**About the Editors**

**Lorenzo Lovisari** (Dr.) is a junior researcher at the Astrophysics and Space Science Observatory (OAS) of Bologna in Italy. He is an X-ray observer interested in the physics of the hot plasma in groups and clusters of galaxies. He obtained his PhD in Physics in 2010 at the University of Innsbruck studying "Metal Distribution in Galaxy Clusters". Since then, he held several postdoctoral positions, at Argelander-Institut fur Astronomie in Bonn (Germany) and at Center for Astrophysics ¨ — Harvard & Smithsonian in Cambridge (USA). His main research interests focused on the scaling properties of galaxy systems and on their chemical enrichment. He is also very involved in scientific activities through giving talks, publishing articles, organizing workshops, and being part of several international collaborations.

**Stefano Ettori** (Dr.) is Primo Ricercatore at the Astrophysics and Space Science Observatory (OAS) of Bologna in Italy. He works on the formation and evolution of galaxy clusters, focusing on the physical properties of the Intra-Cluster Medium and on the use of galaxy clusters as cosmological probes. He obtained his PhD in 1998 at IoA, University of Cambridge (UK). He was a research associate at IoA and then fellow at ESO (Garching, Germany) before becoming staff researcher at INAF OAS in Bologna in 2004. He is involved as key member in several international collaborations (chair of SWG1 for ESA L-mission Athena; co-lead of WP-7 "Astrophysics of Clusters of Galaxies" for ESA M-mission Euclid; co-PI of CHEX-MATE).

## *Editorial* **The Physical Properties of the Groups of Galaxies**

**Lorenzo Lovisari 1,2,\* and Stefano Ettori 1,3**


Galaxy groups consist of a few tens of galaxies bound in a common gravitational potential. They dominate the number count of the halo mass function and contain a significant fraction of the overall universal baryon budget. Being less massive than clusters, the energy that is supplied by supernovae and active galactic nuclei (AGNs) to the hot intragroup medium (IGrM) can easily exceed their gravitational binding energy. Thus, it is expected that these non-gravitational mechanisms have a strong effect on the distribution of of the baryons, making galaxy groups ideal targets to constrain the mechanisms governing the cooling–heating balance. The net effect of the various feedback processes in action in the gravitational potential wells is to change the radial distribution of the energy and mass in groups, affecting the correlations between their observed properties. Therefore, they are key to our understanding of how the bulk of matter in the Universe accretes and forms hierarchical structures and how different sources of feedback affect their gravitational collapse.

Despite their crucial role in cosmic structure formation and evolution, galaxy groups have received less attention compared to massive clusters. This is in part due to the challenges (e.g., faint X-ray emission, low number of galaxies in optical, low S/N in SZ) associated with their detection, observation, and characterization. With the advent of eROSITA (launched in 2019), many thousands of galaxy groups will be detected by X-ray, complementing on-going and future optical (DES, Euclid, Vera Rubin), SZ (SPT-3G, ACT), and radio (LOFAR, MeerKAT) surveys, paving the way towards the exploitation of the next generation of X-ray instruments (onboard XRISM—expected to fly by 2023—and the ESA L2 mission Athena—expected to fly in the 2030s).

To foster progress in the field of the physical properties of galaxy groups, facilitating effective cross-communication among observers, theorists, and simulators, we organized a Special Issue (https://www.mdpi.com/journal/universe/special\_issues/PPGG (accessed on 21 July 2021)) dedicated to the physical properties of galaxy groups. We aimed to collect and organize the latest developments in our understanding of these systems and present future prospects from both observational and theoretical points of view. This Special Issue includes five manuscripts, which we summarize briefly in the following.

• *"Properties of Fossil Groups of Galaxies", by Aguerri and Zarattini [1]*

Fossil groups are an ostensibly special class of galaxy systems. They are objects dominated by a single, bright, elliptical galaxy and are thought to be the latest stage in the evolution of galaxy groups. Their properties differ from the one of other galaxy groups, and since it is likely that they did not experience recent major mergers, they should represent archetypal old undisturbed systems, and are therefore important systems to study. In the review, we show the main observational and theoretical works demonstrating that these systems fall very well in the current theory of structure formation in the Universe.

