###### Abstract

Galaxy groups and poor clusters are more common than rich clusters, and host
the largest fraction of matter content in the Universe. Hence, their studies
are key to understand the gravitational and thermal evolution of the bulk of
the cosmic matter. Moreover, because of their shallower gravitational
potential, galaxy groups are systems where non-gravitational processes (e.g.,
cooling, AGN feedback, star formation) are expected to have a higher impact on
the distribution of baryons, and on the general physical properties, than in
more massive objects, inducing systematic departures from the expected scaling
relations.
Despite their paramount importance from the astrophysical and cosmological
point of view, the challenges in their detection have limited the studies of
galaxy groups. Upcoming large surveys will change this picture, reassigning to
galaxy groups their central role in studying the structure formation and
evolution in the Universe, and in measuring the cosmic baryonic content.
Here, we review the recent literature on various scaling relations between
X-ray and optical properties of these systems, focusing on the observational
measurements, and the progress in our understanding of the deviations from the
self-similar expectations on groups’ scales. We discuss some of the sources of
these deviations, and how feedback from supernovae and/or AGNs impacts the
general properties and the reconstructed scaling laws. Finally, we discuss
future prospects in the study of galaxy groups.

###### keywords:

galaxy groups; X-ray and optical observations; intragroup medium/plasma;
active galactic nuclei; hydrodynamical simulations

7 5 139 Received: 31 March 2021; Accepted: 28 April 2021; Published: 10 May
2021 Scaling Properties of Galaxy Groups Lorenzo Lovisari1,2,‡, Stefano
Ettori1,3,‡, Massimo Gaspari1,4,‡, and Paul A. Giles5,‡ Lorenzo Lovisari,
Stefano Ettori, Massimo Gaspari, and Paul Giles All authors contributed
equally to this work.

##  1 Introduction

Following the hierarchical scenario of structure formation, galaxy systems
form through episodic mergers of small mass units. The less massive ones
(often referred as groups), are the building blocks for the most massive ones
(clusters), and trace the filamentary components of the large scale structure
(e.g., Eke et al. 2004). However, the distinction between groups and clusters
is quite loose and no universal definition exists in literature. Also, because
the halo mass function is continuous, a naive starting point would be to not
single out the low-mass end objects. Nonetheless, these poor systems have some
notable differences (e.g., lack of dominance of the gas mass over the
stellar/galactic component; Giodini et al. 2009) with respect to their more
massive counterpart and they cannot be simply considered their scaled-down
versions.

A conventional "rule of thumb" definition is to label systems of less than 50
galaxies as groups and above as clusters. More in general, galaxy groups have
been broadly classified into three main classes on the basis of their optical
and physical characteristics: poor/loose groups, compact groups, and fossil
groups (e.g., Eigenthaler & Zeilinger 2007). Poor/loose groups are aggregate
of galaxies with a space density of $\sim$10-5 Mpc-3 (e.g., Nolthenius & White
1987). Compact groups are small and relatively isolated systems of typically
4-10 galaxies with a space density of $\sim$10-6 Mpc-3 (e.g., Hickson 1982).
Fossil groups are objects dominated by a single bright elliptical galaxy (a
formal definition is provided in Jones et al. 2003). Early studies (e.g.,
Helsdon & Ponman 2000a) showed that subsamples of loose and compact groups
share the same scaling relations. Thus, in this review, we do not make
distinction between poor/loose and compact groups, and hereafter we simply
refer to them as galaxy groups. The properties of fossil groups are instead
discussed in the companion review by Aguerri & Zarattini (2021). However,
since the optical properties are not always available, a threshold of
M$\sim$1014M⊙, corresponding to a temperature of 2-3 keV, is also often used
to classify these systems. We will show later that this threshold roughly
corresponds to the temperature for which there is a significant change in the
X-ray emissivity.

Despite the crucial role played by groups in cosmic structure formation and
evolution, they have received less attention compared to massive clusters. One
of the reasons is that typical groups contain only a few bright galaxies in
their inner regions, making very difficult to detect them in optical with a
relatively good confidence. A much easier method of detecting them is to study
the X-ray emission from the hot intragroup medium (IGrM). The detection of hot
plasma carries witness that galaxy groups (and clusters) are not simple
conglomerate of galaxies put together by projection effects, but real physical
systems which are undergoing some degree of virialization. Galaxy groups often
show lower and flatter X-ray surface brightness than clusters (e.g., Ponman et
al. 1999, Sanderson et al. 2003). Therefore, the physical properties of the
gas derived for galaxy groups are presumably less robust than the properties
derived for galaxy clusters. Nonetheless, they represent a more common
environment because the mass function of virialized systems, which describes
the number density of clusters above a threshold mass M, is higher at lower
masses (with a factor of $\sim$30/210/1500 more objects in the mass range
M${}_{500}=10^{13}$ M${}_{\odot}-{\rm M}_{1}$ than in M${}_{500}>{\rm M}_{1}$,
and M${}_{1}=1/2/5\times 10^{14}{\rm M}_{\odot}$ at $z=0$; see, e.g., Despali
et al. 2016). Hence, the detection and characterization of galaxy groups is
especially important for astrophysical and cosmological studies.

###  1.1 Galaxy groups and astrophysics

Galaxy groups cover the intermediate mass range between large elliptical
galaxies and galaxy clusters and contain the bulk of all galaxies and baryonic
matter in the local Universe (e.g., Tully 1987; Fukugita et al. 1998, Eke et
al. 2004). Because of that, they are crucial for understanding the effects of
the local environment on galaxy formation and evolution processes. Moreover,
the feedback from supernovae (SNe) and supermassive black holes (SMBHs) is
expected to alter significantly the properties of these systems being the
energy input associated with these sources comparable to the binding energies
of groups (e.g., Brighenti & Mathews 2002; McCarthy et al. 2010; Gaspari et
al. 2012). However, the relative contributions of the different feedback
processes is still a matter of debate and it will constitute a major subject
of research for the next decade. These factors make galaxy groups great
laboratories to understand the complex baryonic physics involved, and to study
the differences with their massive counterpart. For instance, we know that the
fraction of strong cool-cores (CC; i.e., systems with a central cooling time
$t_{\rm cool}<$ 1 Gyr, as described in Hudson et al. 2010), weak cool-cores (1
$<t_{\rm cool}<$ 7.7 Gyr), and non cool-cores (NCC; $t_{\rm cool}>$ 7.7 Gyr)
objects at the group scale are similar to those in galaxy clusters (Bharadwaj
et al. 2014). However, O’Sullivan et al. (2017) found that the CC fraction
increases dramatically when the samples are restricted to low-temperature
systems (i.e., kT$<$1.5 keV) showing a correlation between system temperature
and CC status. Bharadwaj et al. (2014) also found that brightest group
galaxies have a higher stellar mass than brightest cluster galaxies,
suggesting that there is less gas available to feed the SMBHs. Recent results
suggest that the IGrM and intracluster medium (ICM) are also providing a
source of gas which feeds and grows the central SMBHs, in particular leading
to novel scaling relations between the SMBH mass and the X-ray properties of
their host gaseous halos (e.g., Bogdán et al. 2018; Gaspari et al. 2019;
Lakhchaura et al. 2019). These findings imply an interplay between the
feedback mechanisms connected with the SMBHs and the macro-scale halos, which
could explain some features of cosmological simulations driving a relative
break of the Lx–Tx and Lx–M relations at low temperatures (e.g., Sijacki et
al. 2007; Puchwein et al. 2008; Fabjan et al. 2010; McCarthy et al. 2010; Le
Brun et al. 2014). This deviation is often attributed to active galactic
nucleus (AGN) feedback (e.g., Planelles et al. 2014; Gaspari et al. 2014;
Truong et al. 2018).

The properties described above have an important effect on the correlation
between different physical quantities. For instance, it is well established
that CC and NCC objects populate different regions of the X-ray luminosity
space of any scaling relations (e.g., Markevitch 1998; Pratt et al. 2009;
Mittal et al. 2011; Bharadwaj et al. 2015; Mantz et al. 2016b; Lovisari et al.
2020). Therefore, a change in the fraction of CC/NCC systems as a function of
the temperature (mass) will have an impact to the slope, normalization, and
scatter of the observed scaling relations. Hence, it is crucial to have a full
coverage for the whole sample to minimize the systematic errors due to the
incompleteness. In fact, if all the missing objects happen to belong to one of
the subsamples (e.g., NCC), the normalization (and the scatter) of the studied
scaling relations will be wrong. Moreover, the CC/NCC fraction of systems in a
sample depends on the selection function and may not be representative of the
underlying population. For instance, X-ray selected samples are known to be
biased toward centrally peaked and relaxed systems, in particular in the low-
mass regime (Eckert et al. 2011). In fact, recent results by O’Sullivan et al.
(2017), who analyzed a sample of optically selected groups, show that
$\sim$20% of X-ray bright groups (probably the most disturbed ones, or with no
concentrated CC) in the local Universe may have been missed. Thus, the scaling
relations of galaxy groups (and clusters) are the result of the various
processes that govern the formation and evolution of these systems making them
ideal targets for studying the effect of the interplay between galaxy
evolution, the development of the IGrM, and feedback.

###  1.2 Galaxy groups and cosmology

Clusters of galaxies have proven to be remarkably effective probes of
cosmology (e.g., Reiprich & Böhringer 2002; Allen et al. 2008; Vikhlinin et
al. 2009; Rozo et al. 2010; Mantz et al. 2015; de Haan et al. 2016; Planck
Collaboration et al. 2016; Böhringer et al. 2017; Schellenberger & Reiprich
2017b; Pacaud et al. 2018; Bocquet et al. 2019. However, since galaxy groups
represent a large fraction of the number density of virialized systems, their
impact might be relevant (in particular, on the reconstruction of the halo
mass function). For instance, recent results of the Dark Energy Survey (DES)
collaboration show that the $\sigma_{8}$–$\Omega_{\rm M}$ posteriors have a
5.6$\sigma$ tension with Planck CMB results, and a 2.4$\sigma$ tension with
galaxy clustering and cosmic shear results (Abbott et al. 2018). The cause of
this tension is thought to reside at the low-mass (low richness) end of the
cluster population, specifically, clusters with a richness of $\lambda<30$
(corresponding to $\sim$1014 M⊙). The removal of low richness systems from the
analysis significantly reduces the tension with comparative cosmological
probes. However, various tests undertaken in Abbott et al. (2020) suggest that
the discrepancy is probably due to the modelling of the weak lensing signal
rather than the group and cluster abundance. The mass calibration for the
DESY1 analysis is based upon a stacked weak lensing analysis, through
application of the weak lensing–richness relation (McClintock et al. 2019).
This relation is derived over the full richness range, which would not account
for any deviations at the low-mass end. Furthermore, since the mass analysis
relies on stacked quantities, information on scatter in mass with richness is
lost and must be informed from external relations. In the case of the DESY1
analysis, the mass scatter information is inferred from the
temperature–richness relation using X-ray data (Farahi et al. 2019). This
scatter is assumed constant with richness, which again, could evolve as a
function of richness. The investigation of these effects will become of
critical importance as the low-mass end of the mass scales are increasingly
probed by future surveys (e.g., those constructed from the Legacy Survey of
Space and Time undertaken by the Vera C. Rubin Observatory).

Excluding low-mass systems significantly reduces the cosmological parameter
constraints. Thus, despite the important complications present at the group
scale, it is becoming generally appreciated that galaxy groups should be
included in the cosmological analysis. In order to use them to constrain the
cosmological parameters we need a good knowledge of the selection function to
properly correct for the incompleteness, otherwise studies employing the
cluster mass function may find lower $\Omega_{\rm M}$ and/or $\sigma_{8}$
values than the true values. This scenario is supported by the finding of
Schellenberger & Reiprich (2017b), who showed how the increasing
incompleteness of parent samples in the low-mass regime together with a
steeper Lx–M relation observed for groups, can lead to biased cosmological
parameters. It is worth noticing that, if a large fraction of galaxy systems
is missed, than the tension between cluster counts and primary CMB
cosmological constraints may become less severe.

Most of the upcoming large surveys will push the measurements down to the low-
mass regime. Thus, to fully exploit the future datasets to constrain the
cosmological parameters, we need to properly characterize the properties of
galaxy groups and the differences with galaxy clusters, accounting for the
different selection effects, and estimating the amplitude of the various
biases.

###  1.3 This review

In this work, we present an overview of the most recent studies on scaling
relations between a number of integrated observed quantities of galaxy groups,
and complement/update the previous reviews in the field by, e.g., Mulchaey
(2000) and Sun (2012). The review is organized as follows. In Section 2, we
derive the self-similar X-ray scaling relations and overview the observed
deviations. In Section 3, we discuss the relations between X-ray and optical
properties. In Section 4, we discuss the relation between the SMBH mass and
the global group quantities. In Section 5, we shortly discuss the most
relevant upcoming missions and their expected contribution to the field. In
Section 6 we provide our final remarks.

##  2 X-ray scaling relations

###  2.1 Theoretical expectations

The X-ray scaling relations for galaxy systems were derived by Kaiser (1986),
based on the simple assumption that the thermodynamic properties of the ICM
are only determined by gravity (i.e., gas just follow the dark matter
collapse). Since gravity is scale free, this model predicts that objects of
different sizes are the scaled version of each other. For that reason, this
model is often referred as self-similar, and the derivation of the predicted
relations has been extensively covered in the literature (e.g., Kitayama &
Suto 1996; Bryan & Norman 1998; Voit 2005; Maughan et al. 2006; Borgani et al.
2008; Böhringer et al. 2012; Ettori 2013; Giodini et al. 2013; Maughan 2014;
Ettori 2015; Ettori et al. 2020). Here, we only provide a brief review of the
standard derivation of the self-similar scaling relations for massive systems,
and then extend them, when necessary, to the low-mass regime where gas physics
is playing a significant role.

In the self-similar scenario, two galaxy systems which have formed at the same
time have the same mean density. Hence,

$\frac{{\rm M}_{\Delta_{z}}}{{\rm R}^{3}_{\Delta_{z}}}={\rm constant}$ (1)

where M${}_{\Delta_{z}}$ is the mass contained within the radius
R${}_{\Delta_{z}}$, encompassing a mean density ${\Delta_{z}}$ times the
critical density of the Universe $\rho_{c}(z)$, so that
M${}_{\Delta_{z}}\propto\rho_{c}(z){\Delta_{z}}{\rm R}^{3}_{\Delta_{z}}$. The
critical density of the Universe scales with redshift as
$\rho_{c}(z)$=$\rho_{c(z=0)}$E${}^{2}(z)$, where E$(z)$=Hz/H0 describes the
evolution of the Hubble parameter with redshift $z$.

During the gravitational collapse the gas density increases, and a shock
propagates outward from the cluster center and heats the gas. After the
passage of the shock, IGrM and ICM can be considered in hydrostatic
equilibrium, so the temperature Tx provides an estimate of the gravitational
potential well (i.e., T${}_{\rm x}\propto$
GM${}_{\Delta_{z}}$/R${}_{\Delta_{z}}\propto{\rm R}_{\Delta_{z}}^{2}$), and
therefore of the total mass of the cluster:

${\rm M}_{\Delta_{z}}\propto{\rm T_{\rm x}}^{3/2}.$ (2)

In the self-similar scenario, where the gas fraction, fg, of galaxy groups and
clusters is universal, one expects for the total gas mass, Mg, a similar
dependence on the gas temperature: M${}_{\rm g}\propto{\rm T_{\rm x}}^{3/2}$.

