11institutetext: Lorenzo Lovisari 22institutetext: INAF - Osservatorio di
Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, 40129
Bologna, Italia;
Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA; 22email<EMAIL_ADDRESS>33institutetext: Ben J. Maughan 44institutetext: H. H. Wills Physics Laboratory, University of Bristol, Tyndall Ave, Bristol BS8 1TL, UK;
44email<EMAIL_ADDRESS>
# Scaling relations of clusters and groups, and their evolution

Lorenzo Lovisari and Ben J. Maughan Corresponding author; Both authors
contributed equally to this work.

###### Abstract

X-ray observations of the hot intra-cluster medium (ICM) in galaxy groups and
clusters provide quantities such as their gas mass, X-ray luminosity, and
temperature. The analysis of the scaling relations between these observable
properties gives considerable insight into the physical processes taking place
in the ICM. Furthermore, an understanding of the scaling relations between ICM
properties and the total cluster mass is essential for cosmological studies
with clusters. For these reasons, the X-ray scaling relations of groups and
clusters have been a major focus of research over the past several decades. In
this Chapter, after presenting the expectations from the self–similar model,
based on the assumption that only gravity drives the evolution of the ICM, we
discuss how the processes of gas cooling and non–gravitational heating are
believed to be responsible for the observed deviations from the self-similar
scenario. We also describe important complications that must be considered
when measuring and interpreting the scaling relations.

##  1 Keywords

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

##  2 Introduction

Clusters of galaxies are among the most significant cosmic structures, both in
terms of their mass and scale, but also for the depth of insight they can
yield into both cosmology and astrophysics. As their name implies, they appear
in optical wavelengths as collections of galaxies numbering anywhere between a
few to a few thousand. These galaxies, however, account for just a few percent
of the mass of a galaxy cluster. The dominant baryonic component of a cluster
is a hot, tenuous and almost fully ionised gas which pervades the entire
cluster and is known as the intra-cluster medium (ICM). The ICM accounts for
$10$–$20\%$ of the mass of a cluster, and is the source of the bright diffuse
X-ray emission that is characteristic of galaxy clusters. The gravitational
potential is dominated by the dark matter which constitutes $80$–$90\%$ of the
total mass of a cluster.

Galaxy clusters span a mass range of approximately
$10^{13}$–10${}^{15}\text{M}_{\odot}$, although systems at the low mass end of
the range are often referred to as galaxy groups. There is no precise
definition of the boundary between groups and clusters, but a mass of
$10^{14}\text{M}_{\odot}$ is a reasonable working definition. The lower mass
limit of galaxy groups is not sharply defined, since an isolated massive
galaxy with a significant amount of dark matter can be regarded in some ways
as a galaxy group with just one member.

In the current bottom-up scenario for the formation of cosmic structure, tiny
density fluctuations are amplified by gravity to create the massive, dark
matter dominated structures we observe today. Thus, to a first approximation,
clusters of galaxies are simple objects whose properties are defined only by
their mass and redshift. This leads to simple power-law correlations between
different observable properties of clusters, known as scaling relations. The
study of these scaling relations has great utility in advancing our
understanding of both astrophysics and cosmology. The correlations between
observable properties of clusters and their masses provide the possibility of
using easily measured properties as proxies for cluster masses, which are the
essential quantity when using galaxy clusters to derive cosmological
constraints. Meanwhile, the deviations of real clusters from the simple
theoretical scaling relations are diagnostics of important astrophysical
processes which shape the mutual growth and evolution of the different
components of clusters. These two applications are not independent. For
example, the impact of astrophysical processes on the scatter and evolution of
scaling relations must be understood if they are to be used to provide masses
for cosmological studies; neglecting this could mask or mimic an interesting
cosmological signal in the data.

Almost as soon as clusters of galaxies were detected in X-rays, the
correlations between their X-ray properties were investigated, revealing clear
scaling relations between ICM properties (Mitchell et al. 1979)111See Sarazin
(1986) for a comprehensive review of the X-ray properties of galaxy clusters..
This early work set the scene for decades of more detailed investigations to
explore the astrophysics inside the scaling relations, powered by a succession
of ever-improving X-ray telescopes.

The overall picture that has emerged is that the scaling of cluster properties
is dominated by mass, but for lower mass systems other effects become
increasingly important, causing departures from simple scaling relations.
These include the radiative cooling of the ICM; energy injected into the ICM
from supernovae and active galactic nuclei (AGN) in the member galaxies; and
stochastic and transient perturbations of cluster properties due to mergers
with other groups and clusters. However, our understanding of these processes
is far from complete.

Alongside the improvements in observing facilities, developments in numerical
simulations of clusters in cosmological volumes have been critical for
understanding the scaling relations. With increasing computing power, the
resolution and sophistication of simulations has progressed from “N-body”
simulations which model only the gravitational interactions of dark matter
particles, to full hydrodynamical modelling of dark matter and baryons in
cosmological volumes. However, including astrophysical processes in
cosmological simulations is non-trivial because of the significantly different
scales at work: the accreting AGN ($<$1 pc), its host galaxy ($\sim$ several
kpc), the galaxy cluster ($\sim$1 Mpc), and the region that will collapse to
form the final system ($\sim$10 Mpc). Important improvements have been
achieved thanks to advancement in the treatment of physical processes acting
below the resolution scale of the simulation (usually referred to as “sub-
grid” models).

In this Chapter we will focus on the X-ray scaling relations of groups and
clusters of galaxies. We will first present the theoretical background of the
scaling of ICM properties with mass, and the astrophysical processes that are
believed to be responsible for shaping the scaling relations. We will also
describe a number of important biases and other issues that should be
considered when measuring the scaling relations. We will review the current
observational constraints on the scaling relations. The implications of these
observational results for our understanding of the astrophysical processes
responsible for shaping the scaling relations will be discussed, including
comparisons with the results from numerical simulations. Finally, we will
conclude with an outlook on how new X-ray telescopes will help push forward
the frontiers of this field.

##  3 Theoretical background

###  3.1 The X-ray emission from clusters of galaxies

The fundamental physical properties of the ICM are its temperature
($T_{\text{X}}$), density (usually expressed in terms of the number density of
free electrons, $n_{\text{e}}$, and protons, $n_{\text{p}}$), and abundance of
heavy elements ($Z$, defined more precisely below). The main cooling process
for the ICM is the emission of X-ray radiation, whose emissivity (i.e., the
energy emitted per unit time and volume at some photon energy) is equal to

$\epsilon=n_{\text{e}}n_{\text{p}}\Lambda(T_{\text{X}},Z)$ (1)

where $\Lambda$ is known as the cooling function. The form of
$\Lambda(T_{\text{X}},Z)$ depends on the details of the emission mechanisms,
which are calculated by complex emission codes. To a good approximation,
$\Lambda(T_{\text{X}},Z)$ is given by the sum of a thermal bremsstrahlung
continuum and emission lines from collisionally excited metals in the ICM. The
metals in the ICM contribute to the emission both through the obvious lines
and also the continuum component (e.g., Kaastra et al. 2008, Böhringer &
Werner 2010).

The temperature of the ICM can be measured by fitting models to the emitted
X-ray spectrum. Conventionally, X-ray spectral models parametrise the
temperature as $kT$ where $k$ is the Boltzmann constant. X-ray temperatures
are thus usually reported in units of keV, with values ranging from about
$1\text{~{}keV}$ for low mass groups up to about $15\text{~{}keV}$ for very
massive clusters.

The ICM is composed predominantly of fully ionised hydrogen and helium, with
smaller amounts of heavier elements (whose ionisation states depend on the ICM
properties). The heavy element (or metal) abundance in the ICM is usually
measured relative to Solar abundances $Z_{\odot}$. It is common to assume that
the relative abundances of elements heavier than helium is the same as that in
the Sun. The abundance measurement is then the ratio of metals to hydrogen in
the ICM compared to that ratio in the Sun. For example, a typical ICM
abundance of $Z=0.3Z_{\odot}$ means that the ICM has $30\%$ as much metals
relative to hydrogen compared to the Sun (see Mernier et al. - Chapter:
Chemical enrichment in groups and clusters of this volume, for an extensive
discussion on metal abundances in groups and clusters).

For typical ICM metal abundances, then densities of free electrons and protons
are related by $n_{\text{e}}\approx 1.2n_{\text{p}}$. The mass density of the
ICM, $\rho_{\text{gas}}$ can be derived from the number densities via the mean
atomic mass per particle, $\mu\approx 0.6$ for typical ICM composition.

For ICM temperatures above about $3\text{~{}keV}$, the bremsstrahlung
component dominates the X-ray spectrum, but at lower temperatures (i.e., lower
mass clusters), other mechanisms (primarily line emission from collisionally
excited ions) make increasingly important contributions to the X-ray emission
(e.g., Lovisari et al. 2021). This is illustrated in Figure 1, which shows the
enhanced contribution of line emission in lower temperature clusters.

Figure 1: Model X-ray spectra for clusters of different temperatures, assuming
$Z=0.3$. For comparison we show also the spectra with zero metallicity. The
elements and ionisation states responsible for some of the most prominent
emissions lines are indicated.

The X-ray luminosity of the ICM is given by integrating $\epsilon$ over the
X-ray energy band being considered, and over the volume of the cluster222The
volume is generally assumed to be spherical. However, the ICM is optically
thin so the luminosity observed in a circular aperture of projected radius $R$
contains emission from ICM at larger radii in a cylinder projected along the
line of sight. This additional contribution to the luminosity is usually
neglected because the emission is dominated by the denser ICM at smaller radii
due to the $n_{e}$ and $n_{p}$ dependence.. The X-ray emission from the ICM
and the sensitivity of most X-ray observatories both peak around $\sim$1–2
keV, so it is often convenient to measure X-ray luminosities in a “soft”
energy band, $L_{\text{X,soft}}$ (0.1–2.4 keV and 0.5–2 keV are
conventional)333We note that since clusters are generally observed at non-
negligible redshifts, a cluster’s X-ray flux must be “k-corrected” from the
observer’s frame to the desired energy band in the cluster’s rest frame when
calculating its luminosity (see, e.g., Jones et al. 1998).. Alternatively, the
bolometric X-ray luminosity, $L_{\text{X,bol}}$, is given by integrating the
emission across the whole X-ray band (the exact limits don’t matter since the
emissivity drops to zero at high and low energies, but the range 0.01–100  keV
is commonly used).

In addition to the luminosity, X-ray observations allow one to easily derive
the gas temperature and the emission measure444There is some ambiguity over
this term in the literature. The term ”emission integral” may also be used to
refer to this quantity, or sometimes to refer to the related quantity of the
integral of $n_{e}n_{p}$ along the line of sight. (EM=$\int n_{e}n_{p}dV$)
from spectral fits. The latter quantity is proportional to the normalisation
of standard ICM spectral models and provides the means to measure the ICM
density.

Given data of sufficient quality, it is possible to measure the temperature
and density structure of the ICM. For example, radial profiles
$T_{\text{X}}(r)$ and $\rho_{\text{gas}}(r)$ can be recovered assuming
spherical symmetry and accounting for projected emission along the line of
sight. It is then possible to estimate the total gravitating mass of a cluster
under the assumption that the ICM is in hydrostatic equilibrium (HE). The
hydrostatic mass contained within a radius $r$ is given by

$\displaystyle M_{\text{HE}}(<r)$ $\displaystyle=-\frac{kT_{\text{X}}r}{G\mu
m_{p}}\left[\frac{\text{d}\log\rho_{\text{gas}}}{\text{d}\log
r}+\frac{\text{d}\log T_{\text{X}}}{\text{d}\log r}\right].$ (2)

###  3.2 Self-similarity

Under some simplifying assumptions (that galaxy clusters grow from scale-free
initial conditions, form recently in a single spherical collapse, and their
energy content is only obtained from the gravitational potential in this
collapse), Kaiser (1986) showed that clusters should be self-similar objects
(see Böhringer et al. 2012, for a more recent discussion of the self-similar
model). This means that all clusters of the same mass and at the same redshift
should be identical, and that their properties should scale simply with mass
and redshift.

In this self-similar model, the collapsed region that formed the cluster is
virialised (i.e., the kinetic energy of its components have obtained an
equilibrium with the gravitational potential, obeying the virial theorem). One
can then imagine a notional spherical surface separating the virialised
material from the surroundings. The radius of this imaginary surface is known
as the virial radius, and sets the spatial scale of clusters in the self-
similar model.

The virial radius may be expressed in terms of an overdensity, $\Delta$,
relative to the critical density of the Universe at the redshift of the
cluster, $\rho_{c}(z)$, where

$\displaystyle\rho_{c}(z)=\frac{3H(z)^{2}}{8\pi G}.$ (3)

Here, $H(z)$ is the Hubble parameter, which can be expressed in terms of its
present-day value as $H(z)=E(z)H_{0}$, where

$\displaystyle
E(z)=\sqrt{\Omega_{\text{M}}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}},$
(4)

with $\Omega_{\text{M}}$ and $\Omega_{\Lambda}$ being the density parameters
associated to the non-relativistic matter and to the cosmological constant
respectively, and $\Omega_{k}=(1-\Omega_{\text{M}}-\Omega_{\Lambda})$ defines
the curvature of the Universe and is equal to zero for a flat cosmology. Since
in an expanding Universe the Hubble parameter $H$ is a function of time, then
the critical density also changes with time (i.e., the value of $\rho_{c}$
increases with redshift).

In general, we define an overdensity radius $R_{\Delta}$ such that

$\displaystyle\frac{M_{\Delta}}{\frac{4}{3}\pi
R_{\Delta}^{3}}=\Delta\rho_{c}(z),$ (5)

where $M_{\Delta}$ is the mass enclosed within a sphere of radius
$R_{\Delta}$. In a flat, matter-dominated Universe (i.e.,
$\Omega_{\text{M}}$=1 and $\Omega_{\Lambda}$=0), the virial radius corresponds
to $\Delta=18\pi^{2}\approx 178$, but in general the value and redshift-
dependence of the overdensity within the virial radius depend on the assumed
cosmology (e.g., for $\Omega_{\text{M}}$=0.3 and $\Omega_{\Lambda}$=0.7 using
Eq. 6 of Bryan & Norman 1998 one finds $\Delta_{vir}(z=0)\approx 100$). To a
good approximation, however, clusters can be considered self-similar when they
are scaled to a constant overdensity radius at all redshifts. In the case of
X-ray observations of clusters, it is common to adopt $\Delta=500$, since
$R_{\text{500}}$ is the radius to which ICM properties can be measured in
typical observations with the current instruments555When galaxy cluster
properties are used for cosmological studies (or any application in which the
cosmological model is free to vary), one must take into account the various
ways in which the cosmological model impacts the measurements. The most
obvious are the conversion of observed quantities to physical quantities
(i.e., angles to distances, and fluxes to luminosities). However, the use of
an overdensity radius to measure cluster properties also introduces a
cosmology dependence due to the cosmology-dependence of $\rho_{c}(z)$.. An
extensive description of the spherical collapse model and of its parameters
can be found in Clerc et al. - Chapter: X-ray cluster cosmology of this
volume.