#### • *"Scaling Properties of Galaxy Groups", by Lovisari et al. [2]*

The scaling relations are the result of the different physical processes at work in the intracluster medium and provide an important tool to study its thermodynamic history.

**Citation:** Lovisari, L.; Ettori, S. The Physical Properties of the Groups of Galaxies. *Universe* **2021**, *7*, 254. https://doi.org/10.3390/ universe7080254

Received: 14 July 2021 Accepted: 19 July 2021 Published: 21 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In fact, the various processes that govern the formation of galaxy groups are suspected of systematically increasing the intrinsic scatter of the groups relations and changing their integrated properties. We overview the most recent studies on the X-ray scaling relations, obtained at the galaxy group scale, and their relations with optical properties and the supermassive Black Hole (SMBH) mass.

• *"Feedback from Active Galactic Nuclei in Galaxy Groups", by Eckert et al. [3]*

The formation and evolution of the physical properties in groups are a direct consequence of the interplay between galaxy evolution, the development of the intragroup medium, and feedback. Many authors have argued that feedback from SMBHs plays a crucial role in regulating the star formation rates of massive galaxies and suppressing the onset of catastrophic cooling by carving cavities and driving shocks across the medium. We review the current observational evidence for AGN feedback in nearby galaxy groups with observations at X-ray, radio, and millimeter wavelengths and describe the theoretical advances made in recent years to interpret the heating–cooling cycle.

• *"Simulating Groups and the IntraGroup Medium: The Surprisingly Complex and Rich Middle Ground between Clusters and Galaxies", by Oppenheimer et al. [4]*

The influence of the feedback processes is complex and difficult to model and to reproduce in simulations. However, cosmological simulations have enabled breakthroughs in our understanding of the gas and stellar contents of groups and of the impact of groups for cosmological parameter estimation. The review focuses on how groups process their baryons in a cosmological context, discussing the current limitations and the perspectives for improving the theoretical modeling in the near future.

• *"The Metal Content of the Hot Atmospheres of Galaxy Groups", by Gastaldello et al. [5]*

Metals play a central role in the thermodynamic balance of galaxy systems by sustaining the cooling of their environment by means of spectral line emissions. DUe to the shallower gravitational potential of groups, feedback effects leave important marks on their gas and metal contents. Therefore, the shape of the abundance profiles can be used to investigate the impact of the feedback in the IGrM. We review the status of the metal abundance measurements in the IGrM and the progress made by simulations to reproduce and interpret those measurements.

**Funding:** 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".

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Review* **Scaling Properties of Galaxy Groups**

**Lorenzo Lovisari 1,2,\*,†, Stefano Ettori 1,3,†, Massimo Gaspari 1,4,† and Paul A. Giles 5,†**


**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

#### **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. [1]). However, the distinction between groups and clusters is quite loose and no universal definition exists in the 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. [2]) 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 based on their optical and physical characteristics: poor/loose groups, compact groups, and fossil groups (e.g., Eigenthaler and Zeilinger [3]). Poor/loose groups are aggregate of galaxies with a space density of ∼10−<sup>5</sup> Mpc−<sup>3</sup> (e.g., Nolthenius

**Citation:** Lovisari, L.; Ettori, S.; Gaspari, M.; Giles, P.A. Scaling Properties of Galaxy Groups. *Universe* **2021**, *7*, 139. https://doi.org/ 10.3390/universe7050139

