma^{2}_{\ln{Y_{\rm
X}|M}}-2\,t\,\sigma_{\ln{L_{\rm Xc}|M}}\,\sigma_{\ln{Y_{\rm X}|M}}.$ (24)

For a measured intrinsic scatter $\sigma_{\ln{L_{\rm Xc}|M_{\rm Yx}}}\sim
0.13$, a covariance between $L_{\rm Xc}$ and $Y_{\rm X}$ of $t=0.85$ (Farahi
et al. 2019), and assuming a scatter between $Y_{\rm X}$ and true mass $M$ of
$\sigma_{\ln{Y_{\rm X}|M}}\sim 0.16$ motivated by recent numerical simulations
(Planelles et al. 2014; Le Brun et al. 2017; Truong et al. 2018), the
resulting dispersion between core-excised luminosity and true mass is
$\sigma_{\ln{L_{\rm Xc}|M}}\sim 0.22$. This suggests that the measured scatter
is underestimated by a factor of $0.22/0.13\sim 1.7$ due to the use of $Y_{\rm
X}$ 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_{\rm Xc}$$-Y_{\rm SZ}$ relation due to
intrinsic scatter in the $D_{\rm A}^{2}\,Y_{\rm SZ}-M$ relation is completely
negligible, with the slope changing by less than 0.5%. The resulting impact on
the slope of the $L_{\rm Xc}$$-M$ relation is $\Delta B_{\rm L}=-0.01$. This
is important, because it implies that selection effects due to intrinsic
scatter in the $Y_{\rm SZ}$ observable cannot account for the observed steeper
slope of the $L_{\rm Xc}$$-M$ relations seen here.

  * •

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

  * •

Intrinsic scatter in the $L_{\rm Xc}$$-Y_{\rm X}$ relation has the effect of
redressing the slope towards its original value, with $\Delta B_{\rm
L}=+0.02$.

  * •

Covariance between $Y_{\rm X}$ and $L_{\rm Xc}$ again changes the slope. For
$t=0.85$ (Farahi et al. 2019), the impact on the slope of the $L_{\rm Xc}$$-M$
relation is $\Delta B_{\rm L}=+0.04$.

We thus conclude that the slope of the $L_{\rm Xc}$$-M_{\rm Yx}$ relation
found here is robust to selection effects due to intrinsic scatter in the
$Y_{\rm SZ}$ and $Y_{\rm X}$ proxies, and to covariance between quantities.
The dispersion, however, is very sensitive to the covariance. We estimate that
the measured dispersion in the relation between the core-excised luminosity
and true mass, $L_{\rm Xc}$$-M$, is underestimated by a factor of $\sim 1.7$
due to the use of $Y_{\rm X}$ as a mass proxy.

###  5.3 Link to $f_{\rm gas}-M$

The $L_{\rm Xc}$$-M$ relations derived above have a steeper dependence than
self-similar, which cannot be explained by selection effects, intrinsic
scatter, or covariance. There is evidence for a dependence of the gas content
with mass, which has the effect of suppressing the luminosity preferentially
in lower-mass systems, leading to the observed steepening of the relation (see
discussion in Pratt et al. 2009). Interestingly, with the assumption of a
standard dependence of temperature on mass ($T\propto M^{2/3}$, e.g. Arnaud et
al. 2005; Mantz et al. 2016), use of Eqn. 19 with the observed bolometric
$L_{\rm Xc}$$\propto M^{1.7}$ relation yields $f_{\rm gas}\propto M^{0.21}$.

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 $f_{\rm gas}=M_{\rm gas}/M$ on the total mass
$M$, via Eqn. 8. The best-fitting gas density model (Eqn 17) suggests $f_{\rm
gas}\propto M^{0.22\pm 0.01}$. This dependence of the gas content on total
mass is therefore in good agreement with the expected relation given the
observed $L_{\rm Xc}$$-M$ dependence, and with previous findings (e.g. Pratt
et al. 2009; Lovisari et al. 2015; Ettori 2015) based primarily on X-ray-
selected clusters.

##  6 Conclusions

We have examined the gas density profiles and the relation between the core
excised X-ray luminosity $L_{\rm Xc}$ and the total mass derived from the
$Y_{\rm X}$ mass proxy for 118 X-ray and SZE-selected objects covering a mass
range of $M_{500}=[0.5-20]\times 10^{14}\,$ M⊙ and extending in redshift up to
$z\sim 1.13$. We first examined the scaled density profiles:

  * •

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

  * •

Motivated by this finding, we fitted an analytic gas density model to the 93
SZE-selected 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 SZE-selected clusters as a function of mass and redshift. This model is
given in Eqns. 16-18.