The self-similar appearance of galaxy clusters is illustrated in Figure 2,
which shows X-ray images of clusters observed with XMM-Newton. The clusters
span a range in mass of $M_{\text{500}}\approx$ 3–11$\times
10^{14}\text{M}_{\odot}$, but when the images are scaled in size relative to
$R_{\text{500}}$ for each cluster, their overall appearance is very similar
(note that the differences in X-ray emission away from the centres of the
clusters are due to different levels of residual background emission).

Figure 2: Smoothed XMM-Newton images of galaxy clusters of masses
$M_{\text{500}}\approx 3\times 10^{14}\text{M}_{\odot}$ (left),
$M_{\text{500}}\approx 5\times 10^{14}\text{M}_{\odot}$ (centre), and
$M_{\text{500}}\approx 11\times 10^{14}\text{M}_{\odot}$ (right). The images
are scaled so that each is $2.4R_{\text{500}}$ on a side. This Figure is
adapted from CHEX-MATE Collaboration et al. (2021).

Under the assumptions above, the properties of galaxy clusters measured within
overdensity radii are expected to scale with mass following simple power-law
forms. Here we will focus on these scaling relations for the principal
properties of the ICM that can be measured with X-ray observations; namely its
mass ($M_{\text{gas}}$), temperature ($T_{\text{X}}$), and X-ray luminosity
($L_{\text{X}}$), along with quantities derived from combinations of those
properties. Similar relations can also be derived for other properties (such
as the total stellar mass, or the velocity dispersion of the member galaxies).

We note that the use of overdensity radii in the self-similar model means that
clusters of a given mass are denser at higher redshift, having formed in a
denser Universe (see Equation (5)). This leads to the expectation of certain
cluster properties evolving, with a dependence on $E(z)$. So, for example,
clusters of a given mass would be hotter and more luminous at higher
redshifts. This is a simple consequence of defining the mass in terms of an
overdensity, and does not describe any astrophysical evolution of galaxy
clusters; rather it is the baseline against which astrophysical evolution
should be measured.

In the following, all properties are measured within an overdensity radius,
but we drop the $\Delta$ subscript for clarity. While deriving the self-
similar behaviour for the scaling relations, we will also generalise them in
the form

$\displaystyle Y=AE(z)^{\gamma}X^{B}$ (6)

where $A$, $B$, and $\gamma$ describe the normalisation, slope, and evolution
respectively of the scaling relation between some properties $X$ and $Y$. The
values of $B$ and $\gamma$ predicted by the self-similar model for each of the
scaling relations considered are summarised in Table LABEL:table:self.

We note that $E(z)^{\gamma}$ is not a unique way to parametrise the evolution.
A common alternative is to fix $\gamma$ to the self-similar value and then
parametrise additional evolution as a power of $1+z$. Either variation only
describes evolution in the normalisation of the scaling relations, and
extensions include allowing for a redshift dependence of the slope and scatter
of the scaling relations.

#### The Mgas–M relation

For self-similar clusters, the mass of the ICM, $M_{\text{gas}}$, is a
constant fraction, $f_{\text{gas}}$, of the total mass of the cluster:

$\displaystyle M_{\text{gas}}=f_{\text{gas}}M.$ (7)

This can be generalised as

$\displaystyle
M_{\text{gas}}=A_{\text{mM}}E(z)^{\gamma_{\text{mM}}}M^{B_{\text{mM}}}$ (8)

where for self-similarity $\gamma_{\text{mM}}=0$ and $B_{\text{mM}}=1$.

#### The TX–M relation

If the ICM is isothermal with a temperature $T_{\text{X}}$, and its particles
are in virial equilibrium with the cluster potential, then the virial theorem
gives

$\displaystyle T_{\text{X}}\propto\frac{M}{R}.$ (9)

Eliminating $R$ in favour of $M$ using Equation (5), and noting that
$\rho_{c}(z)\propto E(z)^{2}$, we can write the self-similar TX–M relation:

$\displaystyle T_{\text{X}}\propto E(z)^{2/3}M^{2/3}.$ (10)

This generalises to

$\displaystyle
T_{\text{X}}=A_{\text{TM}}E(z)^{\gamma_{\text{TM}}}M^{B_{\text{TM}}}$ (11)

with $\gamma_{\text{TM}}=2/3$ and $B_{\text{TM}}=2/3$ for self-similarity.

#### The LX–M relation

In order to derive the self-similar form of the scaling with mass of the X-ray
luminosity of the ICM ($L_{\text{X}}$), we will make the approximation that
the X-ray emission is due solely to the bremsstrahlung mechanism. Neglecting
the weak temperature dependence of the Gaunt factor in the bremsstrahlung
emissivity (Böhringer & Werner 2010), the bolometric bremsstrahlung luminosity
of the ICM is given by

$\displaystyle L_{\text{X,bol}}\propto\int\epsilon
dV\propto\int\rho_{\text{gas}}^{2}T_{\text{X}}^{1/2}dV,$ (12)

where $\rho_{\text{gas}}$ is the density of the ICM, and $V$ is the volume of
the cluster. The last scaling is strictly valid only for massive systems for
which the cooling function mainly depends only on $T_{\text{X}}$ (i.e.,
$\Lambda(T_{\text{X}},Z)\sim T_{\text{X}}^{1/2}$).

The integral in Equation (12) depends on the temperature and density
distribution of the ICM. However, for self-similar clusters the ICM
distributions are identical, and assuming isothermality, $T_{\text{X}}$ is a
constant. This leads to

$\displaystyle
L_{\text{X,bol}}\propto\rho_{\text{gas}}^{2}T_{\text{X}}^{1/2}R^{3}.$ (13)

For self-similar clusters, $\rho_{\text{gas}}\propto M/R^{3}$, and we can use
Equation (5) again to eliminate $R$ in favour of $M$ and $E(z)$ to give

$\displaystyle L_{\text{X,bol}}\propto E(z)^{2}MT_{\text{X}}^{1/2}.$ (14)

Finally, we can use Equation (10) to eliminate $T_{\text{X}}$ giving the self-
similar bolometric LX–M relation

$\displaystyle L_{\text{X,bol}}\propto E(z)^{7/3}M^{4/3}.$ (15)

The LX–M relation may then be generalised as

$\displaystyle
L_{\text{X}}=A_{\text{LM}}E(z)^{\gamma_{\text{LM}}}M^{B_{\text{LM}}}$ (16)

with $\gamma_{\text{LM}}=7/3$ and $B_{\text{LM}}=4/3$ for self-similar
behaviour and bolometric luminosity.

For soft-band luminosities, the temperature dependence of the bremsstrahlung
emissivity is very small, and can be neglected in Equation (12). This leads to
the self-similar soft band LX–M relation

$\displaystyle L_{\text{X,soft}}\propto E(z)^{2}M$ (17)

i.e., $\gamma_{\text{LM}}=2$ and $B_{\text{LM}}=1$ for soft-band luminosities.

In the preceding derivation, we assumed that the X-ray emission from the ICM
was entirely bremsstrahlung. However, as we saw in §3.1, line emission makes
an increasingly important contribution to the emissivity at lower
temperatures. Taking this into account, the temperature dependence of the
bolometric X-ray luminosity changes from $T_{\text{X}}^{1/2}$ in the
bremsstrahlung-dominated regime (as in Equation (12)) towards approximately
$T_{\text{X}}^{-1/2}$ for $T_{\text{X}}\lesssim 2\text{~{}keV}$ (see Lovisari
et al. 2021 for a more detailed discussion). For the lowest temperature
clusters, the self-similar slope and evolution of the bolometric LX–M relation
then tend towards $B_{\text{LM}}=2/3$ and $\gamma_{\text{LM}}=5/3$.

#### The YX–M relation

Since $T_{\text{X}}$ is proportional to the kinetic energy of an ICM particle,
and $M_{\text{gas}}$ is proportional to the number of particles in the ICM,
their product $Y_{\text{X}}\equiv T_{\text{X}}M_{\text{gas}}$ is proportional
to the total thermal energy of the ICM (Kravtsov et al. 2006). In the self-
similar model, this energy derives from the gravitational collapse of the ICM
so should correlate directly with the total cluster mass.

The self-similar form of the YX–M relation is easy to derive by combining
Equations (7) and (10) to give

$\displaystyle Y_{\text{X}}\propto E(z)^{2/3}M^{5/3}$ (18)

or

$\displaystyle
Y_{\text{X}}=A_{\text{YM}}E(z)^{\gamma_{\text{YM}}}M^{B_{\text{YM}}}$ (19)

with $\gamma_{\text{YM}}=2/3$ and $B_{\text{YM}}=5/3$ for self-similarity.

#### The LX–TX relation

Scaling relations between different combinations of ICM properties can be
derived by combining the relations above to eliminate mass. This is often
desirable since it allows the scaling behaviour of clusters to be studied
using more directly observable quantities. The most important of these is the
scaling relation between $L_{\text{X}}$ and $T_{\text{X}}$.

Combining Equations (10) and (15), we can derive the self-similar form of the
bolometric LX–TX relation:

$\displaystyle L_{\text{X}}\propto E(z)T_{\text{X}}^{2}.$ (20)

Once more we can generalise this as

$\displaystyle
L_{\text{X}}=A_{\text{LT}}E(z)^{\gamma_{\text{LT}}}T_{\text{X}}^{B_{\text{LT}}}$
(21)

with $\gamma_{\text{LT}}=1$ and $B_{\text{LT}}=2$ for self-similar scaling.

As is the case with the LX–M relation, the expected self-similar slope and
evolution of the LX–TX relation will depend on the energy band in which the
luminosity is measured. Furthermore, the increasing contribution of line
emission to the ICM luminosity in lower temperature clusters will also alter
the self-similar slope and evolution for lower temperature clusters. These
effects can be easily incorporated into the LX–TX relation by following the
approach adopted for the LX–M relation.

Table 1: Self-similar values for slope $B$ and redshift evolution $\gamma$ of the scaling relations. As discussed in the text, the slope values for the luminosity scaling relations are appropriate for hotter systems where the X-ray emission is dominated by bremsstrahlung. Relation (Y, X) |  $B_{\text{self}}$ |  $\gamma_{\text{self}}$
---|---|---
$M_{\text{gas}}$–$M_{\text{500}}$ |  1 |  0
$T_{\text{X}}$–$M_{\text{500}}$ |  2/3 |  2/3
$L_{\text{X,bol}}$–$M_{\text{500}}$ |  4/3 |  7/3
$L_{\text{X,soft}}$–$M_{\text{500}}$ |  1 |  2
$L_{\text{X,bol}}$–$T_{\text{X}}$ |  2 |  1
$L_{\text{X,soft}}$–$T_{\text{X}}$ |  3/2 |  1
$Y_{\text{X}}$–$M_{\text{500}}$ |  5/3 |  2/3
$K$–$T_{\text{X}}$ |  1 |  -4/3

#### The entropy of the ICM

The entropy of the ICM is a useful quantity with which to examine the self-
similarity (or lack thereof) of clusters (see Voit 2005a, for a more detailed
introduction to the topic). It is conventional to define entropy of the ICM as

$\displaystyle K=T_{\text{X}}n_{\text{e}}^{-2/3}$ (22)

which is proportional to the logarithm of the traditional thermodynamic
entropy. This quantity can be quite easily measured with X-ray observations.
For self-similar clusters the density is proportional to $E(z)^{2}$ and
therefore is straightforward to derive the K–$T_{\text{X}}$ relation:

$\displaystyle K\propto E(z)^{-4/3}T_{\text{X}}.$ (23)

Once more we can generalise this as

$\displaystyle
K=A_{\text{KT}}E(z)^{\gamma_{\text{KT}}}T_{\text{X}}^{B_{\text{KT}}}$ (24)

with $\gamma_{\text{KT}}=-4/3$ and $B_{\text{KT}}=1$ for self-similar scaling.

Entropy is not changed by adiabatic processes, so has simple behaviour in the
self-similar model, and tracks processes that cause departures from self-
similarity. Low entropy gas will sink in a gravitational potential, while high
entropy gas rises buoyantly. The entropy of the ICM in an undisturbed cluster
will thus simply increase monotonically with radius.

For self-similar clusters, in which the gas that collapses to form the ICM is
cold, the ICM gains entropy at the shock that is formed where the accreting
gas meets the already virialised ICM. The strength of the shock depends on the
mass of the cluster, so layers of ICM with increasing entropy are simply added
as the cluster grows. This leads to a simple radial power law of the form
$K\propto r^{1.1}$ for smooth, spherical accretion (see, e.g., Tozzi & Norman
2001).

If the infalling gas has been “pre-heated” (i.e., has some thermal energy and
hence non-zero entropy prior to accretion), then the accretion shock is weaker
(since the sound speed in the inflowing gas is higher), and so less entropy is
added by the shock. The net effect is that the entropy of the shocked gas is
the same as it would have been without pre-heating, but there is now a minimum
entropy level (the amount provided by the pre-heating), often referred as an
“entropy floor”. Thus, the entropy profile of the ICM will flatten towards the
minimum level in the central parts of the cluster instead of continuing a
power-law form towards $r=0$.

Any energy input in the ICM (either pre-heating or once the gas has accreted)
will raise the entropy of the affected gas which will then change its location
in the cluster. Pre-heated gas will only be able to sink as far into the
cluster as its entropy permits. Similarly, gas whose entropy is increased
within the cluster will rise through convection until it reaches a radius with
a matching entropy. If the entropy were raised enough, then the gas would be
unable to sink down to (or would rise beyond) the inner regions of clusters
where the ICM is bright enough to be observed in X-rays. This would impact ICM
scaling relations by reducing the mass of ICM observed within a cluster, and
hence also its X-ray luminosity.