The hot gas in galaxy systems is typically described as an optically thin
plasma in collisional ionisation equilibrium. Its X-ray emissivity (i.e., the
energy emitted per time and volume) is equal to

$\epsilon={\rm n_{e}}\,{\rm n_{p}}\,\Lambda({\rm T_{x}},{\rm Z}_{\odot}),$ (3)

where ${\rm n_{e}}$ and ${\rm n_{p}}$ are the number densities of electrons
and protons, respectively, that are related to the gas mass density $\rho_{\rm
g}$ through the relation $\rho_{\rm g}$=$\mu{\rm m_{p}}({\rm n_{e}}+{\rm
n_{p}})$, $\mu$ is the mean molecular weight ($\sim$0.6 for a plasma with
solar abundance), ${\rm m_{p}}$ is the proton mass, and $\Lambda({\rm
T_{x}},{\rm Z}_{\odot})$ is the cooling function which depends on the
mechanism of the emission111Three main processes contribute to the X-ray
emission: thermal bremsstrahlung (due to the deflection of a free electron by
the electric field of a ion), recombination (due to the capture of an electron
by an ion), and two-photon decay (due to the changing of the quantum level of
an electron in an ion). See details in the reviews from, e.g., Sarazin (1986),
Peterson & Fabian (2006), Kaastra et al. (2008), and Böhringer & Werner
(2010). and on the considered energy window. At high temperatures (i.e.,
kT$>$3 keV) the main mechanism of emission is thermal bremsstrahlung and the
cooling function in the full energy band mainly depends only on Tx (i.e.,
$\Lambda({\rm T_{x}},{\rm Z}_{\odot})\propto$T${}_{\rm x}^{1/2}$). Thus, for
sufficiently massive systems the bolometric X-ray luminosity (i.e., 0.01-100
keV band) is given by

L${}_{\rm x,bol}\propto\int\epsilon$ dV $\propto$ n${}_{\rm p}^{2}$ T${}_{\rm
x}^{1/2}$R${}^{3}\propto$ f${}_{\rm g}^{2}$ T${}_{\rm x}^{2}\propto$ T${}_{\rm
x}^{2}$ (4)

with the last scaling obtained assuming a constant gas fraction as predicted
by the self-similar scenario. By combining Equations 2 and 4 one obtains the
well known relation between bolometric luminosity and total mass (i.e.,
L${}_{\rm x,bol}\propto$M4/3).

Figure 1: (left panel): total X-ray emissivity as function of the plasma
temperature in different energy bands (bolometric in blue, 0.1–2.4 keV in
magenta, and 0.5–2 keV in green). The curves are calculated using an APEC
(Smith et al. 2001) model (v3.0.9) in XSPEC (Arnaud 1996) for two different
values of metallicity: 1.0 (solid lines) and 0.3 times the solar abundance as
in Asplund et al. (2009). All curves are normalized to the bolometric
emissivity at kBTx = 20 keV with Z = 1 Z⊙. (middle panel): the emissivity
slope as a function of temperature showing the impact of the different
$Z_{\odot}$ and Tx in the low-temperature regime. (right panel):
bremsstrahlung emission fraction (Lbrem/Ltot) as a function of the
temperature, illustrating the increasing contribution of line emission to the
total luminosity for low-temperature plasmas.

In the literature the X-ray luminosities are also often provided in soft
energy bands (e.g., 0.1-2.4 or 0.5-2 keV), more representative of the bandpass
covered by current (and past) X-ray facilities used for the study of groups
and clusters. In Figure 1 (left panel), we show that for massive systems with
typical cluster abundance the X-ray emissivity in soft band is almost
independent of the system temperature (e.g., for Z=0.3Z⊙ the change in
$\epsilon$ between 3 and 10 keV plasmas in the 0.5-2 keV band is $<$10% for
given emission measure), so that L${}_{\rm x,soft}\propto$T${}_{\rm x}^{3/2}$,
and hence using Equation 2, L${}_{\rm x,soft}\propto$M.

However, the gas fraction is not constant, with a difference of almost a
factor of two between groups and clusters (e.g., Vikhlinin et al. 2006;
Gonzalez et al. 2007; Gastaldello et al. 2007, Pratt et al. 2009; Dai et al.
2010; Gonzalez et al. 2013; Lovisari et al. 2015; Eckert et al. 2016; see also
the companion reviews by Eckert et al. 2021 and Oppenheimer et al. 2021).
Moreover, at low temperatures, line cooling becomes very important, and the
emissivity (both in soft and bolometric bands) becomes strongly abundance (Z⊙)
and temperature dependent. In Figure 1 (left and middle panels) we show the
dependence of the emissivity on the temperature and metallicity for widely
used energy bands for scaling relations, clearly showing that a simple scaling
cannot be derived. In Table LABEL:table:em, we provide the dependence for a
set of interesting cases.

The complexity of the emissivity function in the low-temperature regime may
lead to a wrong interpretation of the results of scaling relation studies. In
fact, it is conventional to compare the slopes of the scaling relations
obtained with sample of groups to the self-similar predictions derived for
massive clusters. However, if there is no feedback (i.e., the relations follow
the self-similar predictions), then the Lx–Tx and Lx–M relations should
flatten at low temperatures and masses. Thus, without accounting for the
increasing contribution of the line emission in the low-temperature regime,
one could interpret the agreement between group and cluster relations such
that feedback processes play a negligible role in shaping the IGrM. Thus, the
impact of the feedback could be underestimated. To visualize the contribution
of line emission as function of the temperature, we follow the simple approach
of Zou et al. (2016) in which we measure the luminosity (Ltot) in different
energy bands (i.e., bolometric, 0.1-2.4, and 0.5-2) of APEC spectra with a
metal abundance of Z⊙=1.0 (not rare at the center of galaxy groups, see
companion review by Gastaldello et al. 2021), and then setting Z⊙=0 without
changing any other parameters to approximate the luminosity of the pure
bremsstrahlung component (Lbrem). We repeated the exercise for a more standard
Z⊙=0.3. The results are shown in Figure 1 (right panel) where it is clear the
significant contribution of line emission to the total luminosity in the low-
temperature regime. Thus, the luminosity–temperature and luminosity–mass
relations can be approximated as L${}_{\rm x}\propto$ T${}_{\rm
x}^{1.5+\gamma}$ and L${}_{\rm x}\propto$ M1+γ, where $\gamma$ is the slope of
the X-ray emissivity in the considered energy band (e.g., soft or bolometric)
and temperature range covered by the systems in the studied dataset (see Table
LABEL:table:em). It follows that the self-similar Lx–Tx and Lx–M relations for
galaxy groups are expected to be significantly flatter than the ones for
galaxy clusters. It is also worth noticing that even for massive systems with
Z⊙=0.3 there is a $\sim$5% contribution from line emission. Thus, the
bolometric emissivity slope is smaller than 0.5 (i.e., the value one gets from
pure bremsstrahlung emission) with the net effect being that the correct self-
similar expectation becomes L${}_{\rm x,bol}\propto$T${}_{\rm x}^{\sim 1.9}$.

Table 1: Emissivity dependence on Tx and Z⊙ for different temperature regimes and energy bands. E band | T range |  $\epsilon$ slope (Z=0.3Z⊙) |  $\epsilon$ slope (Z=0.5Z⊙) |  $\epsilon$ slope (Z=1.0Z⊙)
---|---|---|---|---
bol | 0.4-0.7 | +0.20 | +0.16 | +0.11
| 0.4-2.0 | –0.00 | –0.14 | –0.34
| 0.4-3.0 | +0.06 | –0.07 | –0.26
| 0.4-10.0 | +0.20 | +0.11 | –0.03
| 0.7-2.0 | –0.11 | –0.30 | –0.58
| 0.7-3.0 | +0.01 | –0.15 | –0.40
| 0.7-10.0 | +0.20 | +0.10 | –0.06
| 2.0-10.0 | +0.40 | +0.36 | +0.28
| 3.0-10.0 | +0.43 | +0.41 | +0.35
0.1-2.4 | 0.4-0.7 | +0.44 | +0.42 | +0.39
| 0.4-2.0 | –0.04 | –0.19 | –0.42
| 0.4-3.0 | –0.04 | –0.18 | –0.39
| 0.4-10.0 | –0.06 | –0.16 | –0.31
| 0.7-2.0 | –0.29 | –0.52 | –0.84
| 0.7-3.0 | –0.22 | –0.40 | –0.68
| 0.7-10.0 | –0.16 | –0.27 | –0.45
| 2.0-10.0 | –0.08 | –0.12 | –0.20
| 3.0-10.0 | –0.10 | –0.12 | –0.17
0.5-2 | 0.4-0.7 | +0.63 | +0.56 | +0.50
| 0.4-2.0 | –0.03 | –0.23 | –0.48
| 0.4-3.0 | –0.02 | –0.21 | –0.45
| 0.4-10.0 | –0.04 | –0.17 | –0.35
| 0.7-2.0 | –0.38 | –0.65 | –1.00
| 0.7-3.0 | –0.27 | –0.50 | –0.81
| 0.7-10.0 | –0.18 | –0.32 | –0.52
| 2.0-10.0 | –0.06 | –0.11 | –0.22
| 3.0-10.0 | –0.07 | –0.11 | –0.18

The abundance and temperature dependence of the X-ray emissivity at low
temperatures need to be taken into account when determining the luminosities
of galaxy groups. Normally, the luminosities are estimated applying a
conversion factor to the observed count rates to obtain the X-ray fluxes. From
Figure 1 it is clear that this conversion factor in the low-temperature regime
depends strongly on the metallicity of the system. Given the observed
temperature and abundance gradients in groups (e.g., Rasmussen & Ponman 2007;
Sun et al. 2009; Mernier et al. 2017; Lovisari & Reiprich 2019; see also the
companion review by Gastaldello et al. 2021), a possible strategy is to use
the observed profiles of temperature, abundance, and surface brightness to
estimate the luminosity in each radial bin obtained during the spectral
analysis (e.g., Sun 2012, Lovisari et al. 2015). Sun (2012) pointed-out that
although the average luminosities (soft-band or bolometric) only change by
$\sim$5% when the overall values of temperature and abundance are used in the
conversion instead of the profiles, the scatter increases by 10-15%. This is
an important point to keep in mind when using survey data (e.g., ROSAT,
eROSITA) for which simple assumptions like isothermality and single overall
abundance are chosen to obtain an estimate of the luminosity.

Figure 2: Ratio between the rest-frame Lx ( left panels: bolometric, middle
panels: 0.1–2.4 keV band, right panels: 0.5–2 keV band) derived using the
abundances as in Asplund et al. (2009) and Lx obtained with the abundances of
Lodders et al. (2009, turquoise), Anders & Grevesse (1989, orange), and
Grevesse & Sauval (1998, pink), respectively. In the top panels we show the
results for systems at $z$=0.02 with a metallicity of 1.0 (solid lines) or 0.3
times the solar abundance. The dashdot line in the middle panels refers to the
simulations with a modified table of ANGR in which the Fe abundance was set
equal to the value of ASPL, showing that indeed most of the differences arise
from the significant discrepancy in Fe between ANGR and the other tables. The
0.5–2 keV band (i.e., the band used to rescale the APEC normalization) provide
the best agreement between the different tables. In the bottom panels, we show
the impact on Lx of changing abundance table for systems at different
redshifts: $z$=0.02 (solid line), $z$=0.3 (dashed), and $z$=1 (dotted,
dashdotted). The simulations were performed with Z=1Z⊙. The plot shows how the
differences increases with $z$, although the effect is generally smaller than
$\sim$5% unless very high $z$ are considered. The residual difference between
ASPL and the modified ANGR table for high $z$ objects is due to the
differences in elements other than Fe (e.g., C, N, O, Ne, Mg, Si, S).

The dependence of the cooling function on the metallicity also implies that
the use of different abundance tables can lead to different estimates of the
rest-frame X-ray luminosities. Typically, one recovers the source count rate
within a given radius from the surface brightness profile, and then obtain the
X-ray flux by setting the normalization of a thermal model (with proper
temperature and metallicity) to match the observed count rate. However, the
shape of the thermal model (which depends only on the abundance for a given
temperature and column density) can diverge at lower and higher energies than
the ones used to derive the surface brightness. To visualize the impact, we
ran a set of simulations in which the normalization of the thermal model for
systems at $z$=0.02 (i.e., median redshift of the current local group samples,
see Table 2) was set in order to match a count rate of 1 count/sec in the
0.5-2 keV energy band (i.e., the bandpass where many X-ray facilities have
most of their effective area, and often used to derive the surface-brightness
profiles) for each abundance table. Then, we estimated the luminosity in
different energy bands. In Figure 2 (top panels) we show the impact on the
estimated luminosity as function of the system temperature and common
abundance tables. There is a very good agreement in the 0.5-2 keV band
luminosity, regardless of the abundance table used for the analysis. Instead,
small differences (i.e., in the order of a few percent) in the 0.1-2.4 keV
band and bolometric luminosities arise for low temperature systems (i.e.,
kT$\lesssim$1 keV) when the abundances of Grevesse & Sauval (1998, GRSA),
Asplund et al. (2009, ASPL), or Lodders et al. (2009, LODD) are used. The
disagreement is much more significant (i.e., up to $\sim$10%) when the
luminosities are estimated with the abundance table by Anders & Grevesse
(1989, ANGR). Most of the differences are due to the much higher Fe abundance
in ANGR with respect to the other tables investigated here. When the Fe
abundance of ANGR is set to the value of ASPL (leaving unchanged all the other
ANGR abundances) the estimated luminosities are in much better agreement (see
the dashdot lines in the middle panels of Figure 2). The reason for the
differences highlighted in Figure 2 is that by switching the abundance table
we change the emissivity and the relative contribution of the line emission
with respect to the bremsstrahlung emission (see Figure 1). The difference
between the rest-frame luminosity estimated with one or another table tends to
increase at higher redshifts (see bottom panels of Figure 2). However, unless
very high redshifts are considered, the effect is usually smaller than a few
percent. In general, the soft-bands (in particular the 0.5-2 keV band) are the
ones showing a smaller impact on the estimated luminosity by switching
abundance table and should be preferred for galaxy groups studies. Although,
in most cases the effect is relatively small, it can lead to systematic
effects and should be kept in mind when comparing independent literature
results.

Another very useful quantity to describe the IGrM and ICM is the entropy which
is generated during the hierarchical assembly process. In X–ray studies of
galaxy groups and clusters, the entropy is usually defined as

K = kBTx n${}_{\rm e}^{-2/3}$ (5)

where kB is the Boltzmann constant. Entropy is conserved during adiabatic
processes and it is only modified by processes changing the physical
characteristics of the gas. Entropy increases when heat energy is introduced
and decreases when radiative cooling carries heat energy away (e.g., Voit
2005), keeping a record of the energy injection and dissipation in the
intracluster gas. Thus, entropy measurements provide a useful tool for our
understanding of the thermodynamic history of galaxy groups and clusters. Gas
entropy in galaxy groups shows a significant excess to that achievable by pure
gravitational collapse (e.g., Ponman et al. 1999; Lloyd-Davies et al. 2000;
Ponman et al. 2003; Finoguenov et al. 2007b, Sun et al. 2009; Johnson et al.
2009, Panagoulia et al. 2014), indicating a substantial IGrM heating often
ascribed to non-gravitational processes. In fact, due to the shallower
potential well of the small systems, it is expected that the energy released
by past star formation and AGN activities leaves a clear imprint on the
thermodynamic properties of IGrM and ICM (see companion reviews by Eckert et
al. 2021 and Oppenheimer et al. 2021). An effect that can be seen in both the
integrated properties (i.e., in the scaling relations) and in the shape of the
entropy profiles which are expected to follow K$\propto$R1.1.