  * •

The intrinsic dispersion in scaled profiles is greatest in the central
regions, declining to a minimum at $\sim 0.5-0.7\,R_{500}$, and increasing
thereafter. The dispersion is similar for X-ray-selected clusters and for
local SZE-selected clusters, except in the centre, where the X-ray-selected
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 $M_{\rm gas}$ and $R_{500}$
due to the use of $M_{\rm Y_{\rm X}}$ 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
SZE-selected 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 X-ray and
SZE-selected systems at $z<0.3$. The scaled central density of the local
X-ray-selected sample exhibits two peaks. The main peak, corresponding to non-
cool core systems in the X-ray-selected sample, is slightly offset to higher
scaled central density from that of the local SZE-selected sample. The
secondary peak in the X-ray-selected sample, corresponding to the cool core
systems, is not seen in the SZE-selected 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 SZE-selected 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 X-ray luminosity
$L_{\rm Xc}$ and the total mass derived from the $Y_{\rm X,500}$ mass proxy,
$M_{\rm Yx}$.

  * •

This relation is extremely tight, with a logarithmic intrinsic scatter of
$\sigma_{\ln{L_{\rm xc}|M_{\rm Yx}}}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}0.15$ depending on sub-sample and band in
which the luminosity is measured. Importantly, at low redshift, the best-
fitting parameters of this relation do not depend on whether the sample was
selected in X-rays or through the SZE, suggesting that $L_{\rm Xc}$ is a
selection-independent quantity.

  * •

The slope of the bolometric relation fitted to the SZE-selected clusters,
$B\sim 1.74\pm 0.02$, is significantly steeper than self-similar. When left
free to vary, the evolution of $n=-2.5\pm 0.09$ is in agreement with the self-
similar value of $-7/3$ within $<2\sigma$.

  * •

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

  * •

We show explicitly that the scatter in the $L_{\rm Xc}$$-M$ relation can be
accounted for almost entirely by object-to-object 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 X-ray- and SZE-
selected samples. Our overall conclusion is consistent with the view that the
ICM bulk evolves approximately self-similarly, 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 X-ray- and SZE-selected systems. It also supports a view where
the ICM gas mass fraction depends on mass up to high redshift, with a
dependence $f_{\rm gas}\propto M_{500}^{0.22\pm 0.02}$ 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 CHEX-MATE project (CHEX-MATE Collaboration 2021).

###### Acknowledgements.

GWP, MA, and J-BM acknowledge funding from the European Research Council under
the European Union’s Seventh Framework Programme (FP72007-2013) ERC grant
agreement no. 340519, and from the French space agency, CNES. BJM 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 XMM-Newton observatory, an ESA science mission with instruments and
contributions directly funded by ESA Member States and NASA.

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##  Appendix A Sample data

Tables A, A, A, and A contain the sample observation details, including:
cluster name(s), redshift, coordinates, column density, exposure time, and
XMM-Newton OBSID used for the analysis.