The effects of in-situ heating or preheating should also be observable in
radial profiles of thermodynamic properties of the ICM, such as entropy.
However, these can be difficult to measure unambiguously due to effects such
as the clumping of ICM biasing measurements of its density (see Kay et al. -
Chapter: Thermodynamical profiles of clusters and groups, and their evolution,
of this volume).

###  3.3 Heating and cooling the ICM

The self-similar relations derived in the previous sections assume that
gravitational heating dominates in clusters. The basic idea is that due to the
deep gravitational potential wells the gas falling into a cluster (either at
the same time as the dark matter, or subsequently) will cause it to move very
rapidly, colliding with other gas, and being shocked. However, in reality
things are much more complicated and there are a number of heating and cooling
processes at work in galaxy clusters. We will briefly introduce the relevant
processes here, and return to them in more detail when interpreting the
observations of scaling relations in §6.

In contrast to the self-similar model, real clusters do not form in a single
spherical collapse. Instead they form hierarchically through episodic mergers
of smaller mass structures to build larger systems. Clusters grow continuously
through the accretion of smaller structures punctuated by occasional major
mergers with other objects of a similar mass. The shocks produced by major
mergers dissipate kinetic energies of 1063–1064 ergs, and while the ICM will
ultimately approach virial equilibrium with the new gravitational potential
(although it may take a few Gyr to reach equilibrium again; e.g., Nelson et
al. 2012), its temperature and luminosity, and density structure will all be
perturbed significantly during the course of the merger. This would produce
offsets from the mean scaling relations that may change significantly at
different points during the merger.

The emission of X-rays is a strong cooling mechanism for the ICM. The
bolometric emissivity due to bremsstrahlung is proportional to
$n_{\text{e}}^{2}T_{\text{X}}^{1/2}$, and in the dense central regions of
clusters, the cooling time of the ICM may be short compared to the Hubble
time. At high temperatures (where thermal bremsstrahlung dominates), the
cooling time scales as

$t_{cool}\propto T_{\text{X}}^{1/2}n_{\text{e}}^{-1}.$ (25)

At constant pressure the rise in density as the temperature drops means that
cooling accelerates as the gas cools. This leads to the formation of cool
cores: very bright, sharply peaked regions of X-ray emission, associated with
cooler gas in the central $\lesssim 100$ kpc of a cluster. Conversely,
clusters with no central drop of temperature and flatter surface brightness
profile (i.e., with high central entropies) are called non-cool cores.

The X-ray luminosity of a cool core can account for more than $50\%$ of the
total X-ray emission from a cluster, but not all clusters possesses cool cores
(likely due to them being disrupted by significant mergers). This gives rise
to a very significant scatter in X-ray luminosity, which can be significantly
reduced if the core regions (conventionally within a radius of
$0.15R_{\text{500}}$) are excluded. This is not always possible, for example
if the observation lacks the angular resolution to resolve the core, or the
data quality does not permit the exclusion of a significant fraction of the
photons.

In principle this cooling is a runaway process, with gas condensing out of the
X-ray emitting regime in the cores of clusters, and being replaced by a steady
flow of gas cooling from larger radii. However, high resolution observations
of cluster cores have not found the expected amounts of cool gas (e.g.,
Peterson et al. 2001). It is now clear that some form of energy input
regulates the cooling, allowing the formation of stable cool cores.

Active galactic nuclei (AGNs) at the center of galaxy clusters have emerged as
heating agents powerful enough to prevent further cooling. In support of this
scenario, there are many observations of galaxy clusters with striking
cavities in their X-ray emission, which have been interpreted as bubbles of
relativistic plasma inflated by radio jets (see Eckert et al. 2021 and
references therein). Estimates of the energetics of these bubbles show that
there is sufficient power output from the AGN to counterbalance the radiative
cooling666In principle, thermal conduction can transport energy down the
temperature gradients into cool cores (e.g., Narayan & Medvedev 2001), but it
is likely to only increase the central radiative cooling time by a factor of a
few, failing to prevent runaway cooling (e.g., Pope et al. 2005)..

The effects of AGN feedback are expected to be two-fold. In addition to the
ongoing energy input through shocks and bubbles that prevents the runaway
cooling of the ICM in cluster cores, AGN also input significant energy to the
ICM via radiation at high redshift (Fabian, 2012). This “radiative” mode of
AGN feedback is thought to have raised the entropy and hence reduced the gas
content of low-mass clusters (whose shallow gravitational potential cannot
retain the heated gas). This would then be apparent in departures from self-
similar scaling of ICM properties (see §6).

Feedback from supernovae and galaxy winds can operate in a similar way to AGN
feedback (e.g., Menci & Cavaliere 2000), although these may not be energetic
enough to have a significant effect above the scales of low mass groups.

The processes described above (i.e., cooling, AGN and SNe feedback, merging)
are often referred to as “non-gravitational” heating processes, and play an
important role in determining scatter, shape, and evolution of the scaling
relations. Therefore, they can provide information on the physical mechanisms
causing the departures from the self-similar predictions.

##  4 Analysis methods and considerations

Slopes, normalizations, and scatters of cluster scaling relations can vary
notably as a result of different selection functions, observational biases,
observational uncertainties, systematic errors from heterogeneous data
reduction and analysis methods, band-pass sensitivities, and other effects.
Removing all biases and systematic differences is very challenging, but the
correct astrophysical interpretation of the scaling relations, and their
reliable use for cosmological studies, requires that these systematics are
accounted for. In the following we provide a brief summary of the main issues
that should be considered when modelling the cluster scaling relations.

###  4.1 Observational biases

The starting point for any analysis is the definition of the sample to be
used. In astrophysical scenarios, samples are rarely perfectly representative
of the parent population, leading to selection biases. These must be modelled
in order to recover the properties of the true population.

As a simplistic example, we will consider the measurement of the LX–M relation
for a sample defined by selecting clusters in an X-ray survey that have
measured luminosities above some threshold (real samples will have more
complicated selection functions, primarily dependent on observed quantities
like X-ray flux, as discussed below). If there is some degree of scatter in
the luminosity of clusters of the same mass (due to intrinsic scatter777The
intrinsic scatter is the dispersion around the ideal straight line
distribution which would manifest even without measurement error in x and y.
In other words, the intrinsic scatter is the difference between the measured
variance around the best-fit and the variance due to measurement errors. or
measurement noise) then for clusters close to the luminosity threshold, those
with below-average luminosities are more likely to be excluded from the
sample. The result is shown in Fig. 4, where the distribution of observed
clusters (blue points) is obviously not following the underlying relation
indicated by the black line. If one does not account for this effect when
fitting the observed data point, the fitted relation (blue line) will be
flatter.

This type of selection bias will be present in many areas of statistical
analysis, but there are some additional complications that are often important
in astrophysical scenarios.

The most famous is probably the Malmquist bias (Malmquist 1922), an effect
caused by the fact that brighter sources are detectable at greater distances.
This means that clusters with luminosities that are above average for their
mass are detectable over a larger volume of the Universe than those with
luminosities below average for their mass. The brighter clusters will thus be
over-represented in a sample selected by X-ray flux. This is illustrated in
Figure 3, which shows how clusters with a higher luminosity are more likely to
be found in a flux-limited sample.

Figure 3: Illustration showing the effect of the Malmquist bias on a flux-
limited sample. A sample of clusters was simulated by drawing masses and
redshifts from a model mass function for a hypothetical area of sky, and
luminosities were assigned to each cluster assuming a simple scaling relation
with mass. The data points show the clusters plotted on the luminosity-
redshift plane. The full sample is dominated by distant, low luminosity (low
mass) clusters. The blue circles denote the clusters that would be included in
an example flux-limited survey (with the flux limit illustrated by the red
line).

This picture is further complicated by the presence of the Eddington bias
(Eddington 1913, 1940), which arises when a sample is defined by a threshold
in some property which has a noisy correlation with the property of interest.
Suppose a sample of clusters is selected on the basis of their luminosity or
flux. If there is scatter in luminosity at a given mass (due to intrinsic
scatter or measurement noise), there will be some clusters whose average
luminosity for their mass is below the selection threshold, but their scatter
in luminosity carries them above the threshold. Similarly there will be
clusters whose average luminosity for their mass is above the threshold but
are lost from the sample as their scatter in luminosity carries them below the
threshold. This biases the sample towards clusters that have above-average
luminosities. Furthermore, the number of clusters as a function of their mass
(the mass function) decreases rapidly with increasing mass. This means that
there are more low mass clusters available to be scattered into the sample
than high mass clusters to be scattered out of the sample, increasing the
magnitude of the bias. This is illustrated in Figure 4.

Figure 4: Illustration showing the effect of Eddington bias on scaling
relation data. An example LX–M relation was simulated by drawing masses from a
model mass function (resulting in the much larger numbers of points with lower
masses in the plot), and then assigning a luminosity to each point assuming a
reference LX–M relation with $50\%$ intrinsic scatter. This input scaling
relation is shown by the solid black line. A simple luminosity selection was
applied to produce a mock observed sample, indicated by the grey dashed line.
The blue circles show the clusters that were included in the sample and the
grey crosses denote the clusters that were excluded. The solid blue line is a
simple fit to the mock observed sample, and is clearly strongly biased
relative to the relation from which the sample was drawn.

The terms Malquist and Eddington bias are sometimes used interchangeably and
conflated in the literature, and indeed their effects are similar and will
both be present in a sample selected by flux.

These selection biases impact most directly on the property used to define the
sample (luminosity in the above examples). However, any other cluster
properties that have covariance with the property used for selection will also
be impacted. For example, we expect from Equation (12) that clusters with
above-average luminosity relative to their mass will also tend to have above-
average $M_{\text{gas}}$. A luminosity-based selection would thus confer some
degree of bias on scaling relations involving $M_{\text{gas}}$.

On top of the Malmquist and Eddington biases, there is a morphological bias.
As we will see in Section 5, relaxed and disturbed clusters populate different
regions of the residual space with respect to the best-fit of the luminosity-
mass and luminosity-temperature relations, with the relaxed objects on average
above the relations and disturbed systems on average below (e.g., Pratt et al.
2009, Lovisari et al. 2020). This offset is probably associated with the
strength of cool-cores that boost the X-ray luminosity. Mergers also
contribute to the scatter because the total masses can easily be incorrectly
estimated when the clusters are not in HE as happens during cluster mergers,
which can change the X-ray luminosities and temperatures of clusters, both in
a transient and in a long term manner. Thus, if the sample definition depends
directly or indirectly on the morphology of clusters, this could lead to
observed relations that are different in slope, normalization, and scatter for
samples with different selection criteria. For instance, X-ray selected
samples are known to be significantly biased towards cool-core clusters with a
larger effect for low-mass systems (Eckert et al. 2011).

Corrections of the above-described biases can be applied if the selection
function is known. This is clearly not the case for samples constructed from
public archives which are usually over-represented by clusters with peculiar
morphology or special properties (e.g., strong cool core clusters or major
mergers) than more “ordinary” clusters. This is known as archival bias.

The magnitude of these biases differs from sample to sample, and depends on
the amount of true scatter (i.e., likely greater than that observed) in the
relations, with larger values requiring a larger correction. Moreover, it also
depends on the slope of the mass function around the detection threshold of
the sample. Because of the steepening of the mass function at high masses,
surveys with higher mass limits will suffer a larger bias. The situation can
be further complicated by a non-uniform magnitude of the scatter. In fact, the
intrinsic scatter of galaxy groups may be higher than that of clusters due to
the larger impact of the non-gravitational processes on their global X-ray
properties (e.g., Lovisari et al. 2021 and references therein).

We will mention a final type of bias that is independent from the sample
definition, and is often underestimated, the so-called confirmation bias. This
is a type of cognitive bias that involves favoring information that confirms
previously existing beliefs. Confirmation biases impact how information is
collected, and how the results are interpreted. In fact, researchers may tend
to select information that supports their original views, ignoring
inconsistent results, or interpret uncertain evidence as supporting their
original ideas. An example, when studying scaling relations, is to perform a
new (more accurate) analysis on systems that fall far away from the expected
predictions under the (often wrong) assumption that something went wrong in
the analysis. This could, for example, artificially reduce the scatter and
change the slope of the relation. Confirmation bias can be avoided through the
use of blinding in analyses, so that researchers do not view the final results
of the analysis until the methodology is tested and frozen. This approach is
starting to be adopted in astrophysics, but can be very difficult to implement
in practice.

###  4.2 Selection effects and selection functions

In order to account for the selection biases discussed above, and derive the
true scaling relations of the underlying cluster population, as opposed to the
merely observational trends in the studied sample, one needs to understand the
sample definition. This requires the determination of the survey selection
function which describes the probability of detecting an object given a set of
cluster properties (e.g., redshift, luminosity, total mass, dynamical state),
survey and instrumental conditions (e.g., sensitivity, spatial and/or spectral
resolutions), and detection algorithm used (e.g., sliding cell or wavelet
methods in X-ray imaging surveys).

The derivation of a selection function for X-ray galaxy clusters can be split
into two parts: the detection of X-ray sources and their classification as
clusters. X-ray surveys are dominated by unresolved point sources (primarily
AGNs), which outnumber galaxy clusters by of order $100:1$. Sources are
primarily classified as clusters on the basis of being extended (i.e.,
resolved) objects.

Ideally one would like to obtain a sample which is complete (i.e., all objects
fulfilling certain observational criteria are detected), and uncontaminated
(i.e., all objects not fulfilling such observational criteria are excluded).
The detection process for clusters in X-ray images is complicated by their
generally low surface brightness. This reduces their contrast with the
background and results in sensitivity on the background conditions and
modelling. This may lead to clusters with irregular and diffuse morphologies
being missed by a survey, while a more regular cluster that is centrally
peaked due to a cool core would be easier to detect. Moreover, one needs to
deal with spurious (false) detections due to noise fluctuations or
instrumental effects, and the addition of misclassified point sources to the
cluster sample. Finally, point sources can overlap the emission from a cluster
(due to an AGN in the cluster or a chance projection along the line of sight).
Depending on the flux and location of the point source, it may lead to the
cluster being misclassified as an AGN, or it could contribute to the total
flux of the cluster (e.g., Biffi et al. 2018).