###  2.2 Observed scaling relations

Figure 3: XMM-Newton (circles) and Chandra (squares) measurements of the
L0.1-2.4–Tx (left panel) and Lx,bol–Tx (right panel) relations for different
samples of groups: Eckmiller et al. (2011, E11), Lovisari et al. (2015, L15),
Sun et al. (2009, S09), Johnson et al. (2009, J09), Bharadwaj et al. (2015,
B15), Zou et al. (2016, Z16), Pearson et al. (2017, P17). The groups
measurements are compared with the ones from X-ray-selected (Migkas et al.
2020, M20) and SZ-selected (Lovisari et al. 2020, L20) cluster samples. We
note that different studies used different atomic models (including APEC
v1.3.1 which provide a significantly different modeling of the Fe-L line with
respect to newer versions). The luminosities are all within R500 while
temperatures are obtained in different regions (see Table 2). The Chandra
measurements are converted to XMM-Newton-like temperatures using the relations
given in Schellenberger et al. (2015). Empty symbols are from optically-
selected samples. The lines represent the fitted relation for T${}_{\rm x}<$3
keV systems (dotted), T${}_{\rm x}>$3 keV systems (dashed-dotted), all systems
(solid), and are compared with the case predicted by the self-similar scenario
(dashed). The fits have been performed with LIRA (Sereno 2016) assuming self-
similar time evolution and, conservatively, without the scatter on the X
variable, and are meant for visualization purposes only. In brown we provide
the expected values for the slope using the dependence of the emissivity
tabulated in Table LABEL:table:em for different ranges of temperatures (showed
as brown vertical lines).

The Lx–Tx relation involves two of the easiest quantities that can be derived
using X-ray data. It was one of the first X-ray correlations to be studied and
is still one of the most disputable scaling law between integrated observed
properties. In fact, there have been conflicting reports in the literature
about whether the relation for groups behaves as the one derived for massive
clusters (i.e., whether groups are simply scaled-down versions of clusters or
not). It has been clear for many years that the Lx,bol–Tx relation for massive
systems does not scale self-similarly (see, e.g., Giodini et al. 2013 for a
review about the relation for galaxy clusters), with slopes significantly
higher than 2. Although pioneering studies of the relation for galaxy groups
suggested considerably steeper slopes (i.e., slopes larger than 4; Helsdon &
Ponman 2000b, Helsdon & Ponman 2000a, Xue & Wu 2000), later investigations
found relations only slightly steeper than the ones for clusters (e.g., Osmond
& Ponman 2004, Shang & Scharf 2009, Eckmiller et al. 2011, Sun 2012, Lovisari
et al. 2015). In Figure 3, we show a compilation of data for the Lx–Tx
relation taken from recent studies of galaxy groups observed with XMM-Newton
and Chandra, and in Table 2 we list the best-fit slopes from these and other
studies. The results show that indeed the slope obtained for poor systems
(i.e., kT$<$3 keV) is consistent to the one derived for the more massive
clusters (with hints of a slightly different normalization that cause a
flattening when all the systems are fitted together). However, since the Lx–Tx
relation is expected to flatten in the low-mass regime (see Section 2.1) these
results clearly indicate a more significant contribution of the non-
gravitational processes at the group scale. In fact, feedback processes (e.g,
AGN heating) are expected to increase the entropy of the gas reducing its
density (and hence the X-ray luminosity, steepening the relation). For massive
systems the binding energy is so large that only the very central regions are
affected, and the integrated properties of galaxy clusters remain essentially
unchanged. Conversely, at the group scale the gas can be easily removed
towards or beyond the virial radius modifying their global properties. The
agreement between the Lx–Tx relation of groups and clusters seems to stand
also when the Malmquist bias (i.e., the preferential detection of
intrinsically bright objects) is accounted for in galaxy groups studies as
previously done for massive clusters. The Malmquist bias is expected to
flatten the observed X-ray relations because only objects above a certain flux
value are considered (either because one enforce an observational threshold or
because faint systems are not detected). The correction needed to recover the
underlying relation depends on the real intrinsic scatter with larger values
requiring a larger correction (i.e., if the scatter increases in the low-mass
regime then the magnitude of the flattening is larger than for galaxy
clusters). While an attempt to correct this bias in cluster scaling relations
has been provided in many different X-ray studies (e.g., Ikebe et al. 2002,
Stanek et al. 2006, Pacaud et al. 2007, Vikhlinin et al. 2009, Pratt et al.
2009, Mittal et al. 2011, Schellenberger & Reiprich 2017a), there are only a
few papers providing the correction in the galaxy group regime. For instance,
Lovisari et al. (2015) analysing an X-ray flux-selected sample of local groups
showed an increase of the L0.1-2.4–Tx relation slope after correcting for the
Malmquist bias. A similar result was obtained by Bharadwaj et al. (2015), who
estimated a correction for an archival sample of groups observed with Chandra.
In contrast to these results, are the finding by Kettula et al. (2015) and Zou
et al. (2016) who did not find any significant steepening after the bias
correction. However, all the bias corrected relations obtained in the
different studies show a great agreement (once they are converted into the
same energy band). This agreement may suggest that the observational
discrepancies arise from differences in the sample selection (which might
cause the sample to be more or less biased). Once the biases are accounted
for, then the results are not sensitive to the initial choices. Zou et al.
(2016) showed that even once selection biases are taken into account the
Lx,bol–Tx relation at the group scale is consistent with the one for clusters.
This finding confirms the stronger impact of the non-gravitational processes
in the low-mass regime (otherwise a flattening should be observed). Of course,
these corrections work under the assumption that the X-ray selected samples
are representative of the underlying population which might not be the case as
suggested by, e.g., Rasmussen et al. (2006), Anderson et al. (2015), Andreon
et al. (2016), and O’Sullivan et al. (2017) who argued that the X-ray surveys
miss a large fraction of galaxy systems. One possible reason for this
incompleteness is related to the source detection algorithms mostly based on
sliding cell detection methods. These algorithms work efficiently at finding
point-like sources but has difficulties in detecting extended features,
especially for nearby objects and for sources close to the detection limit
(e.g., Valtchanov et al. 2001; see also Šuhada et al. 2012 for a performance
comparison between sliding cell and wavelet detection algorithms). Xu et al.
(2018), using a method optimized for the extended source detection, found a
large number of new group candidates which are not included in any existing
X-ray or Sunyaev-Zel’dovich (SZ) cluster catalogs. If studies are restricted
to groups that are a priori known to be X-ray bright and which properties may
be quite different from those of optically-selected groups, as argued by
Miniati et al. (2016), then our view could be significantly biased.

Table 2: Overview of the most recent published scaling relations for galaxy
groups based on XMM-Newton and Chandra data.

Relation | N | kT (keV) | z range | slopeself | slopeobs | slopeLIRA | reference | Note
---|---|---|---|---|---|---|---|---
L–Texc | 26 | 0.6-3.0 | 0.012-0.049 | [0.9:1.1] | 2.25$\pm$0.21 | 3.27$\pm$0.26 | E11 |  a ${\dagger}$ $\square$ $\upuparrows$
L–Texc BC | 20 | 0.9-2.8 | 0.012-0.034 | [0.7:1.2] | 2.86$\pm$0.29 | - | L15 |  a $\dagger$ $\square$ $\downdownarrows$
L–Texc | 20 | 0.9-2.8 | 0.012-0.034 | [0.7:1.2] | 2.05$\pm$0.32 | 2.90$\pm$0.36 | L15 |  a $\dagger$ $\square$ $\downdownarrows$
Lexc–Texc BC | 12 | 1.7-8.2 | 0.1-0.47 | [1.3:1.5] | 2.52$\pm$0.17 | - | K15 |  a $\ddagger$ $\triangle$ $\downdownarrows$
Lexc–Texc | 12 | 1.7-8.2 | 0.1-0.47 | [1.3:1.5] | 2.65$\pm$0.17 | 2.47$\pm$1.23 | K15 |  a $\ddagger$ $\triangle$ $\downdownarrows$
L–Texc BC | 26 | 0.6-3.6 | 0.012-0.049 | [1.2:1.6] | 3.20$\pm$0.26 | - | B15 |  c $\dagger$ $\square$ $\upuparrows$
L–Texc | 26 | 0.6-3.6 | 0.012-0.049 | [1.2:1.6] | 2.17$\pm$0.26 | 3.11$\pm$0.54 | B15 |  c $\dagger$ $\square$ $\upuparrows$
L–T BC | 23 | 1.0-3.9 | 0.03-0.147 | [0.8:1.3] | 2.79$\pm$0.33 | - | Z16 |  b $\ddagger$ $\Diamond$ $\upuparrows$
L–T BC | 23 | 1.0-3.9 | 0.03-0.147 | [1.2:1.6] | 3.29$\pm$0.33 | - | Z16 |  c $\ddagger$ $\Diamond$ $\upuparrows$
L–T | 23 | 1.0-3.9 | 0.03-0.147 | [1.2:1.6] | 3.28$\pm$0.33 | 2.92$\pm$0.25 | Z16 |  c $\ddagger$ $\upuparrows$
Lexc–Texc | 23 | 1.0-3.9 | 0.03-0.147 | [1.2:1.6] | 3.81$\pm$0.46 | 3.46$\pm$0.45 | Z16 |  c $\ddagger$ $\Diamond$ $\upuparrows$
L–MHE | 26 | 0.6-3.0 | 0.012-0.049 | [0.4:0.8] | 1.34$\pm$0.18 | 1.47$\pm$0.43 | E11 |  a $\dagger$ $\upuparrows$
L–MHE BC | 20 | 0.9-2.8 | 0.012-0.034 | [0.2:0.7] | 1.66$\pm$0.22 | - | L15 |  a $\dagger$ $\downdownarrows$
L–MHE | 20 | 0.9-2.8 | 0.012-0.034 | [0.2:0.7] | 1.32$\pm$0.24 | 1.68$\pm$0.32 | L15 |  a $\dagger$ $\downdownarrows$
Lexc–MWL | 12 | 1.7-8.2 | 0.1-0.47 | [0.8:0.9] | 1.43$\pm$0.16 | 1.52$\pm$0.73 | K15 |  a $\ddagger$ $\downdownarrows$
L–MWL BC | 105 | 0.6-6.0 | 0.054-1.033 | [0.4:0.8] | 1.07$\pm$0.37 | - | S20 |  b $\ddagger$ $\downdownarrows$
MHE–Texc | 43 | 0.7-2.7 | 0.012-0.122 | 1.5 | 1.67$\pm$0.15 | 1.75$\pm$0.14 | S09 |  $\Diamond$ $\upuparrows$
MHE–Texc | 26 | 0.6-3.0 | 0.012-0.049 | 1.5 | 1.68$\pm$0.20 | 1.87$\pm$0.37 | E11 |  $\square$ $\upuparrows$
MWL–Texc | 10 | 1.2-4.6 | 0.124-0.834 | 1.5 | 1.71$\pm$0.49 | 1.46$\pm$0.58 | K13 |  $\circledcirc$ $\downdownarrows$
MHE–Texc | 20 | 0.9-2.8 | 0.012-0.034 | 1.5 | 1.65$\pm$0.07 | 1.61$\pm$0.10 | L15 |  $\square$ $\downdownarrows$
MWL–Texc BC | 12 | 1.7-8.2 | 0.1-0.47 | 1.5 | 1.52$\pm$0.17 | - | K15 |  $\triangle$ $\downdownarrows$
MWL–Texc | 12 | 1.7-8.2 | 0.1-0.47 | 1.5 | 1.68$\pm$0.17 | 1.22$\pm$0.82 | K15 |  $\triangle$ $\downdownarrows$
MWL–T300 | 76 | 0.6-6.0 | 0.044-1.002 | 1.5 | 1.33$\pm$0.75 | 1.14$\pm$0.32 | U20 |  $\boxtimes$ $\downdownarrows$
MHE–YX | 43 | 0.7-2.7 | 0.012-0.122 | 0.6 | 0.56$\pm$0.03 | 0.71$\pm$0.24 | S09 | $\upuparrows$
MHE–YX | 26 | 0.6-3.0 | 0.012-0.049 | 0.6 | 0.53$\pm$0.06 | 0.58$\pm$0.19 | E11 | $\upuparrows$
MHE–YX | 20 | 0.9-2.8 | 0.012-0.034 | 0.6 | 0.60$\pm$0.03 | 0.58$\pm$0.04 | L15 | $\downdownarrows$
Mg–MHE ⋆ | 43 | 0.7-2.7 | 0.012-0.122 | 1 | 1.14$\pm$0.03 | 0.97$\pm$0.21 | S09 | $\upuparrows$
Mg–MHE | 26 | 0.6-3.0 | 0.012-0.049 | 1 | 1.38$\pm$0.18 | 1.22$\pm$0.44 | E11 | $\upuparrows$
Mg–MHE | 20 | 0.9-2.8 | 0.012-0.034 | 1 | 1.09$\pm$0.08 | 1.11$\pm$0.10 | L15 | $\downdownarrows$
Mg–MWL | 118 | 0.6-6.0 | 0.054-1.033 | 1 | 1.35$\pm$0.30 | - | S20 | $\downdownarrows$
K–Texc | 43 | 0.7-2.7 | 0.012-0.122 | 1 | 0.83$\pm$0.20 | - | S09 |  $\Diamond$ $\upuparrows$

  * •

The subscripts exc, 300, HE, and WL indicate properties derived excluding the
core, within R$<$300 kpc, under the assumption of hydrostatic equilibrium, and
with weak-lensing analysis, respectively. BC indicates the relations corrected
for selection effects.
The slope of the Lx–Tx relation predicted by the self-similar scenario have
been obtained as L$\propto$T1.5+γ where $\gamma$ is the slope of the X-ray
emissivity in the considered energy band (e.g., soft or bolometric) and
temperature range covered by the systems analyzed in each work (see Table
LABEL:table:em). Since the X-ray emissivity strongly depends on the
metallicity we provide the extreme values obtained with Z⊙=0.3 and Z⊙=1.0. The
slope of the Lx–M relation is obtained similarly as L${}_{\rm x}\propto$M1+γ.
The values for slopeLIRA have been obtained by fitting each dataset with LIRA
(Sereno 2016) assuming self-similar time evolution with scatter on both
variables and with the following pivot values: 3 keV, 1044 erg/sec,
2$\times$1014 M⊙, 1013 M⊙, 1014 M⊙ for Tx, Lx, M, Mg, and YX respectively.
Before fitting the Lx–Tx relation, Chandra temperatures have been converted
into XMM-Newton-like temperatures.
Note: a, b, and c refer to Lx obtained in the 0.1-2.4 keV, 0.5-2 keV, and
bolometric band; $\dagger$ and $\ddagger$ indicate Lx obtained with ROSAT or
XMM data, while $\upuparrows$ and $\downdownarrows$ indicate if Chandra or XMM
data have been used for the analysis. $\square$, $\triangle$, and $\Diamond$,
indicate that the core-excised region was not a fixed fraction of $R_{500}$,
or fixed to 0.1$R_{500}$ or 0.15$R_{500}$, while $\circledcirc$ and
$\boxtimes$ indicate the region 0.1-0.5R500 and R$<$300 kpc, respectively. ⋆
Relation derived fitting together the groups with a sample of clusters.
References: Sun et al. (2009, S09), Eckmiller et al. (2011, E11), Kettula et
al. (2013b, K13), Lovisari et al. (2015, L15), Bharadwaj et al. (2015, B15),
Kettula et al. (2015, K15), Zou et al. (2016, Z16), Umetsu et al. (2020, U20),
Sereno et al. (2020, S20).