Academic Editor: Francesco Shankar

Received: 31 March 2021 Accepted: 28 April 2021 Published: 10 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and White [4]). Compact groups are small and relatively isolated systems of typically 4–10 galaxies with a space density of ∼10−<sup>6</sup> Mpc−<sup>3</sup> (e.g., Hickson [5]). Fossil groups are objects dominated by a single bright elliptical galaxy (a formal definition is provided in Jones et al. [6]). Early studies (e.g., Helsdon and Ponman [7]) 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 et al. However, since the optical properties are not always available, a threshold of M∼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. [8], Sanderson et al. [9]). 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 <sup>∼</sup>30/210/1500 more objects in the mass range M500 <sup>=</sup> 1013 <sup>M</sup> <sup>−</sup> M1 than in M500 <sup>&</sup>gt; M1, and M1 <sup>=</sup> 1/2/5 <sup>×</sup> 1014M at *<sup>z</sup>* <sup>=</sup> 0; see, e.g., [10]). 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 [11], Fukugita et al. [12], Eke et al. [1]). 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 significantly alter the properties of these systems being the energy input associated with these sources comparable to the binding energies of groups (e.g., Brighenti and Mathews [13], McCarthy et al. [14], Gaspari et al. [15]). However, the relative contributions of the different feedback processes are 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*cool < 1 Gyr, as described in Hudson et al. [16]), weak cool-cores (1 < *t*cool < 7.7 Gyr), and non-cool-cores (NCC; *t*cool > 7.7 Gyr) objects at the group scale are similar to those in galaxy clusters (Bharadwaj et al. [17]). However, O'Sullivan et al. [18] 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. [17] 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. [19], Gaspari et al. [20], Lakhchaura et al. [21]). 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., McCarthy et al. [14], Sijacki et al. [22], Puchwein et al. [23], Fabjan et al. [24], Le Brun et al. [25]). This deviation is often attributed to active galactic nucleus (AGN) feedback (e.g., Planelles et al. [26], Gaspari et al. [27], Truong et al. [28]).

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 [29], Pratt et al. [30], Mittal et al. [31], Bharadwaj et al. [32], Mantz et al. [33], Lovisari et al. [34]). 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. [35]). In fact, recent results by O'Sullivan et al. [18], who analyzed a sample of optically selected groups, show that ∼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., [36–46]). 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 halo mass function). For instance, recent results of the Dark Energy Survey (DES) collaboration show that the *σ*8–Ω<sup>M</sup> posteriors have a 5.6*σ* tension with *Planck* CMB results, and a 2.4*σ* tension with galaxy clustering and cosmic shear results [47]. 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 *<sup>λ</sup>* <sup>&</sup>lt; 30 (corresponding to <sup>∼</sup>1014 <sup>M</sup>). 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. [48] suggest that the discrepancy is probably due to the modeling 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 [49]. 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 [50]. 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. 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 Ω<sup>M</sup> and/or *σ*<sup>8</sup> values than the true values. This scenario is supported by the finding of Schellenberger and Reiprich [44], 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, then 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 lowmass 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 several integrated observed quantities of galaxy groups, and complement/update the previous reviews in the field by, e.g., Mulchaey [51] and Sun [52]. 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 [53], 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 and Suto [54], Bryan and Norman [55], Voit [56], Maughan et al. [57], Borgani et al. [58], Böhringer et al. [59], Ettori [60], Giodini et al. [61], Maughan [62], Ettori [63], Ettori et al. [64]). 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,

$$\frac{\mathbf{M}\_{\Lambda\_{\varepsilon}}}{\mathbf{R}\_{\Lambda\_{\varepsilon}}^3} = \text{constant} \tag{1}$$

where MΔ*<sup>z</sup>* is the mass contained within the radius RΔ*<sup>z</sup>* , encompassing a mean density Δ*<sup>z</sup>* times the critical density of the Universe *ρc*(*z*), so that MΔ*<sup>z</sup>* ∝ *ρc*(*z*)Δ*z*R3 Δ*z* . The critical density of the Universe scales with redshift as *ρc*(*z*) = *ρc*(*z*=0)E2(*z*), where E(*z*) = H*z*/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., Tx ∝ GMΔ*z*/RΔ*<sup>z</sup>* ∝ R<sup>2</sup> Δ*z* ), and therefore of the total mass of the cluster:

$$\mathbf{M}\_{\Lambda\_{\overline{z}}} \propto \mathbf{T}\_{\mathbf{x}}^{\ 3/2}. \tag{2}$$