Understanding the selection function of any survey is not an easy task
because, by its very nature, the selection function is supposed to tell us
about objects that have not been detected. This is often achieved by running
the survey’s detection algorithm on mock observations obtained from realistic
simulations. Since one knows which objects exist in the simulations, this
procedure allows the cluster detection probability to be determined as a
function of some basic input parameters (e.g., fluxes and sizes) and
accounting for a range of instrumental (e.g., PSF, vignetting) and
astrophysical (e.g., different background conditions) effects.

Since the selection effects are specific to a particular catalogue extraction
method, by comparing different codes and cluster detection methods one can
better understand the selection effects and catalogue uncertainties, and their
relation with the cluster properties. In practice, as highlighted by Melin et
al. (2005), a complete understanding of a selection function can only be
obtained by combining such simulations with observations taken under different
conditions and/or in different wavelengths. Thus, our understanding of the
selection function, and therefore of the underlying cluster population, is a
slow process.

###  4.3 X-ray vs Optically and SZ selected samples

Historically, optical and X-ray surveys have provided the primary source of
cluster catalogues. However, in the recent years galaxy clusters have been
also detected through the Sunyaev-Zel’dovich (SZ) effect, the spectral
distortion of the cosmic microwave background (CMB) generated via inverse-
Compton scattering of CMB photons by the hot electrons in the ICM (Sunyaev &
Zeldovich 1972). The strength of the SZ signal is independent of the redshift
of the scattering cluster and is tightly correlated with the cluster’s total
mass. This means SZ-selected samples are as near as possible to being mass-
selected, and as such are considered to be close to unbiased. Comparisons
between properties of samples selected with these different approaches are
giving insight into the impact of the selection effects on the derived cluster
properties.

While the ICM is responsible for both the X-ray light and SZ, the observables
have a different dependence on its distribution with a cluster. The X-ray
emission scales with the square of the ICM density while the SZ scales
linearly. This means that X-ray surveys are expected to detect a larger
fraction of centrally peaked and relaxed clusters than SZ experiments. Indeed,
this is confirmed by several observational studies (e.g., Rossetti et al.
2016, 2017, Andrade-Santos et al. 2017, Lovisari et al. 2017; but see Zenteno
et al. 2020 for a different view).

That said, both the X-ray and SZ signals will depend on the overall amount of
ICM in a cluster, and so may share biases towards detecting clusters with
higher gas fractions (e.g., Andreon et al. 2016). The impact is likely to be
strongest for low mass clusters where the gas fraction is significantly lower
than that of massive clusters (e.g., Lovisari et al. 2015, Eckert et al.
2016), making the selection bias mass-dependent.

One possible solution is to use optical selection to identify groups and
clusters, which can be then observed with X-ray/SZ telescopes. The ICM
properties of these optically-selected clusters can then be compared with
those of X-ray and SZ selected samples. A limitation of this approach is that
current optical selections suffer from projection effects (e.g., Costanzi et
al. 2019 and references therein) which lead to contaminated lists of cluster
candidates. Another independent selection can be achieved through the weak-
lensing signature of clusters, although the lensing signal can be affected by
the projection of foreground and background structures along the line-of-sight
(see Miyazaki et al. 2018 and references therein).

Clusters selected in the optical via overdensities of galaxies or via their
weak lensing signal tend to be X-ray underluminous for a given mass by a
factor of $\sim$2-3 compared with X-ray and SZ selected clusters (e.g., Giles
et al. 2015, Andreon et al. 2016). This may be due to the ICM-based detection
methods being biased against low $f_{\text{gas}}$ (and hence low gas
luminosity and pressure) clusters. However, further investigations are needed
to understand whether these clusters are really X-ray underluminous or their
masses are overestimated (e.g., due to projection effects).

Thanks to the large multiwavelength data-sets which are becoming available, we
are now in the position to compare the cluster detections based on multiple
independent observables in order to cross-validate survey selection functions.

###  4.4 Correlated errors

When modelling the correlations between cluster properties, it is of course
necessary to include the measurement errors on the quantities. One must also
consider whether the errors on the quantities of interest are independent or
correlated. Suppose an observation were repeated a large number of times, and
the same two properties were measured each time, giving a distribution of
values of each. If the errors were correlated, then the distributions would be
correlated (i.e., an above-average value of one quantity would correspond to a
below average value of the second quantity in the case of a negative
correlation).

While independent measurement errors are usually described with a one-
dimensional normal distribution when modelling data, correlated errors require
a multivariate normal distribution (i.e., with one dimension per property
being measured) parameterised by a covariance matrix. The covariance matrix
describes the size of the measurement errors and the degree of correlation
between them. This introduces a degree of complexity to the analysis, and has
been neglected in most analyses of the X-ray scaling relations, but may be
essential if the measurement errors and their correlations are not negligible.

A degree of correlation is expected between the measurements of all of the
X-ray properties derived from a given observation, because the photons used
for the estimation are the same. However, depending on the analysis method
this correlation can be reduced. For instance, if the luminosity and
temperature measurements are both based on the spectroscopic analysis (for
massive clusters the luminosity is given by normalization of the spectrum,
while the temperature is determined from its shape), the correlation of their
measurement errors is high. If a surface brightness profile is used to
estimate the luminosity instead, then the measurements are nearly
uncorrelated.

The correlations between measurements can be complex. Since HE mass
measurements are based on gas density and temperature profiles, correlated
measurement errors exist between $M_{\text{500}}$ and both $M_{\text{gas}}$
and $T_{\text{X}}$. Furthermore, one may then measure the value of cluster
properties within a radius defined from the hydrostatic mass measurement
(e.g., $R_{\text{500}}$), introducing additional correlations in the
measurements (in particular for $M_{\text{gas}}$ and $L_{\text{X}}$). To
minimize the correlation between measurements one can have an independent
estimate of the region within which determining the X-ray properties (e.g.,
Sereno et al. 2020), but this comes at the cost of an increased intrinsic
scatter. In general, the correlation between the errors can be measured
through Monte Carlo simulations and then incorporated into the analysis of the
scaling relations (e.g., Mantz et al. 2016b).

###  4.5 Linear regression and fitting packages

Cluster properties follow power-law correlations, so are linear in log space
and hence the measurement of the scaling relation parameters generally
involves some form of linear regression, typically in the form of Eq. 6 where
$X$ and $Y$ are usually renormalized by values, $X_{0}$ and $Y_{0}$, chosen to
bring the range of $X$ and $Y$ close to unity, which is beneficial for some
modelling approaches. Setting $X_{0}$ and $Y_{0}$ close to the median value of
$X$ and $Y$ for the data being modelled will also reduce the degeneracy
between $A$ and $B$ in the model.

Linear regression is one of the most widely used statistical techniques in
science, but there are numerous complicating factors in the case of
astronomical data. These may include: (i) non-negligible, heteroscedastic,
measurement errors; (ii) correlations between the measurement errors on the
quantities being compared; (iii) no meaningful way to define a dependent or
independent variable; (iv) significant intrinsic scatter of the data; (v) a
non-uniform distribution of data (e.g., many more low-mass clusters than high-
mass clusters); and (vi) missing data due to sample selection (see Sect. 4.2).
These aspects can have a non-negligible effect on the regression results if
not properly accounted for (see, e.g., Akritas & Bershady 1996 for the impact
of measurement errors and intrinsic scatter, and Kelly 2007 for the effect of
non-uniformly distributed variables).

Figure 5: Illustration adapted from Lovisari et al. (2020) showing the
comparison of the fitted results for the LX–M (left) and M–YX (right)
relations obtained using different fitting methods. In green we show the
fitted relation obtained using LIRA (Sereno 2016) allowing all the parameters
to vary (i.e., slope, normalization, evolution, and both scatters). In blue we
show the best-fit results obtained assuming the self-similar redshift
evolution. In black and magenta we show the result from BCES (Akritas &
Bershady 1996), where the redshift evolution was chosen to be self-similar.

Different analyses of scaling relations have used different approaches to
address some or all of the issues above, complicating the comparison and the
interpretation of the results (as highlighted in Lovisari et al. 2020). This
is illustrated in Figure 5. Andreon & Hurn (2013) provide an overview of
common methods for fitting scaling relations, discussing some pros and cons
for each technique. Broadly speaking, the different approaches can be divided
into frequentist and Bayesian. Both methods rely on computing the likelihood
(the probability of obtaining the data that was observed), given a model and
set of parameter values. In the frequentist approach, the probability
distributions of the parameters are derived from the likelihood and
interpreted as describing the frequency with which different parameter values
would be observed if the same experiment were repeated a large number of
times. In the Bayesian approach, the likelihood is combined with prior
probability distributions for the model parameters (summarising what is known
about the parameters before seeing the data) to derive the posterior
probability distributions for the parameters. These are interpreted as the
degree of credence that should be given to the parameter values based on the
current data and any previous knowledge.

The advantage of the frequentist approach is that it is objective (i.e., all
researchers will agree on the $p$-value and confidence limits, whose
interpretation is calibrated). The big disadvantage is that the “objectivity”
is obtained by neglecting any prior knowledge about the properties being
measured, which is waste of information.

The main advantage of Bayesian inference is that all inferences logically
follow from Bayes’ theorem (i.e., given a prior all calculations have the
certainty of deductive logic). In fact, it provides a natural and principled
way to incorporates relevant prior probabilities (i.e., when new observations
become available, the previous posterior distribution can be used as a prior).
The main critique is that the results may strongly depend on the choice of the
priors which are subjective. Since there is no single method for choosing a
prior, different researchers can choose different priors and may therefore
arrive at different posteriors and conclusions. Of course, one can try
different priors (which is an advantage) and see how sensitive are results to
the choice of priors, but that may come with a high computational cost,
especially in models with a large number of parameters.

A growing number of Bayesian methodologies have been developed in recent years
to account for the various complications involved in modelling cluster scaling
relations (see, e.g., Kelly 2007, Andreon & Hurn 2010, Mantz 2016, Sereno
2016).

In most regression analyses, the intrinsic scatter is modelled as a normal
distribution in log space (and this appears to be a good description of the
data). The scatter is usually measured in one direction (i.e., vertically or
horizontally) relative to the regression line (with the scatter in one
direction related to that in the other by the slope of the line). When natural
logarithms are used, the standard deviation of the measured scatter can then
be interpreted as the fractional or percentage scatter in one property
relative to the other. The quality of available data usually requires that the
scatter is treated as a constant across the fitted range, but it is physically
plausible that this is not the case; for example, low mass clusters should be
more impacted by non-gravitational processes and may hence show more scatter
in their properties.

Figure 6: Visualisation of the likelihood of observing cluster properties
given a model of scaling relations (adapted from Sereno et al., 2019). The
lower panel illustrates the prior probability on cluster mass, based on a
theoretical mass function. The scaling relations model predicts that a
hypothetical cluster of mass $M$ would have luminosity $l$ and temperature
$t$, placing it on the LX–TX relation at the black circle in the upper panel.
The intrinsic scatter in $T_{\text{X}}$ and $L_{\text{X}}$ is illustrated by
the blue ellipse, the tilt of which indicates covariance in the scatter. The
blue triangle shows a possible location of the hypothetical cluster given this
scatter. Uncertainties in the measurement of the observed quantities
$T_{\text{obs}}$ and $L_{\text{obs}}$ are illustrated by the red ellipse, with
the red diamond illustrating possible values of the observed cluster
properties given the combination of intrinsic scatter and measurement error.
Finally, the horizontal dashed line illustrates that a detection limit could
be applied. In this case the cluster’s observed luminosity is higher than
$L_{\text{limit}}$, allowing it to be detected in spite of its relatively low
mass.

Figure 6 illustrates the scenario we are dealing with when fitting the
observed data. Let’s consider the LX–TX relation, which is a manifestation of
the correlations of $L_{\text{X}}$ and $T_{\text{X}}$ with mass. We can
consider the true (i.e., unbiased) mass as the fundamental cluster property
(denoted as $M$ in the figure). The true masses of a population of clusters
will be described by the mass function, which serves as a prior on $M$ in the
absence of any observational constraints. This is illustrated in the bottom
panel of Figure 6.

For a hypothetical cluster of mass $M$, our scaling relation model predict the
“theoretical” temperature $t$ and luminosity $l$ of the cluster given a set of
scaling relation parameters (slope, normalisation, evolution). The “true”
values $T$ and $L$ that would be measured given an observation of infinite
accuracy and precision are related stochastically to $t$ and $l$ due to the
intrinsic scatter (illustrated by the blue ellipse in Figure 6). The
“observed” cluster properties $T_{\text{obs}}$ and $L_{\text{obs}}$ are, in
turn, related stochastically to $T$ and $L$ by the measurement uncertainties
(illustrated by the red ellipse in Figure 6). Finally, Figure 6 illustrates a
simple survey selection based on a threshold in $L_{\text{obs}}$. In this
example, an object whose true properties lie below the survey threshold is
included due to the combination of intrinsic and observational uncertainties.

The same approach can be applied to scaling relations between any combination
of observables. The core of the Bayesian analysis methods is then the
construction of a likelihood model that describes the stochastic relations
between the observed, true and theoretical values in the presence of any
selection criteria.

###  4.6 Multivariate analysis

The study of relationships between cluster property pairs provides a valuable
but fundamentally limited view of the data. In fact, there is far more
information that can be extracted by jointly fitting the form of the scaling
relations between multiple observables. For example, the LX–TX relation is
often studied, but in the self-similar model, this two-dimensional correlation
is actually a manifestation of the correlations of $M_{\text{gas}}$ and
$T_{\text{X}}$ with mass.

A (sometimes overlooked) consequence of combining scaling relations in the
derivations of the self-similar relations in Section 3.2 is that their
parameters become linked. For example, since we combined the TX–M and Mgas–M
relations to derive the LX–M relation, the slope and evolution of the LX–M
relation depends on the slope and evolution of those more fundamental
relations. Departures from self-similar behaviour in either or both of the
TX–M and Mgas–M relations would thus lead to non-self-similar scaling of
luminosity with mass.

Similarly, the slope and evolution of the LX–TX relation depends on those of
the more fundamental scaling relations. In particular (and perhaps counter-
intuitively), the algebra leads to a dependence of the expected evolution of
the LX–TX relation on the slope of the LX–M and hence TX–M and Mgas–M
relations. These dependencies can be straightforwardly taken into account when
jointly modelling multiple cluster properties in a multivariate analysis
(Maughan 2014).