Beside the selection biases there are other issues complicating the comparison
between different studies and between systems with different temperatures
(masses). The first is the cross-calibration uncertainty between different
instruments. For instance, Schellenberger et al. (2015, see also ) showed that
the cluster temperatures derived with XMM-Newton are systematically lower than
those obtained with Chandra. To complicate this issue is the temperature
dependence of this difference. Fortunately, 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. 2013b). Although some caution is
still needed, one can expect that calibration differences do not significantly
affect the derived relations at the group scale. However, the impact of the
calibrations needs to be taken into account when comparing the results
obtained for sample of groups and sample of clusters. Another issue, pointed
out by Osmond & Ponman (2004), is related to the flattening of the fitted
relation because the scatter in $\log{({\rm T_{x}})}$ will be asymmetric
(assuming that the scatter in temperature is symmetric) with larger scatter
towards low $\log{({\rm T_{x}})}$. Moreover, if the quality of the data are
homogeneous across the sample, the statistical errors are expected to be
larger in systems with low luminosities, which also tend to flatten the fitted
relation. Finally, each study employ a different fitting algorithm (each with
pros and cons) and treatment of the scatter and selection biases which impact
the final results (see, e.g., Lovisari et al. 2020). In an attempt to remove
this last uncertainty and provide comparable results we fit the published data
with the same fitting method (i.e., using LIRA; Sereno 2016) and assumptions
(e.g., self-similar redshift evolution). The results are given in Table 2.

The correlation between X-ray luminosity and gas temperature reflects the fact
that a deeper potential well (leading to a higher Tx) generally contains more
hot gas (leading to a higher Lx). However, it has been shown that the gas
fraction varies as function of the total mass with galaxy groups showing
almost a factor of two lower gas fraction than galaxy clusters. Since the
X-ray luminosity is proportional to the amount of gas in the IGrM and ICM, a
change in the gas content in low-mass systems translates into a lower
luminosity with the effect of steepening the Lx–Tx relation. Anyway, the mass
dependence of the gas fraction seems to vanish in the outer regions (e.g., Sun
et al. 2009) implying that the low gas fraction observed in groups is mainly
due to the low gas fraction of groups within $\sim$R2500. This weak ability of
the groups to retain the gas in the inner regions is probably a consequence of
their shallow gravitational potential and thus of the increasing contribution
of different non-gravitational processes. These mechanisms are expected to
provide an extra heating to the gas preventing the gas from falling toward the
center, and by that, reducing gas density and X–ray emissivity in the cores.
The effect is expected to play a significant role in poor systems leading to
the steepening of the Lx–Tx relation, as observed, and of the Lx–$\sigma_{\rm
v}$ relation which, however, is not currently supported by observations (see
Section 3.2). Supporting this scenario is the fact that when the core regions
of galaxy clusters are ignored (i.e., by removing the regions where non-
gravitational processes are expected to affect more the gas properties) the
slope of the Lx–Tx relation is more in agreement with the self-similar
prediction. This is also in agreement with the suggestion by Mittal et al.
(2011) that, for low-temperature systems (i.e., kT$<$ 2.5 keV), AGN heating
becomes more important than ICM cooling (which is the dominant mechanism in
massive clusters). Colafrancesco & Giordano (2007) suggested that intracluster
magnetic fields can also affect more strongly the gas properties in the low-
mass regime, resulting in an effective steepening of the scaling relations. In
fact, as shown in Colafrancesco & Giordano (2007), the magnetic pressure tends
to counterbalance part of the gravitational pull of the cluster preventing the
gas from a further infalling. Thus, the presence of a magnetic field
determines the final distribution of the gas density resulting in a less
concentrated core (i.e., leading to a lower luminosity). The effect is mild
for massive systems (due to their large gravitational potential) but is
relevant in the group regime. The presence of the magnetic field is also
expected to decrease the temperature because of the additional magnetic field
energy term that needs to be included in the virial theorem. However, since
galaxy groups and clusters are not isolated systems, the presence of an
external pressure induced by the infalling gas from filaments, would tend to
compensate the decrease of Tx caused by the magnetic field.

The non-gravitational heating implies the existence of an entropy floor (i.e.,
an excess of entropy with respect to the level referable to the gravity only)
calling for some energetic mechanisms that can be summarized in three classes:
preheating, local heating, and cooling (see the companion review by Eckert et
al. 2021 for detailed description of these mechanisms). Indeed, entropy in
excess with respect to that achievable by pure gravitational collapse is
observed in the inner regions of groups and poor clusters (e.g., Mahdavi et
al. 2005, Finoguenov et al. 2007b, Johnson et al. 2009; Sun et al. 2009,
Panagoulia et al. 2014). The excess is found to be radial and mass-dependent,
being smaller for massive systems and extending to larger radii in low-mass
objects. Moreover, Johnson et al. (2009) found that the excess is higher for
groups with higher feedback (roughly estimated assuming that both the
integrated feedback from SNe and AGNs scale with the stellar mass). Since
entropy is expected to remain unchanged when neglecting non-gravitational
processes, in the self-similar scenario it simply scales with the gas
temperature. The finding by Sun et al. (2009) shows that the slope of the
relation depends on the scaled radius at which the measurement is taken, and
groups behave more regularly in the outer regions (e.g., beyond R2500) than in
the core. This was already pointed-out by Ponman et al. (2003). Thus, the
slope of the K-Tx relation approaches the self-similar value at R500, where
there is no significant entropy excess above the entropy baseline (see the
companion review by Eckert et al. 2021). This is in agreement with the finding
by Pratt et al. (2010) for a sample of galaxy clusters.

Figure 4: XMM-Newton (circles) and Chandra (squares) measurements of the
L0.1-2.4–M (left panel) and M–Tx (right panel) relations for different samples
of groups: Eckmiller et al. (2011, E11), Lovisari et al. (2015, L15), Sun et
al. (2009, S09) , Kettula et al. (2013b, K13). The groups measurements are
compared with the ones from an X-ray-selected (Schellenberger & Reiprich
2017a, S17) and an SZ-selected (Lovisari et al. 2020, L20) cluster sample.
Luminosities and masses are all within R500 while temperatures are obtained in
different regions (see Table 2). Note that different studies used different
methods to estimate the total masses. The lines represent the fitted relation
for M$<$1014M⊙ systems (dotted), M$>$1014M⊙ systems (dashed-dotted), all
systems (solid), and are compared with the case predicted by the self-similar
scenario (dashed). Because of the strong covariance between M and Tx we did
not convert the Chandra temperatures to XMM-Newton-like temperatures.

Less studied than the Lx–Tx relation, but of paramount importance for
cosmological studies, is the relationship between X-ray luminosity and total
mass (i.e., Lx-M). This is particularly true for shallow X-ray surveys because
it can be used to directly convert the easiest to derive observable
(luminosity requires only source detection and redshift information) to the
total mass. The calibration of this relation down to the low-mass regime will
allow to break the degeneracy between $\Omega_{\rm m}$ and $\sigma_{8}$ (e.g.,
Reiprich & Böhringer 2002). A large number of observations of galaxy clusters
(e.g., see Mantz et al. 2016a, Schellenberger & Reiprich 2017b, Mantz et al.
2018, Bulbul et al. 2019, and Lovisari et al. 2020 for recent studies; we
refer to Böhringer et al. 2012 and Giodini et al. 2013 for older
investigations) found that most of the values for the relation slope range
from 1.4 to 1.9, steeper than the self-similar prediction of 4/3 suggesting
that the luminosity is affected by non-gravitational processes. Unfortunately,
the literature in the low-mass regime is still quite limited (in the left
panel of Figure 4 we show a compilation of recent galaxy groups studies).
Eckmiller et al. (2011) found a slope of 1.34$\pm$0.18 and suggested that the
single power-law modeling of the relation holds also for low-mass objects (see
also the illustrative fit in Figure 4). However, given the considerations
provided in the previous section, for the sample analyzed by Eckmiller et al.
(2011) the luminosity should scale with the total mass to the power of
[0.4:0.8] (see Table 2) significantly lower than the finding by Eckmiller et
al. (2011). A similar slope was also obtained by Lovisari et al. (2015) but,
after correcting for the selection biases, the corrected slope is steeper
(i.e., 1.66$\pm$0.22) than the observed one (i.e., 1.32$\pm$0.24). making the
deviations from the self-similar prediction even larger than what observed for
galaxy clusters. This behavior can be explained by a gradual steepening of the
true underlying Lx–M relation towards the low-mass regime. This implies that,
as expected, also the luminosity of groups is heavily affected by non-
gravitational processes. However, one should keep in mind that a mass-
dependent bias in the total mass can also affect the shape of the Lx–M
relation. Because of the difficulties to distinguish the low temperature
emitting gas of these systems from the Galactic foreground, the properties of
galaxy groups can usually be observed out to a smaller radial extent than what
is done for galaxy clusters. Thus, an estimate of the group masses at R500
requires an extrapolation for most of the systems making the groups more prone
to biases. Moreover, mass biases can also arise from the assumptions of
hydrostatic equilibrium and spherical symmetry which are not valid for most of
the systems. One way to overcome the problem is to use the weak-lensing masses
which are expected to provide a less biased view of the true masses. Because
of the difficulties to obtain shear maps for low-mass systems the first
attempts have been performed via stacking analysis. Leauthaud et al. (2010)
stacked the weak lensing measurements of a sample of X-ray-selected galaxy
groups and found an Lx–M200 relation in agreement with the finding of galaxy
clusters. This result suggests that the Lx–M200 relation is well described by
a single power-law down to the low-mass regime. However, the lensing analysis
of galaxy groups by Kettula et al. (2015) led to a shallower slope although
the agreement in the low-mass regime of the two relations is fairly good, with
significant tension appearing only at high masses (i.e., above a few 1014 M⊙).
With the advent of multi-wavelength surveys, which uniformly scan large areas
of sky, there has been significant progress in the weak-lensing analysis of
large samples of galaxy groups. For instance, in the XXL framework (see Pierre
et al. 2016), the Lx–MWL relation was investigated by Sereno et al. (2020) who
found a quite good agreement with previous X-ray studies. The results suggest
that the measured hydrostatic bias is consistent with a small role of non-
thermal pressure. However, due to the large uncertainties associated to the
derived weak-lensing masses a large deviation from hydrostatic equilibrium
cannot be completely excluded and further investigations with larger samples
and higher quality data are required to make progress in the field.

In contrast to the Lx–Tx and Lx–M relations, the M–Tx relation is expected to
follow the same behavior for galaxy groups and clusters, under the assumption
that the gas temperature reflects the depth of the underlying potential well.
However, while the assumption is probably reasonable for many clusters (at
least for the most relaxed ones) it may not be strictly true for groups where
the gas has probably been significantly heated by non-gravitational processes.
For this reason, the expectation is that the scatter should increase in the
low-mass regime where the global temperature is not insensitive to the details
of the heating/cooling processes as in the high-mass regime. However, these
processes are definitely more important in the inner regions with their effect
fading at large distances from the center. Nonetheless, some simulations
suggested that the gas removed by AGN activity in groups can affect the gas
properties out to several Mpc (e.g., Schaye et al. 2010), potentially
affecting also cosmic shear measurements (e.g., Semboloni et al. 2011). Thus,
it is important to define the region within which the characteristic cluster
temperature is determined. This is not trivial because, for instance, we have
evidence that the central drop (i.e., the region in the center of relaxed
clusters showing a significant temperature decline, probably caused by
radiative cooling), typically present in relaxed clusters, does not scale
uniformly with the mass (Hudson et al. 2010). However, a common practice is to
exclude the regions within 0.15R500.

There had been a lot of studies investigating the M–Tx relation before the
Chandra and XMM-Newton era. Many of them (e.g., Finoguenov et al. 2001,
Sanderson et al. 2003, and references therein) suggested that the low- and
high-mass end of the relation is characterized by different slopes with the
cross-over temperature between the two regimes at $\sim$3 keV. However, most
of these studies could not constrain the gas properties (i.e., gas density and
temperature gradients) at large radii making the estimated hydrostatic masses
more prone to biases. Thanks to Chandra and XMM-Newton, the measurements could
be extended to larger fraction of R500 reducing the impact of the
extrapolation. Sun et al. (2009) found a relation only slightly steeper than
the prediction of the self-similar scenario. Both Eckmiller et al. (2011) and
Lovisari et al. (2015) found that the slope for galaxy groups is consistent
with the one of galaxy clusters but with a normalization 10-30% lower. The net
effect of this finding is a steepening of the relation when groups and
clusters are fitted together (see, e.g., right panel of Figure 4). Indeed, due
to the limited field-of-view of Chandra and the high and variable XMM-Newton
background level, X-ray measurements are still not tracing well the outer
regions (despite the improvement with respect to previous missions,
measurements extend out to R500 only for a few systems). If the density
profiles of groups are steepening at large radii then the masses could be
underestimated explaining the lower normalization of the M–Tx relation.
Another possible issue is that the samples analyzed in the above-mentioned
papers are biased toward relaxed systems. Thus, if relaxed and disturbed
systems do not share the same relation (e.g., because hydrostatic masses are
more biased for disturbed systems) the relative fraction of relaxed/disturbed
groups can impact the normalization (and possibly the slope) of the observed
M–Tx relations. Lovisari et al. (2020) showed that this is probably not the
case for massive systems, but a dedicated study in the group regime is still
missing. Focusing on the slopes, the results of the most recent papers on
galaxy groups are in agreement with the results for galaxy clusters for which
most of the slope values range from 1.5 to 1.7 (see Table 2). This agreement
suggests a small impact of the non-gravitation processes to this relation.
Again, before overinterpreting these results, one of the key questions is to
asses if the level of mass bias in these systems is similar to the one of
galaxy clusters. For instance, Kettula et al. (2013a, see also ) argued that
the hydrostatic mass bias at 1 keV reaches a level of 30%-50%, higher than
what usually observed for galaxy clusters (e.g., by calibrating the
hydrostatic masses with other mass proxies, like weak-lensing or velocity
dispersion). However, the sample consists of only 10 galaxy groups and the
dynamical state of the systems is not discussed. A much larger sample was
investigated by Umetsu et al. (2020) who found the relation to be consistent
within statistical uncertainties with the self-similar expectations. However,
the uncertainties of the individual weak-lensing mass measurements in the
group regime are still quite large and tighter constraints are needed in the
future to exclude deviations from self-similarity. An increasing mass bias in
the low-mass regime would not be fully unexpected. In fact, in the
unmagnetized case, the viscosity scales as T${}_{\rm x}^{5/2}$ (Spitzer 1962)
favoring the development of strong turbulences. Acting as additional pressure
support against gravity, turbulent motions may increase the mass bias. Anyway,
the IGrM is magnetized and the real magnitude of the turbulence is still
unknown.