Planck name | SPT name | ACT name | Alt. name | z | RA | Dec. | $N_{\mathrm{H}}$ | Exp. | Obs. Id
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  | MOS1-2, PN |
|  |  |  |  | [J2000] | [J2000] | [$10^{20}$cm-3] | [ks] |
|  |  |  RXC J2023.0 -2056, AS0868 | 0.056 | 305.7450 | $-20.9485$ | 5.60 | 17, 9 | 201902301
|  |  |  RXC J2157.4 -0747, A2399 | 0.058 | 329.3673 | $-7.8046$ | 3.50 | 10, 7 | 201902801
PSZ1 G246.01-51.76 |  |  |  RXC J0345.7 -4112, AS0384 | 0.060 | 56.4428 | $-41.2042$ | 1.90 | 17, 8 | 201900801
|  |  | RXC J0225.1 -2928 | 0.060 | 36.2877 | $-29.4773$ | 1.70 | 20, 17 | 302610601
|  |  |  RXC J1236.7 -3354, AS0700 | 0.080 | 189.1712 | $-33.9260$ | 5.60 | 24, 18 | 302610701
PSZ2 G346.86-45.38 |  |  |  RXC J2129.8 -5048, A3771 | 0.080 | 322.4271 | $-50.8167$ | 2.20 | 23, 13 | 201902501
PSZ2 G093.92+34.92 |  |  | A2255 | $0.081$ | $258.2029$ | $64.0636$ | $2.50$ |  $6$, $4$ | 011226081
PSZ2 G222.52+20.58 |  |  |  RXC J0821.8 + 0112, A653 | 0.082 | 125.4614 | 1.1967 | 4.20 | 11, 7 | 201903601
PSZ2 G306.77+58.61 |  |  | A1651 | $0.084$ | $194.8438$ | $-4.1983$ | $1.81$ |  $7$, $4$ | 020302011
PSZ2 G306.66+61.06 |  |  | A1650 | $0.084$ | $194.6734$ | $-1.7622$ | $0.72$ |  $34$, $28$ | 009320011
PSZ2 G308.64+60.26 |  |  |  RXC J1302.8 -0230, A1663 | 0.085 | 195.7217 | $-2.51695$ | 1.70 | 26, 16 | 201901801
PSZ2 G099.57-58.64 |  |  |  RXC J0003.8 +0203, A2700 | 0.092 | 0.9572 | 2.0657 | 3.00 | 26, 19 | 201900101
PSZ2 G321.98-47.96 | SPT-CLJ2249-6426 |  | A3921 | $0.094$ | $342.4917$ | $-64.4294$ | $1.61$ |  $30$, $23$ | 011224011
PSZ2 G336.60-55.43 |  |  | A3911 | $0.097$ | $341.5711$ | $-52.7261$ | $1.50$ |  $23$, $10$ | 014967031
PSZ2 G311.62-42.31 |  |  |  RXC J2319.6 - 7313, A3992 | 0.098 | 349.9168 | $-73.2277$ | 1.90 | 10, 6 | 201903301
PSZ2 G332.23-46.37 | SPT-CLJ2201-5956 |  | A3827 | $0.099$ | $330.4720$ | $-59.9454$ | $2.09$ |  $21$, $11$ | 014967011
|  |  |  RXC J0211.4 - 4017, A2984 | 0.101 | 32.8528 | $-40.2915$ | 1.40 | 29, 22 | 201900601
|  |  |  RXC J0049.4 - 2931, AS0084 | 0.108 | 12.3459 | $-29.5206$ | 1.80 | 20, 13 | 201900401
PSZ2 G053.53+59.52 |  |  | A2034 | $0.113$ | $227.5528$ | $33.5104$ | $1.54$ |  $10$, $6$ | 030393011
PSZ2 G352.28-77.66 |  |  |  RXC J0006.0 - 3443, A2721 | 0.115 | 1.5015 | $-34.7224$ | 1.20 | 12, 6 | 201903801