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: Mg ∝ Tx 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

$$\boldsymbol{\epsilon} = \mathbf{n}\_{\boldsymbol{\epsilon}} \mathbf{n}\_{\mathbb{P}} \, \Lambda(\mathbf{T}\_{\boldsymbol{\chi}}, \mathbf{Z}\_{\left(\bar{\boldsymbol{\chi}}\right)}), \tag{3}$$

where ne and np are the number densities of electrons and protons, respectively, that are related to the gas mass density *ρ*<sup>g</sup> through the relation *ρ*<sup>g</sup> = *μ*mp(ne + np), *μ* is the mean molecular weight (∼0.6 for a plasma with solar abundance), mp is the proton mass, and <sup>Λ</sup>(Tx, Z) is the cooling function which depends on the mechanism of the emission1 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., <sup>Λ</sup>(Tx, Z) <sup>∝</sup> T1/2 <sup>x</sup> ). Thus, for sufficiently massive systems the bolometric X-ray luminosity (i.e., 0.01–100 keV band) is given by

$$\mathbf{L}\_{\mathbf{x},\text{bol}} \propto \int \boldsymbol{\epsilon} \, \mathbf{d} \mathbf{V} \propto \mathbf{n}\_{\mathbf{p}}^{2} \, \mathbf{T}\_{\mathbf{x}}^{1/2} \mathbf{R}^{3} \propto \mathbf{f}\_{\mathbf{g}}^{2} \, \mathbf{T}\_{\mathbf{x}}^{2} \propto \mathbf{T}\_{\mathbf{x}}^{2} \tag{4}$$

with the last scaling obtained assuming a constant gas fraction as predicted by the selfsimilar scenario. By combining Equations (2) and (4) one obtains the well-known relation between bolometric luminosity and total mass (i.e., Lx,bol ∝ M4/3).

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 between 3 and 10 keV plasmas in the 0.5–2 keV band is <10% for given emission measure), so that Lx,soft ∝T3/2 <sup>x</sup> , and hence using Equation (2), Lx,soft ∝ M.

**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. [69]) model (v3.0.9) in XSPEC (Arnaud [70]) for two different values of metallicity: 1.0 (solid lines) and 0.3 times the solar abundance as in Asplund et al. [71]. All curves are normalized to the bolometric emissivity at kBTx = 20 keV with Z=1Z. (**middle panel**): the emissivity slope as a function of temperature showing the impact of the different *Z* 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.

<sup>1</sup> Three 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 [65], Peterson and Fabian [66], Kaastra et al. [67], and Böhringer and Werner [68].

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. [72], Gonzalez et al. [73], Gastaldello et al. [74], Pratt et al. [30], Dai et al. [75], Gonzalez et al. [76], Lovisari et al. [77], Eckert et al. [78]; see also the companion reviews by Eckert et al. and Oppenheimer et al.). 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 1, we provide the dependence for a set of interesting cases.


**Table 1.** Emissivity dependence on T*<sup>x</sup>* and Z for different temperature regimes and energy bands.

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. [79] 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.), 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 Lx ∝ <sup>T</sup>1.5+*<sup>γ</sup>* <sup>x</sup> and Lx <sup>∝</sup> M1<sup>+</sup>*γ*, where *<sup>γ</sup>* 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 1). 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 ∼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 Lx,bol ∝T∼1.9 <sup>x</sup> .

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 and Ponman [80], Sun et al. [81], Mernier et al. [82], Lovisari and Reiprich [83]; see also the companion review by Gastaldello et al.), 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 [52], Lovisari et al. [77]). Sun [52] pointed-out that although the average luminosities (soft band or bolometric) only change by ∼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 such as isothermality and single overall abundance are chosen to obtain an estimate of the luminosity.