Beside the shape of the scaling relations another important information is
given by the scatter (i.e., the dispersion around the best-fit). Minimizing
the scatter of the scaling relations is of paramount importance to obtain
accurate constraints on cosmological parameters which are dominated by
uncertainties in the mass–observable relations. Moreover, understanding the
scatter in the relations is the key to pinpoint the physical processes at play
in the group regime. However, the measurement errors for most of the groups
are large, thus the intrinsic scatter is not well constrained, yet. Because of
that, Sun et al. (2009) and Lovisari et al. (2015) were not able to constrain
the intrinsic scatter in their samples. Eckmiller et al. (2011) instead
suggested that the scatter in galaxy groups (kT$<$3 keV) is much larger than
the one derived for the HIFLUGCS sample (Reiprich & Böhringer 2002).

A mass proxy which has been shown by simulations to bear a low scatter is the
YX parameter (i.e., the product of the gas temperature and the gas mass; see
Kravtsov et al. 2006). Sun et al. (2009) were the first to investigate the
M–YX relation in the low-mass regime finding that a single power law model can
fit very well both galaxy groups and clusters. This result was also confirmed
by Eckmiller et al. (2011) and Lovisari et al. (2015). Unfortunately, given
the sample size and the relatively large measurements errors, the intrinsic
scatter is not properly constrained. However, both Sun et al. (2009) and
Eckmiller et al. (2011) suggested that the scatter of the M–YX relation is
almost half of the M–Tx relation. The findings by Eckmiller et al. (2011) also
suggest that the scatter for galaxy groups is significantly higher than for
galaxy clusters.

The self-similar model also predicts the X-ray scaling relations to be
redshift-dependent (e.g., Giodini et al. 2013 and references therein),
reflecting the decrease with time of the mean density of the Universe. Non-
gravitational processes are expected to affect the evolution of the X-ray
scaling relations because of the increasing importance of such processes to
the energy budget of galaxy systems as a function of redshift. Unfortunately,
although groups are more common than clusters, because of their fainter and
cooler nature it is more difficult to detect them over the background,
especially at higher redshifts. Thus, due to the big challenges to detect
large and representative samples of galaxy groups beyond the local Universe
the literature on this subject is very limited. The few studies (e.g., Jeltema
et al. 2006, Pacaud et al. 2007, Alshino et al. 2010, Umetsu et al. 2020,
Sereno et al. 2020) which have tried to address the evolution of the X-ray
properties of galaxy groups did not find convincing evidence for such
evolution. A characterization of the evolution of the scaling relations also
on galaxy group scales is one of the goal of the next generation instruments
(such as Athena; see Section 5).

##  3 Optical scaling relations

Due to the low X-ray flux at the group scale, there is high probability that
X-ray selected samples are biased toward groups with rich IGrM. Moreover,
since the luminosity strongly depends on the metallicity (see Section 2.1),
variations in the metal abundance between groups (possibly related to their
feedback history) can significantly impact the selection function. Thus, it is
advantageous to explore scaling relations between an X-ray property that can
be measured relatively well in the low count regime (e.g., X-ray luminosity or
gas temperature) and an optical property that can be used as a proxy for the
group mass (e.g., velocity dispersion, optical band luminosity).

###  3.1 Velocity Dispersion

The velocity dispersion ($\sigma_{\rm v}$) of galaxy groups (and indeed
clusters) can be used to estimate dynamical masses via the application of the
virial theorem. Furthermore, the velocity of member galaxies complements X-ray
information about the cluster morphology projected onto the sky. For example,
studying the luminosity–velocity dispersion (Lx–$\sigma_{\rm v}$) relation
provides an understanding of the dynamical properties of galaxy clusters and
their impact on the scaling relations.

One of the most commonly used estimators of the velocity dispersion at the
group regime is via the use of the gapper estimator from Beers et al. (1990).
Of critical importance at the group scale, the gapper estimator is unbiased
when using low numbers of member galaxies (down to $\sim$10 members, e.g.,
Ruel et al. 2014), and is robust against outliers. The gapper velocity
estimator ($\sigma_{\rm v}$) is given by

$\displaystyle\sigma_{\rm v}$
$\displaystyle=\frac{\sqrt{\pi}}{N(N-1)}\sum^{N-1}_{i=1}w_{i}g_{i},$ (6)

where, for ordered velocity measurements, the gaps between each velocity pair
are defined as $g_{i}=v_{i+1}-v_{i}$ (for $i=1,2,3...,N-1$), as well as
Gaussian weights defined as $w_{i}=i(N-i)$.

As stated above, one can study the Lx–$\sigma_{\rm v}$ relation to understand
dynamical properties and the impact on scaling relations. In Equation 4, it is
shown that in the self-similar scenario the bolometric luminosity is expected
to scale with the gas temperature as L${}_{\rm x,bol}\propto$T${}^{2}_{x}$.
Under the consideration that both the cluster/group hot gas and galaxies feel
the same potential, assuming that they both have the same kinetic energy, the
temperature can be converted to velocity dispersion using

$\displaystyle\beta=\frac{\sigma^{2}_{\rm v}\mu{\rm m_{p}}}{\rm k_{B}{\rm
T_{x}}}\approx 1,$ (7)

where the parameter $\beta$ is the ratio of the specific energy in galaxies to
the specific energy in the hot gas. Using Equation 7 and the self-similar
scaling of Lx,bol and Tx above, the self-similar scaling of velocity
dispersion and X-ray properties can be given by

$\displaystyle{\rm L_{x,bol}}\propto\sigma^{4}_{\rm v},$ (8)
$\displaystyle{\rm T_{x}}\propto\sigma^{2}_{\rm v}.$ (9)

However, because of the behavior of the X-ray emissivity in the low-
temperature regime, the dependence of the luminosity on the temperature is
more complicated (see discussion in $\S$2.1) and can be approximated as
L${}_{\rm x}\propto$T${}_{\rm x}^{1.5+\gamma}$, where $\gamma$ is the slope of
the X-ray emissivity in the considered energy band (e.g., soft or bolometric)
and temperature range (see Table LABEL:table:em). Using this $\gamma$
dependent relation, it follows that

$\displaystyle L_{\rm x}\propto\sigma_{\rm v}^{3+2\gamma}.$ (10)

For temperatures lower than 3 keV, the value of $\gamma$ is negative (unless
very cool systems are considered), implying that the expected Lx–$\sigma_{\rm
v}$ relation for galaxy groups is shallower than what is predicted for galaxy
clusters (e.g., Equation 8).

###  3.2 The luminosity-velocity dispersion relation

At the cluster scale, generally, it has been found that the observed
luminosity–velocity dispersion (Lx–$\sigma_{\rm v}$) relation follows, or is
slightly steeper than, the expectation of Equation 8 (e.g., Quintana & Melnick
1982; Mulchaey & Zabludoff 1998; Mahdavi & Geller 2001; Ortiz-Gil et al. 2004;
Zhang et al. 2011a; Nastasi et al. 2014). Furthermore, at the cluster scale,
studies of the Lx–$\sigma_{\rm v}$ relation now utilise samples of clusters
numbering in the high hundreds (e.g., Kirkpatrick et al. 2021, using 755
clusters to investigate the Lx–$\sigma_{\rm v}$ relation). Studies of the
Lx–$\sigma_{\rm v}$ relation at the group scale attempt to compare the form of
the relation at the high-mass regime in order to investigate differences at
these two mass scales (e.g., to probe the effect of AGN feedback processes at
high and low masses). Early studies comparing the slope of the relation
between the two mass regimes provided a mixed picture, with studies finding
groups have a flatter (e.g., Helsdon & Ponman 2000a; Xue & Wu 2000; Osmond &
Ponman 2004) or consistent (e.g., Ponman et al. 1996, Mulchaey & Zabludoff
1998, Mahdavi & Geller 2001) relation than their high-mass counterparts.
Figure 5 (left-panel) provides a (non-comprehensive) compilation of the slope
of the Lx–$\sigma_{\rm v}$ relation from various studies in the literature.
The solid horizontal line represents the dividing line between studies using
clusters (top half) and groups (bottom half). The L${}_{\rm
x,bol}\propto\sigma^{4}_{\rm v}$ expectation is given by the vertical dashed
line. While there appears to be a clear division between the slopes for groups
and clusters, many of the group scale studies compare to the usual L${}_{\rm
x,bol}\propto\sigma^{4}_{\rm v}$ expectation at the cluster scale. As shown in
Equation 10, the scaling can be given by L${}_{\rm x}\propto\sigma_{\rm
v}^{3+2\gamma}$, with $\gamma$ dependent on the X-ray emissivity and energy
band used. If we assume a group temperature range of 0.7–3.0 keV, then given
the range of emissivities in Table LABEL:table:em, the bolometric scaling in
the group regime becomes L${}_{\rm x,bol}\propto\sigma_{\rm v}^{[2.2:3.0]}$.
This range is highlighted by the blue shaded region in Figure 5 (left-panel)
at the group scale. For comparison, using these same arguments, the bolometric
scaling for clusters (assuming 3.0-10.0 keV) becomes L${}_{\rm
x,bol}\propto\sigma_{\rm v}^{[3.7:3.9]}$ (again highlighted by the blue shaded
region in Figure 5, appropriate to the cluster scale). Considering the above,
studies investigating the group scale relation can indeed be considered
consistent with the self-similar expectation (e.g., Osmond & Ponman 2004).
While this is the case, many authors note caveats when studying groups, which
are discussed below.

Figure 5: Compilation of the measured slopes of the luminosity-velocity
dispersion (Lx–$\sigma_{\rm v}$, left panel) and velocity dispersion-
temperature ($\sigma_{\rm v}$–Tx, right panel) relation in the literature. In
each case, the dashed vertical line highlights the usual self-similar
expectation on the slope for each relation (Equation 8, left panel, and
Equation 9, right panel). The horizontal lines represent a dividing line
between the mass scales used for the comparison in each relation. As shown in
Sect. 3.1 when the Lx–$\sigma_{\rm v}$ scaling can be given as L${}_{\rm
x}\propto\sigma_{\rm v}^{3+2\gamma}$ (with $\gamma$ dependant on the energy
band and emissivity). The blue shaded region represents the self-similar
expectation when considering bolometric luminosities for clusters (assuming
Tx=3.0-10.0 keV, L${}_{\rm x}\propto\sigma_{\rm v}^{[3.7:3.9]}$) and groups
(assuming Tx=0.7-3.0 keV, L${}_{\rm x}\propto\sigma_{\rm v}^{[2.2:3.0]}$). The
grey shaded region represents the self-similar expectation when considering
0.1–2.4 keV luminosities for clusters (assuming Tx=3.0-10.0 keV, L${}_{\rm
x}\propto\sigma_{\rm v}^{[2.7:2.8]}$) and groups (assuming Tx=0.7-3.0 keV,
L${}_{\rm x}\propto\sigma_{\rm v}^{[1.6:2.6]}$). Unless otherwise stated,
references consider (bolometric) luminosities and temperatures derived within
an estimate of R500. Please note that the Sohn et al. (2019) relation is based
upon an analysis using 7 bins of using the full sample of 74 systems. * refers
to references using 0.1-2.4 keV band luminosities.

Many early studies were based upon ensemble collections of groups that can
lead to biases in the derived scaling relations. In recent years, studies of
the Lx–$\sigma_{\rm v}$ relation have utilised groups selected over contiguous
survey regions. One such study was performed by Connelly et al. (2012). Groups
were selected from regions of the Canadian Network for Observational Cosmology
Field Galaxy Redshift Survey 2 (CNOC2, Carlberg et al. 1999) that were covered
by XMM-Newton and Chandra observations, totalling 0.2 and 0.3 deg2 contiguous
areas of two fields of the CNOC2 survey. Using X-ray selected groups with high
quality redshift information, they find a slope of the Lx–$\sigma_{\rm v}$ of
2.40${}^{+0.58}_{-0.60}$ (including groups with lower quality redshift
information yields a slope of 1.35${}^{+0.42}_{-0.47}$). While initial
inspection of the value of the slopes would imply the slope is shallower than
the self-similar expectation (as noted in Connelly et al. 2012), we note that
the luminosities are reconstructed in the 0.1–2.4 keV band (from the flux in
the 0.5-2 keV band, and by correcting for extension and K-correction as
described in Finoguenov et al. 2007a). Assuming the scaling follows L${}_{\rm
x}\propto\sigma^{3+2\gamma}_{\rm v}$, then for luminosities in the 0.1–2.4 keV
band, the scaling can be given by L${}_{\rm x}\propto\sigma^{[2.0:2.7]}_{\rm
v}$ (depending on the metallicity of the groups). This expectation is shown in
Figure 5 (left-panel), highlighted by the grey shaded region at the group
regime. The slope determined by Connelly et al. (2012) is coincident with this
scaling. Therefore, if the energy band is considered, the Connelly et al.
(2012) relation is consistent with the self-similar expectation. Hence, it can
be assumed that the groups studied in Connelly et al. (2012) are consistent
with the cluster scale (assuming clusters follow the self-similar
expectation). Another study utilising contiguous fields is presented in Sohn
et al. (2019), using groups selected from the 2 deg2 Cosmic Evolution Survey
(COSMOS, Scoville et al. 2007). This study constructs a catalog of galaxy
groups based upon those identified in George et al. (2011), using XMM-Newton
and Chandra observations of the COSMOS field, reaching an X-ray flux limit of
$\sim$10-15 erg s-1 cm-2. Sohn et al. (2019) use galaxy redshift information
from a wide variety of surveys in the literature and associate them with the
X-ray detected groups, compiling a final sample of 146 groups with at least
three spectroscopic members. Based upon a cleaned sample of 74 groups, Sohn et
al. (2019) showed that the relation for individual groups appears to follow a
shallower relation than clusters (consistent with that found by previous
studies, e.g., Mahdavi & Geller 2001). However, they note that this trend may
be affected by a small number of groups that appear to have anomalously low
velocity dispersions (at $\sigma_{\rm v}\lesssim$125 km s-1) for their
measured X-ray luminosity (discussed further in Sect. 3.4). To overcome this,
Sohn et al. (2019) estimated the median velocity dispersion for 7 bins of
groups created from the 74 groups in their sample, finding a slope of
L${}_{\rm x}\propto\sigma^{4.7\pm 0.7}_{\rm v}$. This study considers
luminosities in the 0.1–2.4 keV band, therefore, as discussed above, the
measured slope is in fact steeper than the self-similar expectation.