Multivariate analyses are becoming more common and can provide important
information about the physics of the ICM, removing ambiguity that may exist in
the interpretation of a simple two-dimensional scaling relation (e.g., does a
cluster fall above the LX–TX relation because it is brighter or cooler, or
both?). For instance Maughan (2014) showed that the observed ICM scaling
relations can be described by a deficit of gas in low-mass clusters
compensated for by higher gas temperatures such that the total thermal energy
of the gas remain constant for a given mass. This could be the result of AGN
feedback removing low entropy gas while heating the remaining gas, an effect
working more efficiently at low masses. Ettori (2015) showed that the
deviations from self-similarity can be explained by three physical mass-
dependent quantities: the gas clumpiness, the gas fraction, and the slope of
the thermal pressure. Sereno et al. (2020) found evidence that the change in
slopes is due to radiative cooling and AGN feedback.

In a multivariate analysis, the intrinsic scatter of the cluster properties is
generally modelled as a multivariate normal distribution, parameterised by a
covariance matrix. The off-diagonal terms of the covariance matrix then
describe the degree of correlation between the scatter of the individual
properties. This encodes useful astrophysical information on the processes
responsible for the scatter. As noted in §4.1, the covariance must also be
modelled to correctly capture the effects of selection biases.

Several studies have started to measure the covariance of X-ray properties
(and those at other wavelengths), finding non-zero covariance between various
quantities (e.g., Evrard et al. 2014, Rozo et al. 2014, Maughan 2014, Mantz et
al. 2015, Sereno et al. 2019). Angulo et al. (2012) argued that these
correlations are due to internal structure, cluster orientation, formation
history, and uncorrelated structures.

A multivariate cluster dataset can also be considered as a cloud of points
which can assume any form of structure, not just linear correlations parallel
to the axes. Therefore, one can use principal component analysis (PCA) to find
a small number of scaling relations (i.e., pair of cluster properties) to
describe most of the variance in the dataset with a small number of new
uncorrelated parameters. PCA finds the direction in the parameter space where
the data are most elongated using least-squares to minimize the variance (the
first principal component). The second component is then found by searching
for the most elongated direction after the first component is removed, and so
forth. Thus, the first principal component provides the smallest distance to a
line in the space of the original variables, the second provides the smallest
distance to a line in the plane perpendicular to the first component, and so
on. Therefore, when the cluster properties lie on a plane defined by the first
two principal components, the scaling relations on this plane are defined by
the third principal component (Efstathiou & Fall 1984).

In a related approach, Ettori et al. (2012) found that all the standard self-
similar scaling relations occur as special cases of a generalized scaling
relation with the form $M\propto A^{\alpha}B^{\beta}$. Here $A$ and $B$
respectively represent ICM properties relating to the depth of the halo
gravitational potential (e.g., the gas temperature) and on the distribution of
the ICM (e.g., the gas mass or X-ray luminosity) which is more affected by the
physical processes determining the ICM global properties (e.g., AGN feedback).
These generalised scaling relations were found to have significantly reduced
intrinsic scatter compared with traditional two-dimensional scaling relations
(Ettori et al. 2012, Ettori 2013, Truong et al. 2018).

###  4.7 X-ray telescope calibration

Beside the selection biases and the differences in the fitting techniques,
there is another issue complicating the comparison between different studies,
namely the calibration uncertainty between different instruments.

The calibration of X-ray instruments is challenging, because there are no
absolute calibration targets (i.e., sources whose absolute flux and spectral
properties are known) and because of the uncertainties in the overall
calibration, which relies on a combination of ground-based calibration
measurements and in-flight calibration programs. Ground-based calibrations are
not perfect because of the time and budget pressure during the mission
development phase, and because of technical limitations in the calibration
facilities. In-flight calibrations are complicated as well because the
instruments performance can change with time due to many different reasons
(e.g., radiation damage, filter deterioration, component failures,
contamination buildup, etc).

In the last decade the International Astronomical Consortium for High Energy
Calibration888http://web.mit.edu/iachec/ supported several studies on clusters
of galaxies in order to investigate the effective area cross-calibration
status between different X-ray instruments. Nevalainen et al. (2010) found
that in the hard band (2-7 keV) the energy dependence of the effective area of
various X-ray telescopes were accurately calibrated but the normalisation of
the effective areas disagreed by 5–10%, with the disagreement becoming
significantly larger in the soft energy band (0.5–2.0 keV).

These calibration differences lead to a systematic difference in the ICM
temperature determined with different instruments. For instance,
Schellenberger et al. (2015, see also ) found that the cluster temperatures
obtained with XMM-Newton are systematically lower than those obtained with
Chandra. The situation is further complicated by the temperature dependence of
this difference. In the low temperature regime the differences are relatively
small (a result that seems to hold also when including Suzaku data; e.g.,
Kettula et al. 2013), but for high temperature clusters the difference can
reach 20% or more.

This has an obvious impact on the scaling relations studies. For instance, it
is obvious that an analysis based on temperatures measured with Chandra would
produce a flatter LX–TX relation than would be obtained by using XMM-Newton
derived temperatures. Other relations are impacted as well. In fact, due to
the temperature-dependence of the discrepancy, this issue has also an impact
on the gradient of the temperature profiles used to calculate the total mass
via the HE equation. Therefore, one need to be careful when comparing the
scaling relations determined with different instruments.

###  4.8 Emission-weighted and spectroscopic-like temperatures

A further complication in the measurement of the temperature of the ICM arises
because it is not isothermal. Due to the limitation of the current
instruments, a single temperature model is often used to fit the observed
X-ray spectra. However, the observed spectrum is the result of an integral of
emission along the line-of-sight through the ICM, which will include multiple
components with different temperatures (and metallicities). A single
temperature fit will then yield an “emission weighted” temperature (i.e., an
average temperature that is weighted by the relative brightness of the
different thermal components). The telescopes and CCD detectors used in
current observatories have a larger effective are in the soft band. This means
that a single-temperature fit will be further weighted towards the lower
temperature components of the ICM that contribute the most emission in softer
bands (e.g., Mazzotta et al. 2004, Vikhlinin 2006).

A related problem arises when comparing the thermal structure of observed and
simulated clusters. To derive a single temperature for a simulated cluster,
one must assign some weighting to each temperature component in its ICM. The
resulting ICM temperature can be written as

$T=\frac{\sum_{i}w_{i}T_{i}}{\sum_{i}w_{i}}$ (26)

where $T_{i}$ is the temperature of the $i$ component of the ICM, which
contributes with a weight $w_{i}$. By assuming that $w_{i}$=$m_{i}$ (where
$m_{i}$ is the mass of ICM in the $i$ component) one obtains $T_{\text{mw}}$,
the so-called mass–weighted temperature. Instead, by weighting the different
temperatures with the radiative emission contributions of the different phases
(i.e., $w_{i}$=$\epsilon_{i}$), one obtains $T_{\text{ew}}$, the emission
weighted temperature (e.g., Mathiesen & Evrard 2001). However, Mazzotta et al.
(2004) showed that the fitted temperature from Chandra or XMM-Newton spectra
is biased low with respect to $T_{\text{ew}}$ and introduced $T_{\text{sl}}$,
the spectroscopic–like temperature which is recovered by Equation 26 using the
weight

$w_{i}=\rho_{i}m_{i}T_{i}^{-0.75}.$ (27)

The spectroscopic–like temperature was generalized to the case of arbitrary
metallicities by Vikhlinin (2006). The two methods provide very similar
results in the high temperature regime (i.e., in the limit where spectra are
continuum-dominated) but significant variations may exist in the low-
temperature regime because of the effects on the temperature introduced by the
metallicity dependent line emission. The form of the weights given in Eq. 27
was obtained for CCDs having a band-pass and a spectral resolution like those
on board Chandra and XMM-Newton. Such resolution blurs the emission lines and
do not allow them to be clearly separated from the continuum. It is likely
that the weighting scheme in Eq. 27 will not be appropriate for high-
resolution spectrometers planned for future missions, and revised weightings
will be required.

##  5 Observational results and deviations from self-similarity

One of the main reasons to study the scaling relations of galaxy clusters is
to test for departures from the self-similar expectations. These would then
give insight into the astrophysical processes that are not included in the
self-similar models, and be used to test the predictions of more realistic
cluster models such as those from numerical simulations. Understanding the
behaviour of the scaling relations is also essential for their use in
cosmological studies. The key predictions of the self-similar model that are
tested observationally are that the slopes of the observed scaling relations
follow the self-similar predictions (sometimes called strong self-similarity),
and that the normalisation follows the predicted evolution (sometimes called
weak self-similarity). The scatter and covariance in scaling relation may also
be measured, giving astrophysical insight while also being crucial for the
application of scaling relations to estimate cluster masses from observable
properties.

In the following, we will review the observational constraints on the slope,
evolution, and scatter of the various X-ray scaling relations. We will also
consider results on the use of ICM scaling relations for mass estimation. In
doing so we will make a distinction between studies that compare X-ray
properties of the ICM with “direct” and “indirect” mass measurements. Direct
measurements are those which measure the mass of a cluster without needing
calibration, such as hydrostatic or weak lensing masses. Indirect measurements
are those which rely on calibration of some observable quantity against direct
mass measurements.

When considering the scaling relation between an X-ray observable and indirect
masses, one is implicitly including a second scaling relation between mass and
some other observed quantity that was used to derive the indirect masses.
Thus, we prefer scaling relations measured using direct masses, but these are
harder to construct (and hence more scarce) as direct mass estimates are
observationally expensive to obtain. Unless stated otherwise, the scaling
relations we discuss are based on direct mass measurements, and properties are
measured within $R_{\text{500}}$.

###  5.1 The slopes of scaling relations

The most widely studied of the ICM scaling relations is that between
$L_{\text{X}}$ and $T_{\text{X}}$, due to the fact that both quantities are
relatively easy to determine from shallow X-ray observations. The earliest
observations of this scaling relation suggested that the slope was steeper
than the self-similar expectation (Mitchell et al. 1979), and these results
have been borne out in a large number of subsequent studies. While the exact
value of the slope found varies (likely due to e.g. differences in modelling,
sample definition and selection effects, instrumentation and calibration), a
broad consensus finds $2.5\lesssim B_{\text{LT}}\lesssim 3.5$ for the
bolometric LX–TX relation (Giodini et al. 2013). Figure 7 shows a compilation
of measurements of the LX–TX relation from the literature. This illustrates
that the observed LX–TX relation is well-described by a power law over a wide
range of temperatures and luminosities, but the slope of the relation is
incompatible with the slope predicted by the self-similar model.

There is some evidence that, when their core regions are excluded, the most
massive and relaxed galaxy clusters do exhibit a self-similar LX–TX relation
(Maughan et al. 2012). The outer parts of these relatively simple systems seem
to conform to the assumptions of the self-similar model and be largely free
from the influence of non-gravitational effects. This is not unreasonable,
since the central regions of clusters are where non-gravitational processes
such as radiative cooling and energy input from AGN and stellar processes are
strongest (see §6).

Figure 7: Figures illustrating compilation of LX,bol–TX and M–TX measurements;
this is adapted from Lovisari et al. (2021). The Chandra measurements
(squares) are converted to XMM-Newton-like temperatures using the relations
given in Schellenberger et al. (2015). The solid lines represent the fitted
relations (with the 1$\sigma$ uncertainties indicated by the shaded areas)
including all systems, and are compared with the case predicted by the self-
similar scenario (dashed). The fits have been performed with the LIRA Bayesian
regression package (Sereno 2016).

The determination of the LX–TX relation in the lower mass regime of galaxy
groups ($T_{\text{X}}\lesssim 2\text{~{}keV}$) is challenging, with a wider
range of observational results, with some evidence that the relation steepened
further for galaxy groups (e.g., Osmond & Ponman 2004). This is likely due to
the larger scatter in ICM properties for groups combined with the difficulty
in forming large samples with good quality data due to their intrinsic
faintness. More recently, the quality of samples has improved, allowing for
analyses that model selection biases (§4.1), and a picture has emerged in
which lower-mass systems follow the same LX–TX relation as high mass clusters,
albeit with larger intrinsic scatter (Lovisari et al. 2021). However, given
the increasing contribution from line emission to the X-ray luminosity for
cooler systems (§3.2), we would expect the LX–TX relation to flatten for
galaxy groups. This may imply that a steepening of the LX–TX relation for
galaxy groups is being masked by the contribution of line emission.

The deviation of the LX–TX relation from the self-similar predictions can also
be explained in terms of entropy variation. Since the overall level of entropy
in the ICM is set by the mass of the cluster (through the strength of the
accretion shock), then a given entropy increase will have a stronger effect on
the ICM in lower-mass clusters. This would have the effect of modifying the
slopes of the scaling relations involving the mass, density and luminosity of
the ICM compared with the self-similar model. This is confirmed
observationally with the cores of galaxy groups exhibiting a higher entropy
than is achievable by pure gravitational collapse (see Eckert et al. 2021 and
references therein) which then leads to a flatter $K$–$T_{\text{X}}$ relation
(e.g., Pratt et al. 2010). However, the slope of the $K$–$T_{\text{X}}$
relation is found to agree better with the self similar value when the inner
regions are removed from the analysis (e.g., Sun et al. 2009) indicating a
connection with the feedback processes at play in the core of galaxy systems.

Observational studies of the scaling of ICM properties with mass are more
challenging due to the difficulty in estimating cluster masses. In the case of
the LX–M relation, the overall picture from the literature (based on both
direct and indirect mass measurements) is that the slope is significantly
steeper than the self-similar expectation, with $1.5\lesssim
B_{\text{LM}}\lesssim 2$ typically found for bolometric luminosities (where
$B_{\text{LM}}=4/3$ is expected). This agrees qualitatively with observations
of the LX–TX relation; the luminosities of low mass (i.e., low temperature
clusters) are lower than expected in the self-similar model.