PSZ2 G255.64-25.30 |  |  | RXC J0616.8 -4748 | 0.116 | 94.2158 | $-47.7950$ | 4.80 | 23, 19 | 302610401
PSZ2 G285.52-62.23 | SPT-CL J0145-5301 | ACT-CL J0145-5301 |  RXC J0145.0 -5300, A2941 | 0.117 | 26.2433 | $-53.02085$ | 2.30 | 1,0 | 201900501
|  |  |  RXC,J1516.3 +0005, A2050 | 0.118 | 229.0747 | 0.0893 | 4.60 | 27, 21 | 201902001
|  |  |  RXC J2149.1 -3041, A3814 | 0.118 | 327.2817 | $-30.7013$ | 2.30 | 25, 18 | 201902601
PSZ2 G277.38+47.07 |  |  |  RXC J1141.4 -1216, A1348 | 0.119 | 175.3513 | $-12.2776$ | 3.30 | 28, 22 | 201901601
PSZ2 G000.04+45.13 |  |  |  RXC J1516.5 -0056, A2051 | 0.120 | 229.1842 | $-0.9696$ | 5.50 | 29, 22, | 201902101
PSZ2 G256.38+44.04 |  |  |  RXC J1044.5 -0704, A1084 | 0.134 | 161.1370 | $-7.0687$ | 3.40 | 26, 18 | 201901501
PSZ2 G241.79-24.01 |  |  |  RXC J0605.8 -3518, A3378 | $0.139$ | $91.4752$ | $-35.3023$ | $4.02$ |  $14$, $11$ | 020190101
PSZ2 G042.81-82.97 |  |  |  RXC J0020.7 -2542, A0022 | 0.141 | 5.1755 | $-25.7080$ | 2.3 | 15, 11 | 201900301
PSZ2 G002.77-56.16 |  |  |  RXC J2218.6 -3853, A3856 | $0.141$ | $334.6677$ | $-38.9018$ | $1.13$ |  $14$, $4$ | 020190301
PSZ2 G226.18+76.79 |  |  | A1413 | $0.143$ | $178.8250$ | $23.4049$ | $1.84$ |  $62$, $42$ | 050269021
PSZ2 G028.77-33.56 |  |  |  RXC J2048.1 $-1750$, A2328 | 0.147 | 312.0419 | -17.8413 | 4.80 | 25, 19 | 201902401
PSZ2 G236.92-26.65 |  |  |  RXC J0547.6 -3152, A3364 | $0.148$ | $86.9081$ | $-31.8727$ | $2.07$ |  $22$, $14$ | 020190091
PSZ2 G008.47-56.34 |  |  |  RXC J2217.7 -3543, A3854 | $0.149$ | $334.4400$ | $-35.7260$ | $1.20$ |  $22$, $14$ | 020190291
PSZ2 G003.93-59.41 |  |  |  RXC J2234.5 -3744, A3888 | $0.151$ | $338.6125$ | $-37.7360$ | $1.32$ |  $20$, $13$ | 040491081
PSZ2 G021.10+33.24 |  |  | A2204 | $0.152$ | $248.1959$ | $5.5754$ | $6.97$ |  $14$, $8$ | 030649021
PSZ2 G244.71+32.50 |  |  | RXC J0945.4 -0839, A0868 | $0.153$ | $146.3575$ | $-8.6557$ | $3.59$ |  $8$, $5$ | 001754011
PSZ2 G018.32-28.50 |  |  | RXC J2014.8 -2430 | 0.154 | 303.7154 | $-24.5059$ | 7.40 | 25, 16 | 201902201
PSZ2 G049.22+30.87 |  |  | RX J1720.1+2638 | $0.164$ | $260.0417$ | $26.6250$ | $5.65$ |  $14$, $8$ | 050067041
PSZ2 G263.68-22.55 | SPT-CLJ0645-5413 | ACT-CL J0645-5413 |  RXC J0645.4 -5413, A3404 | $0.164$ | $101.3712$ | $-54.2273$ | $5.60$ |  $11$, $7$ | 040491041