Although contiguous regions have been used to study the Lx–$\sigma_{\rm v}$
relation, an extremely small number of studies have attempted to correct for
X-ray selection biases. One study that attempts to do so is presented in
Vajgel et al. (2014), using 14 groups with at least 5 galaxy members selected
from the 9 deg2 X-Boötes survey (Murray et al. 2005). They find that the group
scale relation is consistent with the self similar expectation. In order to
test the effects of Malmquist bias on the observed relations, Vajgel et al.
(2014) determined the limiting X-ray luminosity in two survey volumes
($z$=0.20 and $z$=0.35). The resulting relations are consistent with the
sample relation, with the authors concluding the sample may not be dominated
by Malmquist bias effects. However, due to the associated large error on each
relation, making this conclusion is challenging and requires the construction
of larger samples. Another use of contiguous surveys, particularly those
covered by multiple wavelengths, is the possibility to compare the relations
derived using groups selected via multiple selection methods (e.g., X-ray,
optical). Furthermore, the use of optically selected groups allows one to
estimate the form of scaling relations independent of the usual X-ray
selection biases (e.g., Andreon et al. 2016). The study by Connelly et al.
(2012), as detailed above, also constructed a sample of 38 optically
(spectroscopically) selected groups. Using this optically selected sample,
they derive a slope of the Lx–$\sigma_{\rm v}$ relation of
1.78${}^{+0.60}_{-0.54}$. Given the large uncertainties on the measured
slopes, the comparison of the X-ray and optically selected samples is somewhat
limited (note that the comparison is not effected by the energy band used, as
discussed above, since they are consistent between the X-ray and optically
selected samples).

###  3.3 The velocity dispersion-temperature relation

Velocity dispersion and gas temperature are two independent probes of the
depth of the cluster potential well, estimated by using baryons as tracers.
Therefore, this relation can provide useful information about the effect of
non-gravitational processes, which are responsible for the deviation from
thermal equilibrium of the IGrM and ICM. Hence, it is useful to compare group
and cluster relations to investigate the differences between these mass
scales. Figure 5 (right-panel) shows a compilation (again, a non-comprehensive
picture) of the slope of the $\sigma_{\rm v}$-Tx relation from studies in the
literature. The horizontal lines represent the division between studies using
groups (bottom section), clusters (top section) and those using systems which
straddle the group/cluster regime (middle section). Based upon Equation 7, it
is expected that the velocity dispersion of the galaxies should scale with the
square-root of the temperature of the gas, $\sigma_{\rm v}\propto$T${}_{\rm
x}^{1/2}$. In the context of clusters, various studies have found that the
$\sigma_{\rm v}$–Tx relation has a slope steeper than the self-similar
expectation (e.g., Ortiz-Gil et al. 2004; Wilson et al. 2016), with others
finding a steeper slope but with errors too large to confirm a deviation
(e.g., Nastasi et al. 2014). Unfortunately, the study of the $\sigma_{\rm
v}$–Tx relation for groups is somewhat limited in the literature. An early
investigation presented in Osmond & Ponman (2004) showed evidence for
steepening of the relation at the group scale (where they find a slope of
1.15$\pm$0.23). However, they caution that there is both large uncertainties
on the measured X-ray temperatures and a large amount of scatter observed in
the relation, which could be the cause of tension with previous studies
attempting to investigate any steepening of the relation for groups. Although
Osmond & Ponman (2004) found evidence for a steepening of the relation, they
remarked that a comparison cluster based relation passes through the centre of
the group relation data, and represents adequately the cluster based relation.
However, recent studies of the relation, especially at the group scale, become
scarce. One recent study of the $\sigma_{\rm v}$-Tx is presented in Wilson et
al. (2016), making use of groups/clusters detected serendipitously in the XMM
Cluster Survey (Romer et al. 1999). Using 19 groups/clusters having redshifts
z$<$0.5, spanning the temperature range 1.0 $\mathrel{\mathchoice{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}$ Tx
$\mathrel{\mathchoice{\lower 2.5pt\vbox{\halign{$\mathsurround
0pt\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}$ 5.5 keV (with 50% of
clusters having a temperature $<$3 keV), the $\sigma_{\rm v}$–Tx relation is
found to have a slope of 0.89$\pm$0.16. While again steeper than the self-
similar expectation, the result is somewhat shallower (although not
significant) than that presented in Osmond & Ponman (2004). Finally, the last
relation considered is that given in Farahi et al. (2018b), which investigated
the $\sigma_{\rm v}$–Tx relation for a sample of X-ray selected clusters
detected in the XXL survey (Pierre et al. 2016). Clusters were selected from
the 25 deg2 XXL-N region, with spectroscopic data compiled from a range of
surveys (see Adami et al. 2018, for full details of the spectroscopic
coverage). Using a sample of 132 clusters (the majority of which have
T${}_{x}<$3 keV), Farahi et al. (2018b) found a relation of the form
$\sigma_{\rm v}\propto$T${}_{\rm x}^{0.63\pm 0.05}$. Note that the relation is
fitted using an ensemble maximum likelihood method, with fitted slope in
tension with the self-similar expectation. Since both velocity dispersion and
X-ray temperature scales with total mass, one can combine the information to
determine a useful mass calibration (e.g., Farahi et al. 2018b). However,
consideration must be given to velocity anisotropies during mass modelling
using velocity information, which can vary for loose, compact and viralised
groups (Mamon 1993). However, corrections based upon halo concentration have
been developed (e.g., Mamon et al. 2013). The study of the $\sigma_{\rm v}$-Tx
relation can also be probed down to the galaxy scale. Goulding et al. (2016)
used galaxies from the volume-limited MASSIVE survey (Ma et al. 2014), to
study the relation between galaxy kinematics ($\sigma_{\rm e}$) and X-ray
temperature. Goulding et al. (2016) found a relation of the form T${}_{\rm
x}\propto\sigma_{\rm e}^{1.3-1.8}$ (note the inverse of the relation as
discussed above), noted as being marginally flatter than the self-similar
expectation.

As discussed in Section 3.4, AGN feedback and its effects on the ICM could
result in deviations from self-similarity, in particular, the steepening
observed above in the Tx–$\sigma_{\rm v}$. While an observational consensus on
the magnitude of the deviation from self-similarity at the group scale
compared to the cluster scale has yet to be reached, simulations have indeed
shown a mass dependence (e.g., Le Brun et al. 2017, Farahi et al. 2018a,
Truong et al. 2018). The deviations discussed for the Lx–$\sigma_{\rm v}$ and
$\sigma_{\rm v}$–Tx are thought to arise due to the effects of AGN feedback on
the ICM (as shown in simulations, e.g., Kar Chowdhury et al. 2020), which has
little effect on the galaxy velocity dispersions. Furthermore, Goulding et al.
(2016) measured a median value of $\beta$=0.6 for their galaxy sample,
suggesting the galaxies have undergone, or still in the process of, additional
heating due to, e.g., AGN feedback, as discussed above.

###  3.4 Low velocity dispersion groups

One observation made by various authors studying the Lx–$\sigma_{\rm v}$
relation, is the presence of low velocity dispersion groups (appearing at
$\sigma_{\rm v}\lesssim$200 km s-1) that have a high X-ray luminosity in
comparison to their $\sigma_{\rm v}$ (conversely, it can be stated that these
groups have a low $\sigma_{\rm v}$ for their Lx). These low velocity outliers
have been noted in various studies in the literature (e.g., Mahdavi & Geller
2001; Helsdon et al. 2005), attributed as the cause of the flattening of the
Lx–$\sigma_{\rm v}$ relation at the group scale (e.g., Vajgel et al. 2014,
Sohn et al. 2019). While it has been shown in Section 3.2 that the group scale
relations may be consistent with self-similar predictions when accounting for
the differing emissivity, the presence of these outliers are extreme cases. A
physical interpretation of the low $\sigma_{\rm v}$ outliers is therefore
currently lacking. Furthermore, the presence of these outliers remains
somewhat of a mystery if one considers the effects of AGN feedback on the
intragroup medium. During an AGN outburst, gas will be removed from the group
potential, hence lowering the group overall X-ray luminosity. With the group
velocity dispersion unaffected by this process, the expectation would be that
the group should have a lower Lx for a given $\sigma_{\rm v}$, contrary to
this outlier population. It could therefore be argued that it is in fact the
velocity dispersion that have been underestimated for these groups. Potential
explanations for the presence of these low $\sigma_{\rm v}$ outliers were
given in Helsdon et al. (2005). It is postulated the cause could be: (i)
through dynamical friction, energy is transferred from a large orbiting body
to the sea of dark matter particles through which it moves; (ii) due to tidal
interactions, the orbital energy may be converted into internal energy of the
galaxies; and (iii) the orbital motion happens in the plane of the sky,
therefore contributing little to the line of sight velocity dispersion. While
a current physical interpretation is lacking, the presence of low velocity
dispersion outliers could be due to X-ray selections effects (e.g., Eddington
and Malmquist biases). Extreme outliers (e.g., Sohn et al. 2019) may not be
attributed to selection, however various relations involving Lx when using
X-ray selected samples characteristically show a flattening when not account
for selection (e.g., Mantz et al. 2010; Giles et al. 2017). In fact, the
preferential selection of higher luminosity groups for a given velocity
dispersion (i.e., Malmquist bias), leads to the presence of “moderate”
outliers. Andreon et al. (2016) argues that an unbiased sample of clusters can
be obtained when selecting clusters from optical properties and therefore able
to probe the full range of scatter and the true form of the relation. As
stated, Connelly et al. (2012) have used an optically selected sample of
clusters to investigate the form of the Lx–$\sigma_{\rm v}$ relation, however,
they find constancy in both the form and scatter of the X-ray and optically
selected group samples (due to the large errors on the scaling parameters).
This sample only covered an area of 0.5 deg2, therefore the comparison of
optically and X-ray selected samples over overlapping contiguous fields
requires further attention to truly probe the differences in selection.

Figure 6: The Mstars–M relation of various studies in the literature. Two
lines of stellar mass fraction are highlighted by the dashed lines. Please
note that the Ziparo et al. (2016) relation is derived from the conversion of
LK to Mstars assuming a constant mass-to-light ratio of 0.73 (as used in
Ziparo et al. 2016).

###  3.5 Stellar gas content of galaxy groups

The gas mass fraction of clusters can be used as a probe of cosmology (e.g.,
Allen et al. 2008, Ettori et al. 2009, Mantz et al. 2014, Schellenberger &
Reiprich 2017b). However, as mentioned in the previous sections, it has been
shown that the fraction decreases as a function of total mass. Interestingly,
the opposite is true for the stellar mass fraction (fstars=Mstars/Mtot), with
an increasing stellar mass fraction as a function of decreasing total mass
(e.g., Lin et al. 2003, Gonzalez et al. 2007, Giodini et al. 2009, Behroozi et
al. 2010, Zhang et al. 2011b, Leauthaud et al. 2012, Laganá et al. 2013, Chiu
et al. 2018, Decker et al. 2019). To investigate this trend, much effort has
been afforded to the study of the gas mass and stellar mass content in groups
and clusters (e.g., to determine star formation rates). One such observation
is that the stellar mass has a correlation with the halo mass with a slope
$<$1 (see discussion below). This has the implication that at the group scale,
star formation is more efficient. One early study that specifically used
groups to constrain the form of the stellar mass–halo mass relation (Mstars–M)
and the group fstars is that of Giodini et al. (2009). An X-ray selected
sample of groups was constructed from the COSMOS survey, in which X-ray
extended sources were detected based upon a wavelet detection routine
(Vikhlinin et al. 1998). Mean photometric redshifts were assigned to each
candidate and checked against available spectroscopic redshifts from $z$COSMOS
(Lilly et al. 2007). After quality checks, a final sample of 91 groups were
used to constrain the form of the Mstars–M relation. Masses were estimated
based upon a stacked weak lensing analysis (Leauthaud et al. 2010) and the
construction of a Lx–M200 relation, from which the catalog masses were
estimated (note that M500 masses were used in the final analysis, estimated
from the M200 assuming an NFW profile -Navarro et al. 1996, Navarro et al.
1997-, and constant concentration, $c=5$). Within R500 the Mstars–M relation
was found to follow a form of M${}_{\rm stars}\propto$M0.81±0.11 and a stellar
mass fraction of the form f${}_{\rm stars}\propto$M-0.26±0.09 (extending this
to higher masses with the inclusion of clusters, the form follows a relation
of f${}_{\rm stars}\propto$M-0.37±0.04). More recent studies have utilised
increased area X-ray surveys. The XMM Blanco Cosmology Survey (XMM-BCS, Šuhada
et al. 2012) covers 12 deg2 of the sky with XMM-Newton, and was used by Chiu
et al. (2016) to study the form and evolution of the Mstars–M relation using
46 groups/clusters within a mass and redshift range of ($2\lesssim{\rm
M}\lesssim 25)\times 10^{13}~{}{\rm M_{\odot}}$ and 0.1 $\lesssim z\lesssim$
1.02, respectively. The Mstars–M relation is fitted including an evolutionary
redshift term, with parameters estimated by evaluating a likelihood based upon
observing a cluster with observed properties (Lx and Mstars) given a mass,
redshift, Lx–M relation (mass calibration) and the Mstars–M relation. The
likelihood is weighted by the mass function, with full details given in Liu et
al. (2015). The fitted relation has the form M${}_{\rm
stars}\propto$M${}^{0.69\pm 0.15}(1+z)^{-0.04\pm 0.47}$, again consistent with
previous results showing a shallower than unity slope of the relation.
Further, these results indicate little evolution in the stellar content with
stellar mass fraction staying constant out to $z\simeq$1\. In Figure 6, we
plot the Mstars–M relation for various results obtained in the literature
(namely, Giodini et al. 2009; Chiu et al. 2016; Ziparo et al. 2016; Kravtsov
et al. 2018; Pillepich et al. 2018a). We note that no attempt has been made to
correct for the differences in mass calibration used in the various studies.
For reference, two constant stellar mass fractions are given by the black
dashed lines. As discussed above, the relations for Giodini et al. (2009) and
Chiu et al. (2016) are derived at the group scale, whereas Ziparo et al.
(2016) straddles the high-mass groups/low-mass cluster regime (see below) and
Kravtsov et al. (2018) used primarily high-mass clusters (note that the
relation plotted here includes clusters from Gonzalez et al. 2013, as detailed
in Kravtsov et al. 2018). All the relations have a slope less than unity, and
show the trend of decreasing stellar mass fraction from the group to cluster
regime. The observed relations are also consistent with that found in
simulations. Results obtained from the IllustrisTNG simulations show the same
trend in stellar mass (shown by the red dot-dashed line in Figure 6, taken
from Pillepich et al. 2018a).

Due to the difficulty of measuring the stellar masses of groups directly,
requiring deep observations, it is beneficial to use a proxy for the stellar
mass. One such proxy as a tracer of the stellar mass is the K-band luminosity
(LK), as shown in various studies (e.g., Lin et al. 2003; Muzzin et al. 2007).
The use of LK as a stellar mass proxy was investigated in Ziparo et al. (2016)
using a sample of 20 groups/clusters selected from the XXL survey. The
clusters were selected from the overlap of the XXL and CFHTLS, using clusters
with an individual weak lensing mass estimate. Ziparo et al. (2016) found a
relation of the form L${}_{\rm K}\propto$M${}^{0.85^{+0.35}_{-0.27}}_{\rm
WL}$, which while shallower than unity, a slope of 1 cannot be ruled out.
Furthermore, when combined with a sample of high-mass clusters from LoCuSS,
Ziparo et al. (2016) measured a slope of 1.05${}^{+0.16}_{-0.14}$. The
relation derived for the XXL sample is shown in Figure 6 (purple line, with
the shaded light purple region highlighting the 1$\sigma$ uncertainty), which
is derived from the LK–MWL relation assuming a constant mass-to-light ratio of
0.73 (as adopted in Ziparo et al. 2016).