Given that the self-similar LX–TX and LX–M relations are derived from the
Mgas–M and TX–M relations, the fact that the luminosity scaling relations
depart from self similarity imply that one or both of $T_{\text{X}}$ and
$M_{\text{gas}}$ must also depart from self-similar scaling with mass.
Observations of the TX–M relation broadly find slopes that roughly agree with
the self-similar expectation of $B_{\text{TM}}=3/2$ (see right panel of Figure
7), although care must be taken when comparing temperatures with hydrostatic
masses since those are derived from the same temperature measurements. On the
contrary, observations of the Mgas–M relation consistently favour a slope
steeper than $B_{\text{mM}}=1$, which can be interpreted straightforwardly as
a lower gas fraction in less massive clusters. However, when analyses are
restricted to high-mass systems ($M_{\text{500}}\gtrsim 5\times
10^{14}\text{M}_{\odot}$), the gas fraction appears to become constant (i.e.,
$B_{\text{mM}}=1$), implying self-similar behaviour of the most massive
clusters (Mantz et al. 2016b). A constant $f_{\text{gas}}$ for more massive
systems implies that the scaling relations may ultimately be best described by
a broken power law, with self-similar slopes where $f_{\text{gas}}$ is
constant, and a steeper slope at lower masses. However, current observations
tend to be well described by a single power law.

A similar argument can be used when investigating the YX–M relation. In most
studies the slope of this relation is found to be slightly steeper (although
often consistent within the uncertainties) than what is predicted from the
self-similar scenario. However, given the steeper slope of Mgas–M relation,
this is not surprising.

Collectively, observations show that the luminosity of the ICM has a steeper
scaling with mass or temperature than predicted by the self-similar model, due
to the fact that lower mass systems contain less X-ray emitting gas than
predicted. The temperature of the gas, meanwhile, obeys a scaling close to
that expected for an ICM that is in virial equilibrium with the cluster
potential. The principle physical process at work must therefore be to remove
the ICM from the potential well of lower mass systems (or equivalently inhibit
its accumulation into those systems; §3.3).

###  5.2 The evolution of scaling relations

The evolution of cluster scaling has proved difficult to measure due to the
challenge of obtaining good quality data for sufficient numbers of distant
clusters with a well-understood selection function, along with defining a
consistent low-redshift baseline against which to measure the evolution.
Inconsistencies in sample definition, analysis methods or instrumentation
between local and distant samples could create (or obscure) evolution in the
data. Given these limitations, attention has been focussed on testing the
self-similar prediction of an evolving normalisation of the scaling relations.
In principle the slope and scatter may also evolve (not predicted by the self-
similar model), but measuring these is not currently feasible given difficulty
of observing the ICM in distant clusters with current observatories.

The LX–TX relation is again the best-studied of the scaling relations in this
regard, but no consensus exists on its evolution. Measurements in the
literature encompass all possibilities from negative evolution
($\gamma_{\text{LT}}<0$) to evolution that is stronger than self-similar.
Reichert et al. (2011) performed a meta-analysis of samples of clusters out to
$z\approx 1.5$, finding evidence for negative evolution but highlighted the
influence that the choice of local sample can have on the inferred evolution.

The most reliable evolution measurements come from analyses of large samples
spanning a wide redshift range, and which model the effects of selection
biases on the scaling relations. These are now becoming available (e.g., Giles
et al. 2016, Bulbul et al. 2019, Lovisari et al. 2020). Thus far, these do not
provide any strong evidence that the evolution of the LX–TX relation differs
from the self-similar prediction, but the statistical uncertainties remain
significant.

The evolution of the mass-observable scaling relations is even less well-
constrained than the LX–TX relation, due to the difficulty in performing mass
measurements for distant clusters. There is some variation in the measurements
reported, with the most precise constraints tending to come from small
heterogeneous samples for which the systematic uncertainties are largest. A
few studies have been performed with large, self-consistent samples including
modelling of selection biases, and in general these show no strong evidence
for evolution that differs from self-similar expectations for any of the mass-
observable scaling relations (Mantz et al. 2016b, Lovisari et al. 2020, Chiu
et al. 2021), although Sereno et al. (2021) have recently found some hints for
weaker than self-similar evolution of the TX–M relation.

The evolution of the scaling relations (particularly involving $L_{\text{X}}$)
is complicated by the fact that the cores and outer parts of cluster may
plausibly be expected to evolve differently. It is clear that non-
gravitational processes, to which the prediction of self-similar evolution
does not apply, are most important in the centres of clusters. Indeed, the
prevalence of the strongly centrally peaked ICM surface brightness profiles
that we associate with cool cores is found to be lower at high redshift. This
appears to be due to the cool cores being stable over cosmic time, while the
ICM at larger radii evolves self-similarly, reducing in brightness towards the
present time which in turn makes the cores more pronounced (McDonald et al.
2017).

One might expect to see differences in the evolution of $L_{\text{X}}$ scaling
relations when the core regions are, or are not, excluded from the luminosity
measurement. In fact, this is difficult to test, since it is not possible to
self-consistently exclude the core regions and model the selection biases for
an X-ray selected cluster sample, since the core regions significantly
contribute to the detection probability of a cluster. Studies using SZ-
selected clusters have found no indication that evolution in the
$L_{\text{X}}$ scaling relations differed when cores were excluded, but the
uncertainties remain large (Bulbul et al. 2019, Lovisari et al. 2020; with the
former based on indirect mass measurements). There is some limited evidence
that the scatter in the $L_{\text{X}}$ scaling relations is reduced for
distant clusters (Maughan 2007, Mantz et al. 2016b). This is a plausible
consequence of the relative decrease in the strength of cool cores compared to
the surrounding ICM in more distant clusters. The scatter may also be reduced
by non-cool-core clusters being missed in samples of high redshift systems due
to their low contrast against the background.

Overall, current observations do not indicate that the evolution of galaxy
cluster scaling relations is different from self-similar expectations out to
$z\lesssim 1.5$. This implies that clusters have effectively formed (or
regained equilibrium after their most recent disturbance) recently compared to
the redshift at which they are observed. There is no strong evidence for
astrophysical evolution of clusters beyond that due to the use of
overdensities in the self-similar model. However, the uncertainties remain
large, and evolution that is weaker or stronger than the self-similar
predictions cannot be confidently excluded.

###  5.3 Scatter and covariance

The non-zero intrinsic scatter in scaling relations is one very clear way in
which they depart from self-similar predictions. Observational constraints on
the scatter (and covariance) in ICM scaling relations can be used to probe the
physical processes responsible.

The scatter in scaling relations involving $L_{\text{X}}$ is found to be
significant, with e.g. typically $\sim 50\%$ scatter found in $L_{\text{X}}$
at fixed $T_{\text{X}}$. However, this is dominated by the variation in core
properties of clusters, with cool-core clusters having significantly enhanced
luminosities (§6).

Exclusion of the core regions (conventionally the central
$0.15R_{\text{500}}$) from the measurements significantly reduces the scatter
to of order 10–20$\%$ (with the largest reduction for the most relaxed
clusters). This is illustrated in Figure 8, in which clusters are separated
into those hosting a strong cool core and those that do not. The cool core
clusters are clearly offset, but once the core regions are excluded, all
clusters occupy a similar part of the LX–TX plane.

Figure 8: Figure illustrating impact of cool cores on the scatter in the LX–TX
relation, adapted from Maughan et al. (2012). In both plots, blue points
indicate clusters classified as possessing cool cores, while red points denote
those without cool cores (the lines show the best fits to the subsamples). In
the left panel, luminosity and temperature are measured within an aperture of
radius $R_{\text{500}}$. In the right panel, the central $15\%$ of
$R_{\text{500}}$ was excluded from the luminosity and temperature
measurements. For this sample, excluding the core regions reduces the scatter
in luminosity from about $70\%$ to about $30\%$.

The most insight into the physical origin of the scatter in the scaling
relations is gained by measuring the covariance of observables with respect to
mass. This can be assessed by a multivariate analysis in which the mass-
observable relations are modelled simultaneously (see §4.6). Such studies are
becoming more common (e.g., Maughan 2014, Farahi et al. 2019), and one of the
most robust results is a strong positive covariance between $M_{\text{gas}}$
and $L_{\text{X}}$. This is unsurprising given the dependence of
$L_{\text{X}}$ on $M_{\text{gas}}$ (Equation (13)): clusters with a higher
than average mass of ICM relative to their total mass will be correspondingly
more luminous in X-rays.

Most observations have also found a positive covariance between $L_{\text{X}}$
and $T_{\text{X}}$ with respect to mass (i.e., clusters which have higher than
average luminosities for their mass tend also to have higher than average
temperatures, and vice-versa). This is illustrated in Figure 9, which shows
the correlation between the residuals in $L_{\text{X}}$ and $T_{\text{X}}$
relative to their mass scaling relations. There is a positive correlation,
which is associated with the dynamical state of the clusters. Numerical
simulations show that the positive covariances between X-ray properties
probably stands at any redshift (e.g., Stanek et al. 2010, Truong et al.
2018).

Figure 9: The correlation between the residuals in $L_{\text{X}}$ and
$T_{\text{X}}$ relative to their mass scaling relations. The blue and red
datapoints show the most relaxed and disturbed clusters respectively. The
shaded ellipses illustrate the best fitting covariance, with the dark and
light regions indicating the $1\sigma$ and $2\sigma$ intervals, respectively,
of the measured scatter. Figure adapted from Mantz et al. (2016b).

The most relaxed clusters (blue points in Figure 9) tend to be both hotter and
more luminous than average, which is driven by the association of cool cores
with relaxed clusters, which significantly enhance their luminosity while
having a smaller impact on their overall temperature. Clusters undergoing a
merger (red points), meanwhile, tend to be cooler and less luminous than
average. The impact of mergers on ICM properties is complex, with shocks
temporarily and locally boosting both luminosity and temperature. However,
once a merger starts, the mass of the cluster can be considered to have
increased instantly, but the temperature and luminosity of the ICM will not
reach the final values expected for that mass until sufficient time has passed
for the gas to become virialised and settle in the new potential (a few Gyr).
Until this point, the temperature and luminosity will be lower than average
for the new mass. Minor mergers can also contribute to the scatter because
smaller and colder substructures are efficiently stripped and mixed due to
stellar and AGN feedback (Truong et al. 2018).

The covariance of X-ray properties with those measured at other wavelengths
can give further insight. There have been attempts to measure the covariance
between $M_{\text{gas}}$ and the total stellar mass, $M_{\star}$, in member
galaxies of a cluster. The measurements are challenging, since cluster masses
are required in order to provide the scaling relations about which the
covariance is measured, and observational results are mixed. There is some
evidence for negative covariance between $M_{\text{gas}}$ and $M_{\star}$
(Farahi et al. 2019). This can be understood if clusters are considered to be
gravitationally closed, with a fixed proportion of their mass in baryons. A
cluster of a given mass that had formed more of its baryons into stars would
have a correspondingly smaller mass of baryons in the ICM.

###  5.4 Mass proxies

A key application of cluster scaling relations is to infer the masses of
clusters. For this to be done accurately, the slope and normalisation of a
mass-observable scaling relation must be calibrated, which is generally
accomplished by measuring direct masses for a subset of the clusters. Ideally
this would be done for a representative sample of clusters (i.e. subject to
the same selection biases as the parent sample) and in the mass and redshift
range in which the mass proxies were to be applied. However, often this is not
possible, and e.g. a self-similar form for the mass-observable relation is
assumed when applying the mass proxies at redshifts beyond those for which the
proxies were calibrated.

Even if the form and evolution of a mass-observable scaling relation were
known perfectly, the intrinsic scatter about the relation would limit the
precision with which masses could be estimated. Significant effort has thus
been devoted to identifying observables (or their combinations) which possess
the minimum intrinsic scatter with mass999While it is conventional to report
the scatter in a particular mass proxy at a given mass, the primary quantity
of interest for mass estimation is the scatter in mass at a given value of the
proxy. The two scatters are directly related by the slope of the scaling
relation.. The calibration of a low-scatter mass proxy against direct masses
for as large and representative a sample of clusters as possible would then
enable mass estimates for large numbers of clusters.

Simply measuring the scatter between an X-ray property and a direct (or
indirect) mass measurement can be misleading, since this will include
contributions from the scatter of both the X-ray property and the mass
measurement with the true mass.

Hydrostatic masses give the most precise direct mass measurements, but they
are only secure for relaxed clusters for which the assumption of hydrostatic
equilibrium is valid. However, there is not a unique approach to derive the
hydrostatic masses (see, e.g., Ettori et al. 2013 and Pratt et al. 2019 for
advantages and disadvantages of the different methods), and the results are
not always in good agreement.

Weak lensing masses are agnostic to the dynamical state of the cluster, but
have more limited precision with an estimated $\sim 30\%$ scatter between the
weak lensing mass and true mass, due primarily to projected structures and
non-sphericity of the systems (Becker & Kravtsov 2011). Calibrating a low-
scatter proxy via its scaling relation with weak lensing mass would improve
the accuracy of the proxy while avoiding the large scatter in weak lensing
masses, and this is the most promising way to proceed.

As seen in Sect 4.3 the SZ effect probes the hot gas in the ICM independently
from the X-ray observations. Since the integrated SZ flux, $Y_{\rm
SZ}$101010This is related to the Compton parameter
$y=(\sigma_{T}/m_{e}c^{2})\int Pdl$, which is a measure of the thermal
electron pressure of the ICM gas along the line of sight, where P is the ICM
thermal electron pressure, $\sigma_{T}$ is the Thomson cross section, $m_{e}$
is the electron rest mass, and c is the speed of light. $Y_{\rm SZ}$ is then
given by the integral of $y$ over the solid angle of the cluster., from a
cluster directly measures the total thermal energy of the gas (and thereby of
the total mass), it is expected to provide a low-scatter mass proxy. This
expectation is confirmed by both numerical simulations (e.g., da Silva et al.
2004) and indirectly from X-ray observations using $Y_{\text{X}}$ (i.e., the
X-ray counterpart of the SZ signal (e.g., Andrade-Santos et al. 2021).

X-ray luminosity is the most easily measured property of the ICM, and can be
recovered from low signal-to-noise data if the cluster redshift is known.
However, unless core regions are excised, $L_{\text{X}}$ has the largest
scatter with mass of the X-ray properties of the ICM. It is also strongly
influenced by non-gravitational physics, which is apparent in the non-self-
similar slope of the $L_{\text{X}}$ scaling relations. This also means it is
plausible that the evolution of the LX–M relation may differ from self-similar
expectations (although this is not yet strongly tested observationally). All
of this makes $L_{\text{X}}$ a sup-optimal mass proxy. However, the use of
$L_{\text{X}}$ may be necessary if the resolution or data quality prohibit the
exclusion of core regions (typically the case in survey data). Furthermore,
X-ray selection of clusters is based on the entire emission, so the relation
between $L_{\text{X}}$ and mass is needed for analyses that rely on modelling
the selection function of such a sample in detail.