Table 3: Sample observation details. Columns: (1) Planck name; (2) SPT name;
(3) ACT name (4) alternative name (non-exhaustive), 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) XMM-
Newton OBSID.

Planck name | SPT name | ACT name | Alt. name | z | RA | Dec. | $N_{\mathrm{H}}$ | Exp. | Obs. Id
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  | MOS1-2, PN |
|  |  |  |  | [J2000] | [J2000] | [$10^{20}$cm-3] | [ks] |
PSZ2 G249.38+33.26 |  |  |  RXC J0958.3 -1103, A0907 | 0.167 | 149.5929 | $-11.0644$ | 5.10 | 9, 5 | 201903501
PSZ2 G097.72+38.12 |  |  | A2218 | $0.171$ | $248.9597$ | $66.2125$ | $2.60$ |  $17$, $11$ | 011298011
PSZ2 G067.17+67.46 |  |  | A1914 | $0.171$ | $216.5105$ | $37.8243$ | $1.06$ |  $15$, $7$ | 011223021
PSZ2 G149.75+34.68 |  |  | A0665 | $0.182$ | $127.7462$ | $65.8398$ | $4.24$ |  $5$, $2$ | 010989041
PSZ2 G313.33+61.13 |  |  |  RXC J1311.4 -0120, A1689 | $0.183$ | $197.8726$ | $-1.3417$ | $1.52$ |  $36$, $27$ | 009303011
PSZ2 G195.75-24.32 |  |  | A520 | $0.203$ | $73.5402$ | $2.9212$ | $3.74$ |  $20$, $9$ | 020151011
PSZ2 G006.76+30.45 |  |  | A2163 | $0.203$ | $243.9399$ | $-6.1491$ | $16.50$ |  $10$, $6$ | 011223061
PSZ2 G182.59+55.83 |  |  | A963 | $0.206$ | $154.2653$ | $39.0482$ | $1.25$ |  $18$, $12$ | 008423071
PSZ2 G166.09+43.38 |  |  | A773 | $0.217$ | $139.4765$ | $51.7315$ | $1.28$ |  $13$, $14$ | 008423061
PSZ2 G092.71+73.46 |  |  | A1763 | $0.223$ | $203.8298$ | $41.0001$ | $0.94$ |  $12$, $9$ | 008423091
PSZ2 G055.59+31.85 |  |  | A2261 | 0.224 | 260.6127 | 32.1324 | 3.19 | 2, 1 | 0093031001
PSZ2 G072.62+41.46 |  |  | A2219 | $0.228$ | $250.0837$ | $46.7107$ | $1.76$ |  $12$, $6$ | 060500051
PSZ2 G073.97-27.82 |  |  | A2390 | $0.231$ | $328.4031$ | $17.6949$ | $8.66$ |  $10$, $8$ | 011127011
PSZ2 G294.68-37.01 |  |  | RXCJ0303.8-7752 | $0.274$ | $45.9366$ | $-77.8784$ | $8.73$ |  $11$, $8$ | 020533011
PSZ2 G241.76-30.88 |  |  | RXCJ0532.9-3701 | $0.275$ | $83.2323$ | $-37.0270$ | $2.90$ |  $11$, $6$ | 004234181
PSZ2 G259.98-63.43 | SPT-CLJ0232-4421 |  | RXCJ0232.2-4420 | $0.284$ | $38.0772$ | $-44.3464$ | $2.49$ |  $12$, $7$ | 004234031
PSZ2 G244.37-32.15 |  |  | RXC J0528.9-3927 | $0.284$ | $82.2211$ | $-39.4714$ | $2.13$ |  $7$, $4$ | 004234081
PSZ2 G106.87-83.23 |  |  | RXC J0043.4 -2037, A2813 | $0.292$ | $10.8519$ | $-20.6229$ | $1.54$ |  $11$, $5$ | 004234021
PSZ2 G262.27-35.38 | SPT-CL J0516-5430 | ACT-CL J0516-5430 | ACO S520 | 0.295 | 79.1536 | $-54.5089$ | 2.56 | 3, 1 | 0042340701
PSZ2 G266.04-21.25 | SPT-CLJ0658-5556 | ACT-CL J0658-5557 | 1ES0657-558 | $0.296$ | $104.6277$ | $-55.9434$ | $4.17$ |  $21$, $14$ | 011298021
PSZ2 G195.60+44.06 |  |  | A781 | $0.298$ | $140.1018$ | $30.5028$ | $1.94$ |  $57$, $47$ | 040117011
PSZ2 G125.71+53.86 |  |  | A1576 | $0.302$ | $189.2441$ | $63.1871$ | $1.68$ |  $6$, $1$ | 040225011
PSZ2 G008.94-81.22 |  |  | A2744 | $0.307$ | $3.5775$ | $-30.3863$ | $1.60$ |  $14$, $10$ | 004234011
PSZ2 G278.58+39.16 |  |  | A1300 | $0.308$ | $172.9775$ | $-19.9285$ | $4.50$ |  $11$, $9$ | 004234101
PSZ1 G103.58+24.78 |  |  |  | 0.334 | 286.1007 | 72.4622 | 7.48 | 28, 15 | 0693660501
|  |  |  |  |  |  |  |  | 0693663101
PSZ2 G349.46-59.95 | SPT-CLJ2248-4431 |  | AS 1063 | $0.347$ | $342.1824$ | $-44.5305$ | $1.84$ |  $25$, $15$ | 050463011
PSZ2 G083.29-31.03 |  |  | RXCJ2228+2037 | $0.412$ | $337.1405$ | $20.6204$ | $4.26$ |  $24$, $15$ | 014789011
PSZ2 G284.41+52.45 |  |  | MACSJ1206.2-0848 | $0.441$ | $181.5521$ | $-8.8002$ | $4.35$ |  $30$, $21$ | 050243041
PSZ2 G056.93-55.08 |  |  | MACSJ2243.3-0935 | $0.444$ | $340.8387$ | $-9.5947$ | $3.11$ |  $103$, $78$ | 050349021
PSZ2 G254.08-58.45 | SPT-CL J0304-4401 |  | SMACS J0304.3-4402 | 0.458 | 46.0705 | $-44.0257$ | 1.29 | 17, 12 | 0700182201
PSZ2 G265.10-59.50 | SPT-CLJ0243-4833 |  | RXCJ0243.6-4834 | $0.500$ | $40.9129$ | $-48.5611$ | $2.15$ | $15,6$ | 0672090501
|  |  |  |  |  |  |  |  | 0723780801
PSZ2 G044.77-51.30 |  |  | MACSJ2214.9-1359 | $0.503$ | $333.7383$ | $-14.0045$ | $2.88$ | $17,7$ | 0693661901
PSZ2 G211.21+38.66 |  |  | MACSJ0911.2+1746 | $0.505$ | $137.7970$ | $17.7760$ | $3.28$ | $33,25$ | 0693662501
PSZ2 G004.45-19.55 |  |  |  | $0.516$ | $289.2692$ | $-33.5228$ | $5.90$ | $14,5$ | 0656201001
PSZ2 G110.28-87.48 |  |  |  | $0.520$ | $12.2939$ | $-24.6792$ | $1.50$ | $26,4$ | 0693662101
|  |  |  |  |  |  |  |  | 0723780201
PSZ2 G212.44+63.19 |  |  | RMJ105252.4+241530.0 | $0.529$ | $163.2159$ | $24.2584$ | $1.89$ | $34,20$ | 0693660701
|  |  |  |  |  |  |  |  | 0723780701