##  4 The role of SMBHs: observed scaling relations and predictions via HD
simulations

As introduced in Section 1, the evolution of the IGrM filling galaxy groups
can not be merely understood in isolation as giant self-similar gaseous
spheres. Particularly in the last decade, a wide range of evidences have
accumulated showing that the SMBHs at the center of each galaxy group are
tightly co-evolving with the hot X-ray halo. Such co-evolution works in both
directions: the hot halo acts as an active atmosphere and reservoir of gas
which recurrently feeds the central SMBH (Gaspari et al. 2013; Prasad et al.
2015; Voit et al. 2017; Temi et al. 2018; Tremblay et al. 2018; Gaspari et al.
2018; Rose et al. 2019; Storchi-Bergmann & Schnorr-Müller 2019). In turn, the
SMBH re-ejects back large amount of mass and energy (in particular via jets
and outflows; e.g., Tombesi et al. 2013; Sa̧dowski & Gaspari 2017; Fiore et
al. 2017), thus re-heating and re-shaping the IGrM via bubbles, shocks, and
turbulence up to the group outskirts (McNamara & Nulsen 2012; Fabian 2012;
Gitti et al. 2012; Brighenti et al. 2015; Gaspari 2015; Liu et al. 2019; Yang
et al. 2019; Wittor & Gaspari 2020; Voit et al. 2020). While the small-scale
AGN self-regulation thermodynamics/kinematics is respectively tackled in the
companion Eckert et al. (2021) and Gastaldello et al. (2021) reviews, here we
focus on the macro-scale integrated (X-ray) IGrM properties and group scaling
relations, which complete and complement Sections 2 and 3. Furthermore, we
compare with high-resolution and hydrodynamical (HD) simulations, in
particular to discuss what the X-ray scaling relations can constrain and tell
us in terms of the baryonic physics shaping the IGrM.

Figure 7: Scaling relations between the central (dynamical/direct) SMBH mass
and key macro X-ray halo properties — adapted from Gaspari et al. (2019). Top-
left to bottom-right panels: gas X-ray luminosity (in the 0.3 - 7.0 keV band),
gas mass, total mass (dominated by the dark matter component), gas electron
density, Compton parameter, and gas fraction. The extraction radius is within
the macro-scale $r<{\rm R_{500}}$. The 85 circles include the observed
direct/dynamical SMBH mass with the X-ray halo detected in the host galaxy
groups (and a handful of clusters). The circle colors depict the morphological
type of the central galaxy: elliptical/blue, lenticular/green, spiral/cyan.
The employed Bayesian analysis accounts for the observed errors, providing a
statistically robust estimate of the intercept and slope of the linear fitting
(top-left inset). The solid red line with dark band shows the 16-84 percentile
interval of the fit, while the 1-$\sigma$ intrinsic scatter is shown via the
wider light red band (dotted lines are the 3-$\sigma$ levels).

Figure 7 shows several key macro X-ray halo scaling relations, which are
usually employed in cosmological studies (see Section 2), but now plotted
against the SMBH mass M∙, which is also an integrated property. These SMBH
masses are retrieved only via robust direct measurements, i.e. resolving the
stellar or gas kinematics within the SMBH influence region (e.g., via HST).
The current largest sample correlated with the available X-ray hot gas
properties is presented by Gaspari et al. (2019), which includes central
galaxies and satellites, with morphological types such as ellipticals (blue
circles), lenticulars (green), and a few spirals (cyan). The 85 systems span a
range of M${}_{500}\sim 3\times 10^{12}-3\times 10^{14}$ M⊙, with the majority
of systems in the group regime (T${}_{\rm x}\sim 1$ keV) and a few in the poor
or cluster tails. The companion Eckert et al. (2021) review shows that the M∙
correlation with $\rm T_{x}$ is significantly tighter than the classical
optical scalings, such as the Magorrian relation (e.g., Kormendy & Ho 2013;
Saglia et al. 2016), with intrinsic scatter down to 0.2 dex, in particular
within the circumgalactic and core region. Here, in Figure 7 we show the other
key X-ray properties integrated up to R500, namely the plasma X-ray luminosity
(in the 0.3 - 7.0 keV band), gas mass, total mass (gas plus stars plus dark
matter), gas density, Compton parameter, and gas fraction. All the fits
parameter – including the intercept, slope, scatter, and correlation
coefficient – are shown in the top-left inset. The related Bayesian analysis
(Gaspari et al. 2019) shows the 1-$\sigma$ intrinsic scatter as light red
bands with the dotted lines enveloping the rare 3-$\sigma$ loci. As indicated
by all correlation coefficients, even the macro-scale IGrM (several 100 kpc to
Mpc scale) is tightly linked to the central M∙. The tighter correlations are
those involving the gas mass/luminosity, X-ray Compton parameter Yx, and total
mass, while the loosest one is that with the gas density. It is interesting to
note that using the core radius (or smaller) as extraction radius (not shown)
leads to similar results, except that the total mass scatter increases by 0.1
dex, with the gas properties emerging as dominant drivers (in particular Mgas
and fgas). In other words, we suggest to use the R500 scaling to probe the
total mass, while smaller extraction radii to probe gas mass (and related
properties).

Figure 8: Effects of different baryonic models in shaping the evolution of the
hot halos, in particular the X-ray luminosity – temperature relation (top
panels) and cool-coreness via the central cooling time (bottom panels);
adapted from Anderson et al. (2015) and Gaspari et al. (2014). The colored
individual objects in the top panels are from a wide range of observational
works (Helsdon & Ponman 2000b; Mulchaey et al. 2003; Osmond & Ponman 2004, Sun
et al. 2009; Pratt et al. 2009, Maughan et al. 2012; luminosities are
extracted mostly in the 0.5 - 2 keV band). The empty circles and solid line
show the raw Anderson et al. (2015) stacking analysis and the unbiased fit,
respectively. The filled black points show evolutionary tracks in large-scale
HD simulations (Gaspari et al. 2014) implementing self-regulated AGN jets
(left) or strong thermal blast feedback (right), preserving or evacuating the
surrounding diffuse gaseous halo, respectively. The initial state is marked
with magenta contour. Evacuation and overheating becomes particularly dramatic
in low-mass, less-bound groups.

The X-ray correlations shown in Figure 7 are important to probe models of
galaxy group evolution. A key debated topic in current extragalactic
astrophysics is which mode of accretion feeds internally the IGrM and
eventually the central SMBH. In hot accretion modes (usually Bondi or ADAF;
e.g., Bondi 1952; Narayan & Fabian 2011), the larger the thermal entropy of
the gas, the stronger the feeding is stifled, as the inflowing gas has to
overcome the hot-halo thermal pressure, increasing toward the center. This
would induce negative correlations with the IGrM properties, which are ruled
out by the slopes shown in Figure 7. Conversely, cold-mode accretion –
typically in chaotic form due to the turbulent IGrM condensation generating
randomly colliding clouds (e.g., Gaspari et al. 2013; Prasad et al. 2015; Voit
2018; Olivares et al. 2019) – would produce major positive and tight
correlations with the gas mass and X-ray luminosity (e.g., the cooling rate is
$\propto$ Lx). Therefore, X-ray correlations favour chaotic cold accretion
(CCA) over hot mode accretion. Hierarchical mergers (of both SMBHs and
galaxies) are another channel to potentially grow such correlations. However,
cosmological simulations (Bassini et al. 2019; Truong et al. 2021) show this
to be effective only at the high-mass end. Moreover, Figure 7 shows that all
the mass scalings are either sub- or super-linear, far off from any simple
self-similar expectation. In other words, a positive baseline due to
hierarchical assembly is present, but gas feeding (dominated by CCA, in terms
of mass) substantially shapes the slope and scatter of such M∙ correlations
over the long-term evolution. Overall, observed scaling relations of macro
X-ray halo properties (shown in Sections 2 and 3) cannot be thought as
disjointed from scaling relations of micro properties (e.g., M∙), since both
systems are tightly co-evolving and intertwined through the several billion
years evolution and over 10 orders of magnitude in spatial scale (cf. the
diagram in Gaspari et al. 2020 linking the micro, meso, and macro scales). As
striking as it appears, such scaling relations allow us to convert back and
forth between vastly different scales, depending on the availability of either
the micro (Section 4) or macro (Section 2) properties for each detected galaxy
group.

The X-ray scaling relations presented in Section 2 can be also leveraged to
test feedback models in large-scale simulations or to calibrate semi-analytic
models of group evolution, thus giving us hints on the dominant baryonic
processes in the IGrM (e.g., Puchwein et al. 2008, McCarthy et al. 2010;
Kravtsov & Borgani 2012; Tremmel et al. 2017). Figure 8 shows the key impact
of archetypal feedback models on the evolution of the diffuse hot atmospheres
(Gaspari et al. 2014). The filled black points indicate the Gyr evolution of
the hot halos IGrM and ICM as it suffers recurrent injections of either
anisotropic mechanical energy via jets (left column) or a strong impulsive
thermal quasar-like blast (right column). Evidently, the latter model has a
dramatic impact on the main Lx–Tx relation (even when the core is excised),
producing a catastrophic evacuation of gas that lowers luminosities by 3
orders of magnitude, especially toward lower-mass group regime ($\rm T_{x}<1$
keV). Such quasar-like models are inconsistent with the observed X-ray scaling
relations, in particular those probing the very low-mass regime via stacking
analysis (e.g., Anderson et al. 2015 shown via empty circles and solid line
fit). Conversely, a tight self-regulation (e.g., achieved via CCA feeding) and
a flickering injection via gentle AGN jets can preserve the hot halo
throughout the several 100 outburst cycles. The bottom panels show indeed that
the initial cool core (magenta contour) can be preserved even in less bound
halos, such as poor galaxy groups, without becoming overheated above half of
the Hubble time. Such overheating is instead catastrophic for an impulsive AGN
blast injection, transforming all hot halos into perennial non-cool-core
systems, which is ruled out by observations finding groups to have almost
universally a low central $t_{\rm cool}$ (Sun et al. 2009; Babyk et al. 2018).
Such self-regulated, gentle SMBH feedback has thus become a staple for subgrid
models of cosmological simulations which can reproduce other tight scaling
relations without any major break at the group scale, such as the M–Yx or M–Tx
computed over R500 (e.g., Planelles et al. 2017; Truong et al. 2018;
Weinberger et al. 2018). For comparisons with further cosmological
simulations, we refer the interested reader to the companion reviews
Oppenheimer et al. (2021) and Eckert et al. (2021) (section 5).

##  5 Galaxy groups with the next generation instruments

Over the next decade, dedicated survey instruments will increase the number of
known groups and clusters out to high redshift, constraining the scenario for
their formation and evolution. Examples include eROSITA in X-rays, Vera Rubin
Observatory and Euclid in the Optical/Infrared, and several “Stage 3” ground-
based mm-wave observatories. The SZ-effect surveys, in particular, will break
new ground by providing robustly selected, large catalogs of clusters at
$z>1.5$, as well as the first informative absolute mass calibration from CMB-
cluster lensing. All future observatories list the baryonic mass and energy
distribution on groups’ scales resolved up to redshift $\sim 2$ and beyond,
when they first appeared as collapsed X-ray bright structures, as one of their
main scientific goals.

Currently, a big step forward in the collection and characterization of low-
mass systems is expected from the ongoing observations of the all X-ray sky
with the extended ROentgen Survey with an Imaging Telescope Array
(eROSITA222http: //www.mpe.mpg.de/erosita/, Predehl et al. 2021). eROSITA is
operating in the X-ray energy band (0.2-10 keV) at L2 orbit on-board the
‘Spectrum-Roentgen-Gamma’ (SRG) satellite. eROSITA has a spatial resolution
comparable to the XMM–Newton one, a similar effective area at low energies,
but a wider field of view, while it will be 20–30 times more sensitive than
the ROSAT sky survey in the soft band and will provide the first all sky
imaging survey in the hard band. Optimizing galaxy group and cluster detection
has been one of the most important tasks during the mission preparation (e.g.,
Clerc et al. 2018, and Käfer et al. 2020 in particular for a detection and
characterization through ICM outskirts that reduces possible biases due the
peaked X-ray emission associated to cool cores). During its 4-yr-long all-sky
survey, with an average exposure of 2.5 ks (whereas the average exposure in
the ecliptic plane region is $\sim$1.6 ks), eROSITA is planned to deliver a
sample of about 3 million active galactic nuclei (AGNs) and about 125,000
galaxy systems (mostly groups) detected with more than 50 photons and
M${}_{\rm 500c}>10^{13}{\rm M}_{\odot}/h$ up to redshift $\sim 1$ (median:
$z\sim 0.3$) (Merloni et al. 2012; Pillepich et al. 2012; Borm et al. 2014;
Zandanel et al. 2018; Pillepich et al. 2018b). Almost all groups (and
clusters) detected with eROSITA will lack sufficient X-ray photons to
accurately constrain temperature and mass profiles (Borm et al. 2014). Thus,
cosmological studies using group and cluster of galaxies to be detected with
eROSITA, will rely heavily on a detailed understanding of the scaling
relations where systematic effects would have to be factored in to ensure that
the cosmological applications of these relations are not hampered. Hence, a
thorough investigation of these systems, to understand the interplay between
the development of the hot IGrM and feedback processes, becomes highly
important, not only for cosmology but also to understand complex baryonic
physics. Moreover, to reach the planned goals of $1\sigma$ errors of 1%, 1%,
7%, and 25% on $\sigma_{8}$, $\Omega_{\rm m}$, $w_{\rm 0}$, and $w_{\rm a}$,
respectively, the critical passages will be: i) a better knowledge (by a
factor of $\sim 4$) of the parameters describing the Lx–M relation to improve
the constraints on $\sigma_{8}$ and $\Omega_{\rm m}$, and ii) a lower mass
threshold to enlarge the analyzed sample to reduce the statistical
uncertainties in DE sector.

The physics of IGrM and ICM will be the main scientific driver for the
exposures with the Advanced Telescope for High ENergy Astrophysics
(Athena333https://www.the-athena-x-ray-observatory.eu/), the X-ray observatory
mission selected by ESA as the second L(large)-class mission (due for launch
in early 2030s) within its Cosmic Vision programme to address the Hot and
Energetic Universe scientific theme. Among the main scientific goals, Athena
will have the capabilities to find evolved groups of galaxies with M${}_{\rm
500c}>5\times 10^{13}$M⊙ and hot gaseous atmospheres at $z>2$. For about ten
of those, a global gas temperature estimate is expected to be measurable
(Pointecouteau et al. 2013). Athena will determine the magnitude of the
injection of non-gravitational energy into the IGrM and ICM as a function of
cosmic epoch by measuring the structural properties (e.g., the entropy
profiles) out to R500, and their evolution up to $z\sim 2$, for a sample of
galaxy groups and clusters, improving significantly the constraints, presently
unknown, on the evolution of the scaling relations between bulk properties of
the hot gas (Ettori et al. 2013; Pointecouteau et al. 2013). In local systems,
Athena will be also in condition to determine the occurrence and impact of AGN
feedback phenomena by searching for ripples in surface brightness in the cores
of a statistical sample of objects. Using temperature-sensitive line ratios,
Athena’s observations will trace how much gas is at each temperature in the
cores of these systems, providing a complete description of the gas heating-
cooling balance (Croston et al. 2013) and transport processes such as
turbulence and diffusion (Cucchetti et al. 2018; Roncarelli et al. 2018;
Mernier et al. 2020).