If the cores are excised, the luminosity ($L_{\text{X,ce}}$) becomes a much
better mass proxy, with e.g. Mantz et al. (2016a, 2018) estimating a scatter
of $10-20\%$ relative to the true mass, based on comparisons with either
hydrostatic or weak lensing masses, which include the scatter between those
quantities and true mass (although Farahi et al. 2019 report a significantly
larger scatter in $L_{\text{X,ce}}$ relative to true mass, based on weak
lensing masses). Relatively low scatters are also found when comparing
$L_{\text{X,ce}}$ with indirect mass measurements (e.g., Maughan 2007, Pratt
et al. 2009).

Historically, $T_{\text{X}}$ has been regarded as a useful mass proxy, due to
its simple connection to cluster mass via the virial theorem. Comparisons of
temperatures with hydrostatic masses for relaxed clusters indicate that the
scatter is low ($\lesssim 10\%$ scatter in $T_{\text{X}}$; Vikhlinin et al.
2006, Arnaud et al. 2007, Sun et al. 2009, Mantz et al. 2016a). However, these
analyses may underestimate the scatter since they tend to neglect the
covariance between $T_{\text{X}}$ and hydrostatic mass due to the physical
dependence of the latter on the former, and also due to correlated measurement
errors if both quantities are determine from the same X-ray data. Studies that
utilise weak lensing masses instead, tend to measure scatters that are a
little larger (typically $10-20\%$ in $T_{\text{X}}$; Mantz et al. 2016b,
Farahi et al. 2019, Chiu et al. 2021).

The mass of the ICM, $M_{\text{gas}}$, has been found to be have a scatter
with mass that is similar to, or smaller than that of $T_{\text{X}}$.
$M_{\text{gas}}$ has the additional advantage as a mass proxy in that it can
be measured reliably with lower quality data than $T_{\text{X}}$. For relaxed
clusters with hydrostatic masses, the scatter in $M_{\text{gas}}$ relative to
mass is found to be $\lesssim 10\%$ (Arnaud et al. 2007, Mantz et al. 2016a).
Studies based on weak lensing masses again tend to report slightly higher
scatters of $10-20\%$ (even when modelling the scatter between lensing mass
and true mass; Farahi et al. 2019, Chiu et al. 2021).

The final X-ray mass proxy we will consider is $Y_{\text{X}}$, which is
expected to be a low-scatter tracer of cluster mass due to its relation to the
total thermal energy content of the ICM. This is because $M_{\text{gas}}$ and
$T_{\text{X}}$ tend to react in opposite ways to non-gravitational processes,
with the effects approximately cancelling to reduce the scatter of
$Y_{\text{X}}$ with mass. In case of mergers the gas mass increases while the
measured temperature may be lower due to the presence of colder structures or
incomplete virialisation of the ICM. In case of AGN feedback, the gas mass may
decreases because it is expelled by the AGN activity which at the same time is
heating the gas. Observational studies based on hydrostatic masses support
this, finding a scatter that is comparable to (or a little smaller than)
$M_{\text{gas}}$ (i.e., $\lesssim 10\%$; Arnaud et al. 2007, Sun et al. 2009).
Once again, measurements utilising weak lensing masses tend to find higher
scatter, spanning a range from about $10\%$ to $30\%$ (e.g., Mahdavi et al.
2013, Farahi et al. 2019, Chiu et al. 2021). This could be due in part to
covariance between hydrostatic mass and $Y_{\text{X}}$ that is neglected in
X-ray studies suppressing the scatter to some degree.

In summary, all of the X-ray properties of clusters have some use as mass
proxies, with strengths and weaknesses. If the data quality are sufficient,
then $Y_{\text{X}}$ or $M_{\text{gas}}$ alone are probably the most precise
mass proxies. For lower quality data, $L_{\text{X,ce}}$ is a good alternative.
It should also be noted that for low temperature clusters, the computation of
$M_{\text{gas}}$ becomes quite sensitive to the metal abundance of the ICM,
which cannot generally be well constrained in lower quality data. Luminosities
less strongly affected by this additional systematic uncertainty (Eckert et
al. 2020).

Of course, uncertainties on all of the proxies increase at lower masses and
higher redshifts where they are largely untested. This is principally due to
the observational difficulties in obtaining direct mass measurements in those
parts of the parameter space.

##  6 Interpretation of scaling relations

It should come as no surprise that real clusters do not perfectly conform to
the simplistic self-similar scaling relations. However, the self-similar model
provides a useful reference with which to compare observed cluster scaling
relations. As we have seen in §5, it provides a good description of some, but
by no means all, of the observed scaling behaviour of clusters.

For instance, all the relations involving quantities strongly connected with
the gas distribution (i.e., luminosity, gas mass and therefore $Y_{\text{X}}$)
show a steepening with respect to the self-similar expectations. On the
contrary, the M–TX relation is found to be largely consistent with the
predictions. While there is a general consensus on the steeping of the
relations, it is not yet clear if the evolution is weaker or stronger than the
self-similar predictions. Part of the reason is the poor sampling of groups
and clusters at high redshift which lead to very large uncertainties in the
estimation of $\gamma$, but the large intrinsic scatter can also easily hide
weak effects of the evolution. Now, the obvious question is what are the
possible mechanisms that could explain the observed relations in term of
deviation from the self-similar slopes, scatter, and (lack of) evolution?

Departures from self-similarity are signs of astrophysical processes beyond
those included in the simplifying assumptions underpinning the self-similar
model. These are often referred to as “non-gravitational” processes because
the self-similar properties of the ICM are dictated by the gravitational
potential of the cluster, and can be broadly separated into the categories of
cooling and heating of the ICM (see §3.3).

There are a number of indications that these non-gravitational processes have
changed the properties of the gas, particularly in poor clusters and groups
(e.g., Eckert et al. 2021). For instance, due to the increasing complexity of
the emissivity function in the low-temperature regime, in absence of any
feedback the LX–TX and LX–M relations are expected to gradually flatten at low
temperatures and masses, contrary to observations. When comparing relations
obtained in the group and cluster regimes, one must account for the increasing
contribution of the line emission at low-temperatures, otherwise the impact of
feedback processes could be underestimated.

Since the observed cluster X-ray properties do not agree very well with the
self-similar predictions (i.e., scaling relations for purely gravitational
heating), extra physics is required for a proper description of the ICM. As
discussed in §3.3, a heating source is therefore required to raise the entropy
of the ICM. This heating could happen before or after the cluster formation.
If the non-gravitational heating occurred just prior to the cluster collapse,
then the amount of heat needed is a few keV per particle (e.g., Loewenstein
2000). This minimum entropy in the pre-collapse intergalactic medium is
expected to break self-similarity at the scale of groups.

An alternative to heating is that cooling could effectively remove low entropy
gas from the centres of groups and clusters. This low entropy gas has the
shortest cooling time, and once it has cooled sufficiently it will no longer
emit in the X-ray band. This means that the average temperature (and entropy)
of the gas that remains visible in X-rays will be increased (e.g., Voit
2005b).

Ponman et al. (1999) was the first to show that the entropy does not scale
linearly with temperature (as expected from self-similar scenario). Later
studies, confirmed that poor systems shows a significant excess to that
achievable by pure gravitational collapse (e.g., Sun et al. 2009, Panagoulia
et al. 2014). Also the observed entropy profiles appears to be inconsistent
with purely gravitational heating, showing a significant flattening in the
cores (e.g, Ponman et al. 2003, Sun et al. 2009).

Currently, the favored mechanism is represented by energy release by central
AGNs which are thought to heat and displace the thermal gas changing the
radial and global properties of the clusters. It is an inside-out process
which affects more significantly the regions at small radii. Moreover, the
effect is expected to increase at lower masses impacting the shape of the
scaling relations. The injection is likely provided by outflows from AGN (i.e,
jets of plasma inflate bubbles as large as $\sim 100$ kpc in radius). This is
sometimes referred to as the “kinetic” mode of AGN feedback (Fabian 2012). In
this mode, outflows of relativistic material interact with the surrounding ICM
through inflating bubbles and by driving shocks through the ICM. The shocks
heat the ICM directly, while bubbles initiate motions in the ICM that they
displace as they rise buoyantly, which are then dissipated into heating via
turbulence. The observed energetics of such bubbles are sufficient to maintain
cool cores in an approximately steady state (McNamara & Nulsen 2007).

This kinetic-mode feedback is likely to shape the scatter of the scaling
relations (particularly those involving luminosity), by virtue of its
influence on cool cores. However, the feedback appears sufficiently well
regulated that there is not excess energy that would remove gas from low mass
systems and hence modify the slopes of scaling relations.

Instead, the slopes of scaling relations are likely to be set by radiative-
mode AGN feedback operating at high redshift (peaking at around $z\sim 2-3$).
This is a form of pre-heating, increasing the gas entropy at early stages in
the growth of clusters. Many AGN will be located in galaxies that will
ultimately reside at the centres of groups and clusters of galaxies as the
surrounding structures collapse. The energetic impact of AGN in this mode is
primarily in radiation coupling to the surrounding gas and dust, and either
heating or pushing the gas sufficiently that it escapes from the dark matter
halo in which the AGN resides (such that it is not re-accreted again over the
lifetime of the cluster). This feedback acts on the lowest entropy gas closest
to the AGN, effectively removing it from the halo, and leaving behind higher-
entropy gas that takes longer to cool and settle to the centre of the halo
(McCarthy et al. 2011). This effect is naturally stronger in lower mass halos
whose gravitational potentials are less able to hold on to the heated gas (or
equivalently, a smaller entropy increase is needed to prevent the gas sinking
to the inner regions). The hot gas content of these systems is thus reduced,
altering the slopes of the ICM scaling relations (e.g., Gaspari et al. 2014).

###  6.1 Comparison with simulations

The comparison between the observed scaling relations with the results from
hydrodynamical simulations offer a great opportunity to constrain the physical
processes at work in the ICM and to enable the investigation of the selection
effects in surveys of groups and clusters. In simulations one has the
advantage of being able to turn on and off different physical processes (e.g.,
galactic winds, AGN feedback) keeping the same initial conditions. By over-
plotting the results obtained from different simulations with the observed
scaling relations it is possible to obtain hints about the processes which are
driving the evolution of groups and clusters. In fact, it is frequent to
obtain a significant offset between observed and simulated relations (e.g.,
Truong et al. 2018). This could reflect a different gas distribution (over- or
under-peaked), or a different masking in observations and simulations of
substructures present within $R_{\text{500}}$ (since there is not a unique
definition of what constitutes a distinct substructure).

For instance, Puchwein et al. (2008) showed that simulations including only
stellar feedback were unable to reproduce the observed
$f_{\text{gas}}$–$T_{\text{X}}$ relation. However, by adding AGN feedback they
were able to reduce the gas fraction within $R_{\text{500}}$, and match the
observational values. The intensity of the AGN feedback regulates the amount
of gas that is ejected from the potential well. The role of the AGN feedback
in reproducing the observed properties of galaxy clusters have been confirmed
in many other studies (e.g., McCarthy et al. 2010, Fabjan et al. 2010, Le Brun
et al. 2014, Gaspari et al. 2014, McCarthy et al. 2017). It is possible to
obtain reasonable gas mass fractions with models without AGN feedback, but the
$f_{\text{gas}}$–$T_{\text{X}}$ relation is flatter than observed (see the
reviews by Eckert et al. 2021 and Oppenheimer et al. 2021 for more details).
Reproducing the correct $f_{\text{gas}}$–$T_{\text{X}}$ (or the Mgas–M)
relation is crucial for a proper comparison between observations and
simulations. The gas fraction is found to be mass-dependent (e.g., Vikhlinin
et al. 2006, Gonzalez et al. 2007, Pratt et al. 2009, Lovisari et al. 2015,
Eckert et al. 2016; see also the reviews by Eckert et al. 2021 and Oppenheimer
et al. 2021). A lower gas fraction leads to lower luminosity and
$Y_{\text{X}}$ parameter impacting the shape of the relations. Therefore, as
highlighted in Lovisari et al. (2020) the mass range of the investigated
sample plays a significant role in both the slope and normalization of the
derived scaling relations.

The gas fraction of low mass systems is an extremely sensitive to the feedback
scheme implemented in simulations (see Oppenheimer et al. 2021). However, in
general when feedback processes are included in the simulations the derived
scaling relations are in better agreement with the observed properties. This
confirms the gravitational processes are insufficient to fully describe the
ICM properties.

##  7 Summary and Future outlook

Until the 90s the largest cluster samples with X-ray data were mostly drawn
from optical surveys. In most cases the samples were completely inhomogeneous
and the selection function unknown. A big step was achieved with ROSAT that
detected a few thousand galaxy clusters in its all sky-survey, and allowed the
construction of flux-selected samples of several tens to hundreds of clusters
(e.g., HIFLUGCS, Reiprich & Böhringer 2002; REXCESS, Böhringer et al. 2007).
More recently, the SZ surveys provided new catalogs of clusters which have
been followed-up in X-ray.

In most cases the samples contained of order of one hundred objects, many of
them in the local Universe, with only a few objects above redshift 1. In the
near future, surveys across the electromagnetic spectrum will produce samples
of tens of thousands objects out to redshift 2-3.

In the optical and near-infrared bands, Nancy Grace Roman Space Telescope,
Vera Rubin Observatory, and Euclid will provide cluster detections and precise
redshift and weak-lensing measurements to complement studies of the ICM
properties.

Next-generation SZ surveys (e.g., the Simons observatory, CMB-S4) are forecast
to detect over 1,000 clusters at $z>2$. This will allow studies into the epoch
of cluster formation and an evolutionary timeline that are inaccessible with
current catalogs.

Meanwhile, large radio surveys (e.g., LOFAR, MWA, SKA) can also provide useful
information to investigate the dynamic and thermodynamic structures observed
in the other wavelengths. Moreover, the detection of radio halos and relics
can provide information on the acceleration of particles during cluster
mergers providing further insights into the virialization process.

Studies of the X-ray scaling relations will be advanced by dedicated X-ray
surveys along with follow-up observations of the ICM in distant clusters
detected at other wavelengths.