Table 4: Continued.

Planck name | SPT name | ACT name | Alt. name | z | RA | Dec. | $N_{\mathrm{H}}$ | Exp. | Obs. Id
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  | MOS1-2, PN |
|  |  |  |  | [J2000] | [J2000] | [$10^{20}$cm-3] | [ks] |
PSZ2 G201.50-27.31 |  |  | MACSJ0454.1-0300 | $0.538$ | $73.5471$ | $-3.0162$ | $3.92$ | $22,17$ | 0205670101
PSZ2 G094.56+51.03 |  |  | WHL J227.050+57.90 | $0.539$ | $227.0821$ | $57.9164$ | $1.50$ | $26,20$ | 0693660101
|  |  |  |  |  |  |  |  | 0723780501
PSZ2 G228.16+75.20 |  |  | MACSJ1149.5+2223 | $0.544$ | $177.3976$ | $22.4011$ | $1.92$ | $13,2$ | 0693661701
PSZ2 G111.61-45.71 |  |  | CL0016+16 | $0.546$ | $4.6399$ | $16.4362$ | $3.99$ | $33,24$ | 0111000101
|  |  |  |  |  |  |  |  | 0111000201
PSZ2 G180.25+21.03 |  |  | MACSJ0717.5+3745 | $0.546$ | $109.3800$ | $37.7587$ | $6.63$ | $156,116$ | 0672420101
|  |  |  |  |  |  |  |  | 0672420201
|  |  |  |  |  |  |  |  | 0672420301
PSZ2 G183.90+42.99 |  |  | WHL J137.713+38.83 | $0.559$ | $137.7032$ | $38.8357$ | $1.63$ | $14,8$ | 0723780101
PSZ2 G155.27-68.42 |  |  | WHL J24.3324-8.477 | $0.567$ | $24.3536$ | $-8.4557$ | $3.52$ | $28,18$ | 0693662801
|  |  |  |  |  |  |  |  | 0700180201
PSZ2 G046.13+30.72 |  |  | WHL J171705.5+240424 | $0.569$ | $259.2742$ | $24.0737$ | $5.18$ | $26,2$ | 0693661401
PSZ2 G239.93-39.97 |  |  |  | $0.580$ | $71.6966$ | $-37.0625$ | $1.44$ | $34,23$ | 0679181001
|  |  |  |  |  |  |  |  | 0693661201
PSZ2 G254.64-45.20 | SPT-CLJ0417-4748 |  |  | $0.581$ | $64.3464$ | $-47.8134$ | $1.34$ | $20,9$ | 0700182401
PSZ2 G144.83+25.11 |  |  | MACSJ20647.7+7015 | $0.584$ | $101.9590$ | $70.2481$ | $5.40$ | $72,46$ | 0551850401
|  |  |  |  |  |  |  |  | 0551851301
PSZ2 G045.32-38.46 |  |  | MACSJ2129.4-0741 | $0.589$ | $322.3591$ | $-7.6913$ | $4.32$ | $9,3$ | 0700182001
PSZ2 G070.89+49.26 |  |  | WHL J155625.2+444042 | $0.602$ | $239.1098$ | $44.6772$ | $1.23$ | $47,25$ | 0693661301
PSZ2 G045.87+57.70 |  |  | WHL J151820.6+292740 | $0.609$ | $229.5866$ | $29.4603$ | $2.12$ | $23,15$ | 0693661101
PSZ2 G073.31+67.52 |  |  | WHL J215.168+39.91 | $0.609$ | $215.1709$ | $39.9187$ | $0.82$ | $26,19$ | 0693661001
PSZ2 G099.86+58.45 |  |  | WHL J213.697+54.78 | $0.615$ | $213.6952$ | $54.7840$ | $1.50$ | $21,9$ | 0693660601
|  |  |  |  |  |  |  |  | 0693662701
|  |  |  |  |  |  |  |  | 0723780301
PSZ2 G193.31-46.13 |  |  |  | $0.634$ | $53.9644$ | $-6.9758$ | $4.15$ | $67,48$ | 0658200401
|  |  |  |  |  |  |  |  | 0693661501
PLCK G147.32-16.59 |  |  |  | $0.645$ | $44.1056$ | $40.2885$ | $8.29$ | $59,39$ | 0679181301
|  |  |  |  |  |  |  |  | 0693661601
PLCK G260.7-26.3 | SPT-CLJ0616-5227 | ACT-CL J0616-5227 |  | $0.680$ | $94.1429$ | $-52.4518$ | $4.25$ | $35,20$ | 0693662301
PSZ2 G097.52+51.70 |  |  |  | 0.700 | 223.8379 | 58.8718 | 1.06 | 24, 17 | 0783881301
PSZ2 G219.89-34.39 |  |  |  | $0.700$ | $73.6894$ | $-20.2851$ | $3.25$ | $51,34$ | 0679180501
|  |  |  |  |  |  |  |  | 0693660301
PSZ1 G080.66-57.87 |  | ACT-CL J2327.4-0204 | RCS2 J2327-0204 | 0.699 | 351.8654 | $-2.0771$ | 4.62 | 25, 75, 71 | 7355
|  |  |  |  |  |  |  |  | 14025
|  |  |  |  |  |  |  |  | 14361
PSZ2 G208.61-74.39 |  |  |  | $0.711$ | $30.0695$ | $-24.9132$ | $1.37$ | $47,28$ | 0693662901
|  |  |  |  |  |  |  |  | 0723780601
PSZ1 G226.65+28.43 |  |  |  | 0.724 | 134.0858 | 1.7803 | 3.38 | 44, 27 | 0783881001
PSZ2 G084.10+58.72 |  |  |  | 0.731 | 222.2535 | 48.5569 | 2.29 | 72, 58 | 0783880901

Table 5: Continued.