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 - 1 keV) when the abundances of (Grevesse and Sauval [84], GRSA), (Asplund et al. [71], ASPL), or (Lodders et al. [85], LODD) are used. The disagreement is much more significant (i.e., up to ∼10%) when the luminosities are estimated with the abundance table by (Anders and Grevesse [86], 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

$$\mathbf{K} = \mathbf{k}\_{\rm B} \mathbf{T}\_{\rm x} \; \mathbf{n}\_{\rm e}^{-2/3} \tag{5}$$

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., [56]), 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. [8], Lloyd-Davies et al. [87], Ponman et al. [88], Finoguenov et al. [89], Sun et al. [81], Johnson et al. [90], Panagoulia et al. [91]), 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. and Oppenheimer et al.). 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∝R1.1.

**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. [71] and Lx obtained with the abundances of (Grevesse and Sauval [84], pink), (Lodders et al. [85], turquoise), and (Anders and Grevesse [86], orange), 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 ∼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).

#### *2.2. Observed Scaling Relations*

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. [61] 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 and Ponman [92], Helsdon and Ponman [7], Xue and Wu [93]), later investigations found relations only slightly steeper than the ones for clusters (e.g., Osmond

and Ponman [94], Shang and Scharf [95], Eckmiller et al. [96], Sun [52], Lovisari et al. [77]). 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 bestfit 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 enforces 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). Although an attempt to correct this bias in cluster scaling relations has been provided in many different X-ray studies (e.g., Ikebe et al. [97], Stanek et al. [98], Pacaud et al. [99], Vikhlinin et al. [38], Pratt et al. [30], Mittal et al. [31], Schellenberger and Reiprich [100]), there are only a few papers providing the correction in the galaxy group regime. For instance, Lovisari et al. [77] analyzing 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. [32], 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. [101] and Zou et al. [79] 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. [79] 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. [102], Anderson et al. [103], Andreon et al. [104], and O'Sullivan et al. [18] 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. [105]; see also Šuhada et al. [106] for a performance comparison between sliding cell and wavelet detection algorithms). Xu et al. [107], 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. [108], then our view could be significantly biased.

**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. [96], E11), (Lovisari et al. [77], L15), (Sun et al. [81], S09), (Johnson et al. [90], J09), (Bharadwaj et al. [32], B15), (Zou et al. [79], Z16), (Pearson et al. [109], P17). The groups measurements are compared with the ones from X-ray-selected (Migkas et al. [110], M20) and SZ-selected (Lovisari et al. [34], 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. [111]. Empty symbols are from optically selected samples. The lines represent the fitted relation for Tx < 3 keV systems (dotted), Tx > 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 [112]) 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 1 for different ranges of temperatures (showed as brown vertical lines).


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

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∝T1.5<sup>+</sup>*<sup>γ</sup>* where *γ* 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 1). 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 Lx ∝ M1+*γ*. The values for slope*LIRA* have been obtained by fitting each dataset with LIRA (Sereno [112]) assuming self-similar time evolution with scatter on both variables and with the following pivot values: 3 keV, 10<sup>44</sup> erg/sec, 2 × <sup>10</sup><sup>14</sup> <sup>M</sup>, 1013 <sup>M</sup>, 1014 <sup>M</sup> 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; † and ‡ indicate L*<sup>x</sup>* obtained with *ROSAT* or *XMM* data, while and indicate if *Chandra* or *XMM* data have been used for the analysis. , , and ♦, indicate that the core-excised region was not a fixed fraction of *R*500, or fixed to 0.1*R*<sup>500</sup> or 0.15*R*500, while and 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. [81], S09), (Eckmiller et al. [96], E11), (Kettula et al. [113], K13), (Lovisari et al. [77], L15), (Bharadwaj et al. [32], B15), (Kettula et al. [101], K15), (Zou et al. [79], Z16), (Umetsu et al. [114], U20), (Sereno et al. [115], 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. [111], see also Nevalainen et al. [116]) 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. [113]). 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 and Ponman [94], is related to the flattening of the fitted relation because the scatter in log (Tx) will be asymmetric (assuming that the scatter in temperature is symmetric) with larger scatter towards low log (Tx). Moreover, if the quality of the data is 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 employs 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. [34]). To remove this last uncertainty and provide comparable results we fit the published data with the same fitting method (i.e., using LIRA; Sereno [112]) 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. [81]) implying that the low gas fraction observed in groups is mainly due to the low gas fraction of groups within ∼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–*σ*<sup>v</sup> 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. [31] 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 and Giordano [117] 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 and Giordano [117], 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. 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. [118],

Finoguenov et al. [89], Sun et al. [81], Johnson et al. [90], Panagoulia et al. [91]). 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. [90] 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 selfsimilar scenario it simply scales with the gas temperature. The finding by Sun et al. [81] 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. [88]. 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.). This agrees with the finding by Pratt et al. [119] for a sample of galaxy clusters.