###  7.1 eROSITA

The extended ROentgen Survey with an Imaging Telescope Array (eROSITA, Predehl
et al. 2021) is operating at L2 orbit on-board the ’Spectrum-Roentgen-Gamma’
satellite. eROSITA consists of seven co-aligned X-ray mirror modules covering
the soft band (i.e, 0.2-10 keV), with a large field of view and moderate
spatial resolution. Thanks to the combination of these characteristics eROSITA
is 20–30 times more sensitive than the ROSAT sky survey in the soft band and
is providing the first all sky imaging survey in the hard band (i.e., 2-10
keV). eROSITA is planned to perform a 4-yr-long all-sky survey (started in
December 2019) and is expected to detect $>$105 groups and clusters with mass
$M_{\text{500}}$$>$1013M⊙ out to redshift $\sim 1$ (median: $z\sim$0.3; e.g.,
Pillepich et al. 2012, Borm et al. 2014). This large number of clusters will
allow to significantly improve the constraints of several cosmological
parameters. However, these goals will be achieved only if we will be able to
improve significantly our knowledge of the LX–M (and other) scaling relation.
Most of the systems detected with eROSITA will have too few photons to measure
temperature profiles (Borm et al. 2014), and so the masses required for
cosmology will need to be estimated through scaling relations.

###  7.2 ATHENA

To be launched in the 2030s, the Advanced Telescope for High ENergy
Astrophysics (ATHENA) will comprise two instruments: the X-ray Integral Field
Unit (X-IFU; Barret et al. 2016) and the Wide Field Imager (WFI; Rau et al.
2013). While the former will allow spectroscopic investigation with
unprecedented resolving power the latter will offer unrivaled survey power
(over small to medium sized fields) with X-ray imaging with good spatial
resolution. The WFI survey is expected to detect 104 objects at $z>$0.5
including a few tens of clusters at $z>2$, and measure their temperature to
better than 25% accuracy (e.g., Zhang et al. 2020).

Perhaps the most exciting aspect of ATHENA will be its ability to study the
ICM properties in follow-up observations of the most distant clusters detected
in other surveys. With such observations it will be possible to transform the
currently poor constraints on the evolution of the scaling relations and to
determine the history of the physical processes that dominate the injection of
non-gravitational energy in the ICM.

##  8 Acknowledgements

The authors thank A. Mantz and S. Ettori for providing an adapted version of
previous published figures. We also thank S. Ettori for useful comments and
suggestions that helped to improve and clarify the presentation of this work.
LL acknowledges financial contribution from the contracts ASI-INAF Athena
2019-27-HH.0, “Attività di Studio per la comunità scientifica di Astrofisica
delle Alte Energie e Fisica Astroparticellare” (Accordo Attuativo ASI-INAF n.
2017-14-H.0), and from INAF “Call per interventi aggiuntivi a sostegno della
ricerca di main stream di INAF”. BJM acknowledges support from the Science and
Technology Facilities Council (grant number ST/V000454/1).

## References

  * Akritas & Bershady (1996) Akritas, M. G. & Bershady, M. A. 1996, ApJ, 470, 706
  * Andrade-Santos et al. (2017) Andrade-Santos, F., Jones, C., Forman, W. R., et al. 2017, ApJ, 843, 76
  * Andrade-Santos et al. (2021) Andrade-Santos, F., Pratt, G. W., Melin, J.-B., et al. 2021, ApJ, 914, 58
  * Andreon & Hurn (2013) Andreon, S. & Hurn, M. 2013, Statistical Analysis and Data Mining: The ASA Data Science Journal, 9, 15
  * Andreon & Hurn (2010) Andreon, S. & Hurn, M. A. 2010, MNRAS, 404, 1922
  * Andreon et al. (2016) Andreon, S., Serra, A. L., Moretti, A., & Trinchieri, G. 2016, A&A, 585, A147
  * Angulo et al. (2012) Angulo, R. E., Springel, V., White, S. D. M., et al. 2012, MNRAS, 426, 2046
  * Arnaud et al. (2007) Arnaud, M., Pointecouteau, E., & Pratt, G. W. 2007, A&A, 474, L37
  * Barret et al. (2016) Barret, D., Lam Trong, T., den Herder, J.-W., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99052F
  * Becker & Kravtsov (2011) Becker, M. R. & Kravtsov, A. V. 2011, ApJ, 740, 25
  * Biffi et al. (2018) Biffi, V., Dolag, K., & Merloni, A. 2018, MNRAS, 481, 2213
  * Böhringer et al. (2012) Böhringer, H., Dolag, K., & Chon, G. 2012, A&A, 539, A120
  * Böhringer et al. (2007) Böhringer, H., Schuecker, P., Pratt, G. W., et al. 2007, A&A, 469, 363
  * Böhringer & Werner (2010) Böhringer, H. & Werner, N. 2010, A&A Rev., 18, 127
  * Borm et al. (2014) Borm, K., Reiprich, T. H., Mohammed, I., & Lovisari, L. 2014, A&A, 567, A65
  * Bryan & Norman (1998) Bryan, G. L. & Norman, M. L. 1998, ApJ, 495, 80
  * Bulbul et al. (2019) Bulbul, E., Chiu, I.-N., Mohr, J. J., et al. 2019, ApJ, 871, 50
  * CHEX-MATE Collaboration et al. (2021) CHEX-MATE Collaboration, Arnaud, M., Ettori, S., et al. 2021, A&A, 650, A104
  * Chiu et al. (2021) Chiu, I.-N., Ghirardini, V., Liu, A., et al. 2021, arXiv e-prints, arXiv:2107.05652
  * Costanzi et al. (2019) Costanzi, M., Rozo, E., Rykoff, E. S., et al. 2019, MNRAS, 482, 490
  * da Silva et al. (2004) da Silva, A. C., Kay, S. T., Liddle, A. R., & Thomas, P. A. 2004, MNRAS, 348, 1401
  * Eckert et al. (2016) Eckert, D., Ettori, S., Coupon, J., et al. 2016, A&A, 592, A12
  * Eckert et al. (2020) Eckert, D., Finoguenov, A., Ghirardini, V., et al. 2020, The Open Journal of Astrophysics, 3, 12
  * Eckert et al. (2021) Eckert, D., Gaspari, M., Gastaldello, F., Le Brun, A. M. C., & O’Sullivan, E. 2021, Universe, 7, 142
  * Eckert et al. (2011) Eckert, D., Molendi, S., & Paltani, S. 2011, A&A, 526, A79
  * Eddington (1913) Eddington, A. S. 1913, MNRAS, 73, 359
  * Eddington (1940) Eddington, A. S., S. 1940, MNRAS, 100, 354
  * Efstathiou & Fall (1984) Efstathiou, G. & Fall, S. M. 1984, MNRAS, 206, 453
  * Ettori (2013) Ettori, S. 2013, MNRAS, 435, 1265
  * Ettori (2015) Ettori, S. 2015, MNRAS, 446, 2629
  * Ettori et al. (2013) Ettori, S., Donnarumma, A., Pointecouteau, E., et al. 2013, Space Sci. Rev., 177, 119
  * Ettori et al. (2012) Ettori, S., Rasia, E., Fabjan, D., Borgani, S., & Dolag, K. 2012, MNRAS, 420, 2058
  * Evrard et al. (2014) Evrard, A. E., Arnault, P., Huterer, D., & Farahi, A. 2014, MNRAS, 441, 3562
  * Fabian (2012) Fabian, A. C. 2012, ARA&A, 50, 455
  * Fabjan et al. (2010) Fabjan, D., Borgani, S., Tornatore, L., et al. 2010, MNRAS, 401, 1670
  * Farahi et al. (2019) Farahi, A., Mulroy, S. L., Evrard, A. E., et al. 2019, Nature Communications, 10, 2504
  * Gaspari et al. (2014) Gaspari, M., Brighenti, F., Temi, P., & Ettori, S. 2014, ApJ, 783, L10
  * Giles et al. (2015) Giles, P. A., Maughan, B. J., Hamana, T., et al. 2015, MNRAS, 447, 3044
  * Giles et al. (2016) Giles, P. A., Maughan, B. J., Pacaud, F., et al. 2016, A&A, 592, A3
  * Giodini et al. (2013) Giodini, S., Lovisari, L., Pointecouteau, E., et al. 2013, Space Sci. Rev., 177, 247
  * Gonzalez et al. (2007) Gonzalez, A. H., Zaritsky, D., & Zabludoff, A. I. 2007, ApJ, 666, 147
  * Jones et al. (1998) Jones, L. R., Scharf, C., Ebeling, H., et al. 1998, ApJ, 495, 100
  * Kaastra et al. (2008) Kaastra, J. S., Paerels, F. B. S., Durret, F., Schindler, S., & Richter, P. 2008, Space Sci. Rev., 134, 155
  * Kaiser (1986) Kaiser, N. 1986, MNRAS, 222, 323
  * Kelly (2007) Kelly, B. C. 2007, ApJ, 665, 1489
  * Kettula et al. (2013) Kettula, K., Nevalainen, J., & Miller, E. D. 2013, A&A, 552, A47
  * Kravtsov et al. (2006) Kravtsov, A. V., Vikhlinin, A., & Nagai, D. 2006, ApJ, 650, 128
  * Le Brun et al. (2014) Le Brun, A. M. C., McCarthy, I. G., Schaye, J., & Ponman, T. J. 2014, MNRAS, 441, 1270
  * Loewenstein (2000) Loewenstein, M. 2000, ApJ, 532, 17
  * Lovisari et al. (2021) Lovisari, L., Ettori, S., Gaspari, M., & Giles, P. A. 2021, Universe, 7, 139
  * Lovisari et al. (2017) Lovisari, L., Forman, W. R., Jones, C., et al. 2017, ApJ, 846, 51
  * Lovisari et al. (2015) Lovisari, L., Reiprich, T. H., & Schellenberger, G. 2015, A&A, 573, A118
  * Lovisari et al. (2020) Lovisari, L., Schellenberger, G., Sereno, M., et al. 2020, ApJ, 892, 102
  * Mahdavi et al. (2013) Mahdavi, A., Hoekstra, H., Babul, A., et al. 2013, ApJ, 767, 116
  * Malmquist (1922) Malmquist, K. G. 1922, Meddelanden fran Lunds Astronomiska Observatorium Serie I, 100, 1
  * Mantz (2016) Mantz, A. B. 2016, MNRAS, 457, 1279
  * Mantz et al. (2016a) Mantz, A. B., Allen, S. W., Morris, R. G., & Schmidt, R. W. 2016a, MNRAS, 456, 4020
  * Mantz et al. (2018) Mantz, A. B., Allen, S. W., Morris, R. G., & von der Linden, A. 2018, MNRAS, 473, 3072
  * Mantz et al. (2016b) Mantz, A. B., Allen, S. W., Morris, R. G., et al. 2016b, MNRAS, 463, 3582
  * Mantz et al. (2015) Mantz, A. B., von der Linden, A., Allen, S. W., et al. 2015, MNRAS, 446, 2205
  * Mathiesen & Evrard (2001) Mathiesen, B. F. & Evrard, A. E. 2001, ApJ, 546, 100
  * Maughan (2007) Maughan, B. J. 2007, ApJ, 668, 772
  * Maughan (2014) Maughan, B. J. 2014, MNRAS, 437, 1171
  * Maughan et al. (2012) Maughan, B. J., Giles, P. A., Randall, S. W., Jones, C., & Forman, W. R. 2012, MNRAS, 421, 1583
  * Mazzotta et al. (2004) Mazzotta, P., Rasia, E., Moscardini, L., & Tormen, G. 2004, MNRAS, 354, 10
  * McCarthy et al. (2017) McCarthy, I. G., Schaye, J., Bird, S., & Le Brun, A. M. C. 2017, MNRAS, 465, 2936
  * McCarthy et al. (2011) McCarthy, I. G., Schaye, J., Bower, R. G., et al. 2011, MNRAS, 412, 1965
  * McCarthy et al. (2010) McCarthy, I. G., Schaye, J., Ponman, T. J., et al. 2010, MNRAS, 406, 822
  * McDonald et al. (2017) McDonald, M., Allen, S. W., Bayliss, M., et al. 2017, ApJ, 843, 28
  * McNamara & Nulsen (2007) McNamara, B. R. & Nulsen, P. E. J. 2007, ARA&A, 45, 117
  * Melin et al. (2005) Melin, J. B., Bartlett, J. G., & Delabrouille, J. 2005, A&A, 429, 417
  * Menci & Cavaliere (2000) Menci, N. & Cavaliere, A. 2000, MNRAS, 311, 50
  * Mitchell et al. (1979) Mitchell, R. J., Dickens, R. J., Burnell, S. J. B., & Culhane, J. L. 1979, MNRAS, 189, 329
  * Miyazaki et al. (2018) Miyazaki, S., Oguri, M., Hamana, T., et al. 2018, PASJ, 70, S27
  * Narayan & Medvedev (2001) Narayan, R. & Medvedev, M. V. 2001, ApJ, 562, L129
  * Nelson et al. (2012) Nelson, K., Rudd, D. H., Shaw, L., & Nagai, D. 2012, ApJ, 751, 121
  * Nevalainen et al. (2010) Nevalainen, J., David, L., & Guainazzi, M. 2010, A&A, 523, A22
  * Oppenheimer et al. (2021) Oppenheimer, B. D., Babul, A., Bahé, Y., Butsky, I. S., & McCarthy, I. G. 2021, Universe, 7, 209
  * Osmond & Ponman (2004) Osmond, J. P. F. & Ponman, T. J. 2004, MNRAS, 350, 1511
  * Panagoulia et al. (2014) Panagoulia, E. K., Fabian, A. C., & Sanders, J. S. 2014, MNRAS, 438, 2341
  * Peterson et al. (2001) Peterson, J. R., Paerels, F. B. S., Kaastra, J. S., et al. 2001, A&A, 365, L104
  * Pillepich et al. (2012) Pillepich, A., Porciani, C., & Reiprich, T. H. 2012, MNRAS, 422, 44
  * Ponman et al. (1999) Ponman, T. J., Cannon, D. B., & Navarro, J. F. 1999, Nature, 397, 135
  * Ponman et al. (2003) Ponman, T. J., Sanderson, A. J. R., & Finoguenov, A. 2003, MNRAS, 343, 331