Planck name | SPT name | ACT name | Alt. name | z | RA | Dec. | $N_{\mathrm{H}}$ | Exp. | Obs. Id
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  | MOS1-2, PN |
|  |  |  |  | [J2000] | [J2000] | [$10^{20}$cm-3] | [ks] |
PSZ2 G087.39+50.92 |  |  |  | 0.748 | 231.6379 | 54.1524 | 1.21 | 20, 11 | 0783881201
PSZ2 G088.98+55.07 |  |  |  | 0.754 | 224.7448 | 52.8167 | 1.83 | 44, 30 | 0783880801
PSZ2 G086.93+53.18 |  |  |  |  |  |  |  |  |
|  |  |  | 0.771 | 228.5022 | 52.8035 | 1.62 | 57, 37 | 0783880701
PLCK G079.95+46.96 |  |  |  | 0.790 | 240.5461 | 51.0615 | 1.56 | 81, 51 | 0783880601
PSZ2 G352.05-24.01 |  |  |  | $0.798$ | $290.2490$ | $-45.8500$ | $6.03$ | $52,28$ | 0679180201
|  |  |  |  |  |  |  |  | 0693660401
PLCK G227.99+38.11 |  |  |  | 0.810 | 143.0891 | 5.6899 | 3.40 | 26, 10 | 0783880501
PSZ2 G091.83+26.11 |  |  |  | 0.822 | 277.7925 | 62.2429 | 4.30 | 23, 13 | 0762801001
PSZ2G 208.57-44.31 |  |  |  | 0.830 | 60.6477 | $-15.6783$ | 2.13 | 60, 47 | 0783880301
PSZ2 G071.82-56.55 |  |  |  | 0.870 | 347.3914 | $-4.1707$ | 3.59 | 46, 27 | 0783880201
PSZ2 G160.83+81.66 |  |  | WARPS J 1226.9+3332 | 0.892 | 186.7420 | 33.5463 | 1.83 | 63, 47 | 0200340101
PSZ1 G254.58-32.16 | SPT-CL J0535-4801 |  |  | 0.900 | 83.9566 | -48.0247 | 3.11 | 56, 30 | 0783880101
| SPT-CLJ2146-4633 |  |  | $0.933$ | $326.6447$ | $-46.5475$ | $1.64$ | $153,102$ | 0744400501
|  |  |  |  |  |  |  |  | 0744401301
PSZ2 G266.54-27.31 | SPT-CLJ0615-5746 |  |  | $0.972$ | $93.9660$ | $-57.7796$ | $4.32$ | $12,3$ | 0658200101
| SPT-CLJ2341-5119 |  |  | $1.003$ | $355.3010$ | $-51.3286$ | $1.21$ | $91,45$ | 0744400401
|  |  |  |  |  |  |  |  | 0763670201
| SPT-CLJ0546-5345 | ACT-CL J0546-5345 |  | $1.066$ | $86.6551$ | $-53.7596$ | $6.79$ | $127,113$ | 0744400201
|  |  |  |  |  |  |  |  | 0744400301
| SPT-CLJ2106-5844 |  |  | $1.132$ | $316.5221$ | $-58.7421$ | $4.33$ | $26,16$ | 0763670301

Table 6: Continued.

##  Appendix B Sample selection bias and covariance tests

We used similar simulations as those produced for the Andrade-Santos 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 $Y_{\rm SZ}$ value for each object was drawn from the $Y_{\rm SZ}-M$
relation of Arnaud et al. (2010), with a bias between the X-ray 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 $Y_{\rm
SZ}-Y_{\rm X}$ 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 $Y_{\rm X}-M$ and $Y_{\rm SZ}-M$ relations are the same.

We draw the $Y_{\rm SZ}$, $Y_{\rm X}$ and $L_{\rm X}$ quantities associated to
each simulated cluster following a correlated Gaussian distribution with
covariance matrix. If $\mathcal{Q}_{M}$ is the latent value of $\mathcal{Q}$
at a given mass, obtained if there were no scatter, one can write

$\displaystyle Y_{\rm M}=A_{\rm Y}E(z)^{2/3}M^{B_{\rm Y}}$ (25) $\displaystyle
L_{\rm M}=A_{\rm L}E(z)^{2}M^{B_{\rm L}}$ (26) $\displaystyle P(D_{\rm
A}^{2}Y_{\rm SZ},Y_{\rm X},L_{\rm Xc}|Y_{\rm M},Y_{\rm M},L_{\rm
M})=\mathcal{N}[(Y_{\rm M},Y_{\rm M},L_{\rm M}),V_{\sigma}],$ (27)

where $Y_{\rm M}$ and $L_{\rm M}$ are the latent values of $Y$ and $L$ at a
given mass (the values obtained if there were no scatter), and $D_{\rm
A}^{2}Y_{\rm SZ}$, $Y_{\rm X}$, and $L_{\rm Xc}$ are the true values. For
these simulations we assume $B_{\rm Y}=1.79$ (Arnaud et al. 2010) and $B_{\rm
L}=1.37$ (Table 4). $\mathcal{N}$ is a Gaussian log-normal correlated
distribution at fixed mass, where