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 the breaking of the degeneracy between Ω<sup>m</sup> and *σ*<sup>8</sup> (e.g., Reiprich and Böhringer [36]). A large number of observations of galaxy clusters (e.g., see Mantz et al. [120], Schellenberger and Reiprich [44], Mantz et al. [121], Bulbul et al. [122], and Lovisari et al. [34] for recent studies; we refer to Böhringer et al. [59] and Giodini et al. [61] 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. [96] found a slope of 1.34 ± 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. [96] 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. [96]. A similar slope was also obtained by Lovisari et al. [77] but, after correcting for the selection biases, the corrected slope is steeper (i.e., 1.66 ± 0.22) than the observed one (i.e., 1.32 ± 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. [123] 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 [101] 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 <sup>M</sup>). With the advent of multi-wavelength surveys, which uniformly scan large areas of sky, there has been significant progress in the weaklensing analysis of large samples of galaxy groups. For instance, in the XXL framework (see Pierre et al. [124]), the Lx–MWL relation was investigated by Sereno et al. [115] 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 with 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.

**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. [96], E11), (Lovisari et al. [77], L15), (Sun et al. [81], S09), (Kettula et al. [113], K13). The groups measurements are compared with the ones from an X-ray-selected (Schellenberger and Reiprich [100], S17) and an SZ-selected (Lovisari et al. [34], L20) cluster sample. Luminosities and masses are all within R500 while temperatures are obtained in different regions (see Table 2). Please note that different studies used different methods to estimate the total masses. The lines represent the fitted relation for M <sup>&</sup>lt; <sup>10</sup>14M systems (dotted), <sup>M</sup> <sup>&</sup>gt; 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.

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 nongravitational 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. [125]), potentially affecting also cosmic shear measurements (e.g., Semboloni et al. [126]). 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. [16]). 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. [127], Sanderson et al. [9], 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 ∼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. [81] found a relation only slightly steeper than the prediction of the self-similar scenario. Both Eckmiller et al. [96] and Lovisari et al. [77] 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. [34] 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 agree 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 assess if the level of mass bias in these systems is similar to the one of galaxy clusters. For instance, (Kettula et al. [128], see also Kettula et al. [101]) 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, such as 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. [114] who found the relation to be consistent within statistical uncertainties with the self-similar expectations. However, the uncertainties of the individual weaklensing 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 T5/2 <sup>x</sup> (Spitzer [129]) 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, refs. [77,81] were not able to constrain the intrinsic scatter in their samples. [96] instead suggested

that the scatter in galaxy groups (kT < 3 keV) is much larger than the one derived for the HIFLUGCS sample (Reiprich and Böhringer [36]).

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. [130]). Sun et al. [81] 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. [96] and Lovisari et al. [77]. Unfortunately, given the sample size and the relatively large measurements errors, the intrinsic scatter is not properly constrained. However, both Sun et al. [81] and Eckmiller et al. [96] suggested that the scatter of the M–YX relation is almost half of the M–Tx relation. The findings by Eckmiller et al. [96] 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. [61] 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. [131], Pacaud et al. [99], Alshino et al. [132], Umetsu et al. [114], Sereno et al. [115]) 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 (*σ*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–*σ*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 [133]. Of critical importance at the group scale, the *gapper* estimator is unbiased when using low numbers of member galaxies (down to ∼10 members, e.g., [134]), and is robust against outliers. The *gapper* velocity estimator (*σ*v) is given by

$$\sigma\_{\mathbf{v}} = \frac{\sqrt{\pi t}}{N(N-1)} \sum\_{i=1}^{N-1} w\_i \mathbf{g}\_{i\prime} \tag{6}$$

where, for ordered velocity measurements, the gaps between each velocity pair are defined as *gi* = *vi*+<sup>1</sup> − *vi* (for *i* = 1, 2, 3..., *N* − 1), as well as Gaussian weights defined as *wi* = *i*(*N* − *i*).