Presently, concepts funded for study by NASA for consideration in the 2020
Astrophysics Decadal Survey, Lynx444https://www.lynxobservatory.com/ (as high-
energy flagship mission) and AXIS555http://axis.astro.umd.edu (as probe-class
mission) are proposing to investigate with sub-arcsecond resolution over a FoV
of 400-500 arcmin2 the X-ray sky, improving this capability of a factor $\sim
100$ with respect to Chandra ACIS-I. Their predicted low background level and
capability to resolve embedded and background AGN will allow to track group
and cluster emission at very low surface brightness values. For example, AXIS
is expected to reach a flux limit of $\sim 1\times 10^{-16}$ erg/s/cm2 (0.5–2
keV) over the 50 deg2 of the proposed Wide Survey (e.g., Marchesi et al.
2020), providing the detection of thousands of groups and clusters, and
evidences of merging and effects of feedback resolved even at high-$z$. With a
larger collection of instruments, Lynx will be also able to resolve the
thermodynamic and kinematic structure of systems at $z\approx 2$, as well as
determine the role of feedback from AGN and stars.

Complementary data will be provided from the ongoing (and planned) SZ surveys.
SPT-3G666https://pole.uchicago.edu/ will extend the work of SPT-SZ by covering
a nearly identical area of 2,500 deg2 but with noise levels about 12, 7, and
20 times lower at 95, 150, and 220 GHz, respectively. This will enhance the
sensitivity, allowing to reduce the mass limit and extending the redshift
coverage with respect to SPT-SZ. About 5000 clusters with M${}_{\rm
500c}\mathrel{\mathchoice{\lower 2.5pt\vbox{\halign{$\mathsurround
0pt\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
2.5pt\vbox{\halign{$\mathsurround
0pt\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}10^{14}$M⊙ at a signal-
to-noise $>4.5$ (corresponding to a 97% purity threshold) are expected by the
completion of the survey (2023; Benson et al. 2014). The next-generation
ground-based cosmic microwave background experiment
CMB-S4777https://cmb-s4.org/, with a planned beginning of science operations
in 2029, will build catalogs more than an order of magnitude larger than
current ones, lowering the mass limit M500c to 6-8 $\times 10^{13}$ M⊙ at
$z>0.3$ and being especially adept at finding the most distant groups and
clusters. Large catalogs of low-mass systems together with the progress on the
measurement of the thermal SZ power spectrum will open a new window into
groups.

In the optical and near-infrared bands, space missions
(Euclid888https://www.euclid-ec.org/ -from 2022- and Nancy Grace Roman Space
Telescope999https://roman.gsfc.nasa.gov/ \- formerly WFIRST; launch date:
2025) and ground based missions (Vera Rubin
Observatory101010https://www.lsst.org and the 4-metre Multi-Object
Spectroscopic Telescope, 4MOST111111https://www.4most.eu/cms/) will map the
large scale structures over more than 15,000 deg2, extending the current
catalogs of systems with M${}_{\rm 500c}>5\times 10^{13}$M⊙ (see, e.g.,
results from DES121212https://www.darkenergysurvey.org/ in Abbott et al. 2020)
by orders of magnitude, in particular at high ($z>1$) redshifts (e.g.,
Sartoris et al. 2016). Of particular interest for the measurement of the
velocity dispersions, is the WFIRST and 4MOST observatories. The 4MOST
observatory has been designed as a survey instrument at the forefront, with
plans underway to combine the power of 4MOST with eROSITA (Finoguenov et al.
2019) in order to provide dynamical mass estimates for $\sim$10000 clusters at
redshift z$<$0.6 and masses $>10^{14}$ M⊙. $4MOST$ will also provide
spectroscopic confirmation of $eROSITA$ detected groups at redshifts $<$0.2
down to a mass limit of 1013 M⊙. Additionally, the wide area vista
extragalactic survey (WAVES, Driver et al. 2019) being planned using $4MOST$,
is aiming to perform the WAVES-Wide and WAVES-Deep surveys, allowing for the
construction of optically selected groups catalogs. The WAVES-WIDE(-DEEP)
surveys aiming to cover an area of $\sim$70 (1200) deg2, identifying
$\sim$50000 (20000) dark matter halos down to a mass of $10^{14}$ ($10^{11}$)
M⊙ and out to a redshift of $z_{\rm phot}\lesssim$0.2 (0.8). Similarly to
X-ray selected objects, optically-selected groups are physically heterogenous
systems (e.g., see the dynamical analysis by Zheng & Shen (2021) for a sample
of compact groups). However, it is possible that the physical processes at
work in the IGrM of optically– and X-ray–selected systems are different. Thus,
the comparison of group samples selected via distinct methods can shed light
on these physical phenomena.

##  6 Final remarks

Bridging the gap in mass between field galaxies and massive clusters, galaxy
groups are key systems to make progress in our understanding of structure
formation and evolution. Thanks to the current generation of X-ray satellites,
together with dedicated hydrodynamical simulations, there have been
significant improvement in our comprehension of the interplay between the hot
ambient gas, radiative cooling and feedback due to, e.g., AGN activity and SNe
winds, in particular in the central regions. Indeed, the thermodynamic
structure of galaxy groups is more complex than in massive galaxy clusters,
with the physics associated to non gravitational processes playing a
significant role in shaping their general properties. The scaling relations
capture the result of the various (thermal and non thermal) processes and show
that galaxy groups are not simply the scaled-down versions of rich clusters.
Thanks to the enlarged catalogs of low-mass systems that the current (and
upcoming) wide surveys at X-ray, millimeter, and optical wavelengths will
provide, such scaling relations can be measured with very high precision. The
comparison between results obtained from differently selected samples will
shed light on the intrinsic properties of the groups’ population.

Due to the complexity of the X-ray emitting processes in the low-temperature
regime, and of how AGN heating impacts the general properties of the core of
poor systems, the interpretation will depend on the specific choices of the
individual analyses. For instance, many X-ray studies on groups (and clusters)
provide integrated measurements within a certain aperture. However, the
definition of such aperture is often quite different, and the comparison
between the various works is not always straightforward. In the future, it is
desirable to provide the global properties using an unified definition of the
regions, that are efficiently accessible from observations. This will ease the
comparison between different observational and theoretical studies, improving
our understanding of the physical processes at work in the complex group
regime. Of course, in each study there are good reasons to use a specific
energy band or definition of region of interest. However, regardless of the
choices made in each paper to reach specific goals, we suggest to also
provide, whenever possible, both global and core-excised properties within
R500. Although there are evidences that the cool-core radius does not scale
uniformly with the virial radius, we think that the common choice of excising
$r<$0.15R500 is a good starting point. For the rest-frame luminosities, we
have shown that the 0.5-2 keV band is less sensitive to the choice of the
abundance table and is easily accessible for all the current and future
facilities (differently from the 0.1-2.4 keV band which extend to a regime
where, for instance, Chandra and XMM-Newton are not well calibrated and also
the choice of the abundance table start to play a role as discussed in Section
2.1). However, to ease the comparison with the literature it is useful to
provide also the rest-frame bolometric and 0.1-2.4 keV band luminosities.
Finally, until R500 will be routinely achieved for most of the systems in the
low-mass regime, we also suggest to provide the properties at R2500 (i.e.,
$\sim$0.5R500). Of course, there are further complications (e.g., the impact
on the temperature of using a certain abundance table or spectral code,
Lovisari et al. 2015; the choice of the column density, Lovisari & Reiprich
2019; the fitting technique, Balestra et al. 2007) which play a relevant role
in the low-mass (but not only) systems. Nonetheless, starting to set standard
definitions will definitely help the analysis in this critical regime.

Conceptualization, L.L, S.E., M.G., and P.A.G.; resources, L.L, S.E., M.G.,
and P.A.G.; data curation, L.L., S.E., M.G., and P.A.G.; writing-original
draft preparation, L.L, S.E., M.G., and P.A.G.; writing-review and editing,
L.L, S.E., M.G., and P.A.G.; visualization, L.L, S.E., M.G., and P.A.G.;
supervision, L.L; project administration, L.L. All authors have read and
agreed to the published version of the manuscript.

L.L. and S.E. acknowledge financial contribution from the contracts ASI-INAF
Athena 2015-046-R.0, 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". M.G. acknowledges partial support by NASA Chandra GO8-19104X/GO9-20114X
and HST GO-15890.020-A grants. P.A.G. acknowledges support from the UK Science
and Technology Facilities Council via grants ST/P000525/1 and ST/T000473/1.

###### Acknowledgements.

The authors thank the anonymous referees for useful comments and suggestions
that helped to improve and clarify the presentation of this work. We thank M.
Sun for providing the luminosity values for the group sample analysed in his
work. We also thank MNRAS and the AAS, together with the authors of the
corresponding publications, for granting permission to use images published in
their journals. The authors declare no conflict of interest. no References yes

## References

  * Abbott et al. (2018) Abbott, T. M. C., Abdalla, F. B., Alarcon, A., et al. 2018, Phys. Rev. D, 98, 043526
  * Abbott et al. (2020) Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2020, Phys. Rev. D, 102, 023509
  * Adami et al. (2018) Adami, C., Giles, P., Koulouridis, E., et al. 2018, A&A, 620, A5
  * Aguerri & Zarattini (2021) Aguerri, J. A. L. & Zarattini, S. 2021, Universe, 7, 132
  * Allen et al. (2008) Allen, S. W., Rapetti, D. A., Schmidt, R. W., et al. 2008, MRAS, 383, 879
  * Alshino et al. (2010) Alshino, A., Ponman, T., Pacaud, F., & Pierre, M. 2010, MNRAS, 407, 2543
  * Anders & Grevesse (1989) Anders, E. & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197
  * Anderson et al. (2015) Anderson, M. E., Gaspari, M., White, S. D. M., Wang, W., & Dai, X. 2015, MNRAS, 449, 3806
  * Andreon et al. (2016) Andreon, S., Serra, A. L., Moretti, A., & Trinchieri, G. 2016, A&A, 585, A147
  * Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Babyk et al. (2018) Babyk, I. V., McNamara, B. R., Nulsen, P. E. J., et al. 2018, ApJ, 862, 39
  * Balestra et al. (2007) Balestra, I., Tozzi, P., Ettori, S., et al. 2007, A&A, 462, 429
  * Bassini et al. (2019) Bassini, L., Rasia, E., Borgani, S., et al. 2019, A&A, 630, A144
  * Beers et al. (1990) Beers, T. C., Flynn, K., & Gebhardt, K. 1990, AJ, 100, 32
  * Behroozi et al. (2010) Behroozi, P. S., Conroy, C., & Wechsler, R. H. 2010, ApJ, 717, 379
  * Benson et al. (2014) Benson, B. A., Ade, P. A. R., Ahmed, Z., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9153, Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII, ed. W. S. Holland & J. Zmuidzinas, 91531P
  * Bharadwaj et al. (2015) Bharadwaj, V., Reiprich, T. H., Lovisari, L., & Eckmiller, H. J. 2015, A&A, 573, A75
  * Bharadwaj et al. (2014) Bharadwaj, V., Reiprich, T. H., Schellenberger, G., et al. 2014, A&A, 572, A46
  * Bocquet et al. (2019) Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55
  * Bogdán et al. (2018) Bogdán, Á., Lovisari, L., Volonteri, M., & Dubois, Y. 2018, ApJ, 852, 131
  * Böhringer et al. (2017) Böhringer, H., Chon, G., & Fukugita, M. 2017, A&A, 608, A65
  * Böhringer et al. (2012) Böhringer, H., Dolag, K., & Chon, G. 2012, A&A, 539, A120
  * Böhringer & Werner (2010) Böhringer, H. & Werner, N. 2010, A&A Rev., 18, 127
  * Bondi (1952) Bondi, H. 1952, MNRAS, 112, 195
  * Borgani et al. (2008) Borgani, S., Diaferio, A., Dolag, K., & Schindler, S. 2008, Space Sci. Rev., 134, 269
  * Borm et al. (2014) Borm, K., Reiprich, T. H., Mohammed, I., & Lovisari, L. 2014, A&A, 567, A65
  * Brighenti & Mathews (2002) Brighenti, F. & Mathews, W. G. 2002, ApJ, 573, 542
  * Brighenti et al. (2015) Brighenti, F., Mathews, W. G., & Temi, P. 2015, ApJ, 802, 118
  * 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
  * Carlberg et al. (1999) Carlberg, R. G., Yee, H. K. C., Morris, S. L., et al. 1999, Philosophical Transactions of the Royal Society of London Series A, 357, 167
  * Chiu et al. (2018) Chiu, I., Mohr, J. J., McDonald, M., et al. 2018, MNRAS, 478, 3072
  * Chiu et al. (2016) Chiu, I., Saro, A., Mohr, J., et al. 2016, MNRAS, 458, 379
  * Clerc et al. (2018) Clerc, N., Ramos-Ceja, M. E., Ridl, J., et al. 2018, A&A, 617, A92
  * Colafrancesco & Giordano (2007) Colafrancesco, S. & Giordano, F. 2007, A&A, 466, 421
  * Connelly et al. (2012) Connelly, J. L., Wilman, D. J., Finoguenov, A., et al. 2012, ApJ, 756, 139
  * Croston et al. (2013) Croston, J. H., Sanders, J. S., Heinz, S., et al. 2013, arXiv e-prints, arXiv:1306.2323
  * Cucchetti et al. (2018) Cucchetti, E., Pointecouteau, E., Peille, P., et al. 2018, A&A, 620, A173
  * Dai et al. (2010) Dai, X., Bregman, J. N., Kochanek, C. S., & Rasia, E. 2010, ApJ, 719, 119
  * de Haan et al. (2016) de Haan, T., Benson, B. A., Bleem, L. E., et al. 2016, ApJ, 832, 95
  * Decker et al. (2019) Decker, B., Brodwin, M., Abdulla, Z., et al. 2019, ApJ, 878, 72
  * Despali et al. (2016) Despali, G., Giocoli, C., Angulo, R. E., et al. 2016, MNRAS, 456, 2486
  * Driver et al. (2019) Driver, S. P., Liske, J., Davies, L. J. M., et al. 2019, The Messenger, 175, 46
  * Eckert et al. (2016) Eckert, D., Ettori, S., Coupon, J., et al. 2016, A&A, 592, A12
  * 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
  * Eckmiller et al. (2011) Eckmiller, H. J., Hudson, D. S., & Reiprich, T. H. 2011, A&A, 535, A105
  * Eigenthaler & Zeilinger (2007) Eigenthaler, P. & Zeilinger, W. W. 2007, Astronomische Nachrichten, 328, 699
  * Eke et al. (2004) Eke, V. R., Baugh, C. M., Cole, S., et al. 2004, MNRAS, 348, 866
  * Ettori (2013) Ettori, S. 2013, MNRAS, 435, 1265
  * Ettori (2015) Ettori, S. 2015, MNRAS, 446, 2629
  * Ettori et al. (2020) Ettori, S., Lovisari, L., & Sereno, M. 2020, A&A, 644, A111