$\displaystyle V_{\sigma}=\ \begin{pmatrix}\sigma^{2}_{\ln{Y_{\rm
SZ}}}&r\,\sigma_{\ln{Y_{\rm SZ}}}\sigma_{\ln{Y_{\rm X}}}&s\,\sigma_{\ln{Y_{\rm
SZ}}}\sigma_{\ln{L_{\rm Xc}}}\\\ r\,\sigma_{\ln{Y_{\rm SZ}}}\sigma_{\ln{Y_{\rm
X}}}&\sigma^{2}_{\ln{Y_{\rm X}}}&t\,\sigma_{\ln{Y_{\rm X}}}\sigma_{\ln{L_{\rm
Xc}}}\\\ s\,\sigma_{\ln{Y_{\rm SZ}}}\sigma_{\ln{L_{\rm
Xc}}}&t\,\sigma_{\ln{Y_{\rm X}}}\sigma_{\ln{L_{\rm
Xc}}}&\sigma^{2}_{\ln{L_{\rm Xc}}}\end{pmatrix}.$ (28)

For the covariance matrix $V_{\sigma}$, we assume $\sigma_{\ln{Y_{\rm
SZ}}}=0.12$ (Kay et al. 2012; Le Brun et al. 2017); $\sigma_{\ln{Y_{\rm
X}}}=0.16$ (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 first-order estimate of $\sigma_{\ln{L_{\rm
Xc}}}=0.22$ in the absence of selection effects.

Figure 13: Simulated $Y_{\rm SZ}$ reextracted from PSZ2 maps, plotted as a
function of $L_{\rm Xc}$. The quantity $\mathcal{Q}_{\rm M}$ 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
$Y_{\rm SZ},Y_{\rm X}$, and $L_{\rm Xc}$ with $M$. The blue points include
covariance of $t=0.85$ between $Y_{\rm X}$ and $L_{\rm Xc}$.

The Multi-Matched Filter extraction algorithm (Melin et al. 2006) was then
applied, to obtain SZ detections at ${\rm S/N}>6$, 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 $Y_{\rm M}-L_{\rm M}$ relation, which assumes that there
is zero intrinsic scatter in either of the observables with respect to the
mass. The scatter in the $Y$-axis seen in Fig. 13 is then entirely due to the
observational uncertainties in the SZE measurements. The resulting slope of
$0.759\pm 0.001$ implies a change in the $L_{\rm Xc}$$-M$ relation of
$\Delta\,B_{\rm L}=-0.01$. Adding intrinsic scatter of $\sigma_{\ln{Y_{\rm
SZ}}}=0.12$ (Kay et al. 2012; Le Brun et al. 2017), the slope of the $Y_{\rm
SZ}-L_{\rm M}$ relation is $0.759\pm 0.001$, again implying a negligible
change of $\Delta\,B_{\rm L}=-0.01$ on the slope of the $L_{\rm Xc}$$-M$
relation. These results imply that Malmquist bias in the $Y_{\rm SZ}$
observable are negligible, and cannot account for the significantly steeper
than self-similar slope we find for the $L_{\rm Xc}$$-M$ relation.

We next studied the robustness of the recovery of the $L_{\rm Xc}$$-M$
relation slope to intrinsic scatter in the $Y_{\rm X}$ proxy, due to $Y_{\rm
X}$ being a scattered estimates of $Y_{\rm SZ}$. An intrinsic scatter of
$\sigma_{\ln{Y_{\rm X}}}=0.16$ (Planelles et al. 2014; Le Brun et al. 2017;
Truong et al. 2018) pushes the slope of the $Y_{\rm X}-L_{\rm M}$ relation to
$0.741\pm 0.002$, in turn changing the $L_{\rm Xc}$$-M$ relation slope by
$\Delta\,B_{\rm L}=-0.04$.

We then added an intrinsic scatter of $\sigma_{\ln{L_{\rm Xc}}}=0.22$, finding
that this redresses the slope to $0.775\pm 0.005$, implying a change of
$\Delta\,B_{\rm L}=+0.02$ in the slope of the $L_{\rm Xc}$$-M$ relation.
Finally, we added a covariance of $t=0.85$ between $Y_{\rm X}$ and $L_{\rm
Xc}$ (Farahi et al. 2019). This pushes the slope to a slightly steeper value
of $0.785\pm 0.003$, changing the $L_{\rm Xc}$$-M$ relation slope by
$\Delta\,B_{\rm L}=+0.04$. The covariance significantly reduces the dispersion
in the $Y_{\rm X}-$$L_{\rm 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. 13.

In conclusion, we find that selection effects and intrinsic scatter have a
negligible effect on the slope of the $L_{\rm Xc}$$-M$ relation, and cannot
account for the significantly steeper than self-similar value that we find in
this work. The dispersion of the $L_{\rm Xc}$$-M$ relation that we derive is
significantly underestimated, most likely by a factor of $\sim 1.7$, due to
the covariance between $L_{\rm Xc}$ and the mass proxy, $Y_{\rm X}$.