As stated above, one can study the Lx–*σ*<sup>v</sup> 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 Lx,bol ∝T2 *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

$$
\beta = \frac{\sigma\_{\text{v}}^2 \mu \mathbf{m}\_{\text{P}}}{\mathbf{k}\_{\text{B}} \mathbf{T}\_{\text{x}}} \approx 1,\tag{7}
$$

where the parameter *β* 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

$$\mathcal{L}\_{\infty,\text{bol}} \propto \sigma\_{\text{V}'}^4 \tag{8}$$

$$\text{T}\_{\text{X}} \propto \sigma\_{\text{v}}^{2}. \tag{9}$$

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 Section 2.1) and can be approximated as Lx <sup>∝</sup>T1.5+*<sup>γ</sup>* <sup>x</sup> , where *<sup>γ</sup>* is the slope of the X-ray emissivity in the considered energy band (e.g., soft or bolometric) and temperature range (see Table 1). Using this *γ* dependent relation, it follows that

$$L\_{\chi} \propto \sigma\_{\text{V}}^{3+2\gamma}.\tag{10}$$

For temperatures lower than 3 keV, the value of *γ* is negative (unless very cool systems are considered), implying that the expected Lx–*σ*<sup>v</sup> 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–*σ*v) relation follows, or is slightly steeper than, the expectation of Equation (8) (e.g., [135–140]). Furthermore, at the cluster scale, studies of the Lx–*σ*<sup>v</sup> relation now use samples of clusters numbering in the high hundreds (e.g., [141], using 755 clusters to investigate the Lx–*σ*<sup>v</sup> relation). Studies of the Lx–*σ*<sup>v</sup> relation at the group scale attempt to compare the form of the relation at the high-mass regime 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., [7,93,94]) or consistent (e.g., Ponman et al. [142], Mulchaey and Zabludoff [136], Mahdavi and Geller [137]) relation than their high-mass counterparts. Figure 5 (left-panel) provides a (non-comprehensive) compilation of the slope of the Lx–*σ*<sup>v</sup> 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 Lx,bol ∝ *σ*<sup>4</sup> <sup>v</sup> expectation is given by the vertical dashed line. Although there appears to be a clear division between the slopes for groups and clusters, many of the group scale studies compare to the usual Lx,bol ∝ *σ*<sup>4</sup> <sup>v</sup> expectation at the cluster scale. As shown in Equation (10), the scaling can be given by Lx <sup>∝</sup> *<sup>σ</sup>*3+2*<sup>γ</sup>* <sup>v</sup> , with *<sup>γ</sup>* 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 1, the bolometric scaling in the group regime becomes Lx,bol <sup>∝</sup> *<sup>σ</sup>*[2.2:3.0] <sup>v</sup> . 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 Lx,bol <sup>∝</sup> *<sup>σ</sup>*[3.7:3.9] <sup>v</sup> (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., [94]). Although 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–*σ*v, **left panel**) and velocity dispersion-temperature (*σ*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 Section 3.1 when the Lx–*σ*<sup>v</sup> scaling can be given as Lx <sup>∝</sup> *<sup>σ</sup>*3+2*<sup>γ</sup>* <sup>v</sup> (with *<sup>γ</sup>* dependent 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, Lx <sup>∝</sup> *<sup>σ</sup>*[3.7:3.9] <sup>v</sup> ) and groups (assuming Tx = 0.7–3.0 keV, Lx <sup>∝</sup> *<sup>σ</sup>*[2.2:3.0] <sup>v</sup> ). 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, Lx <sup>∝</sup> *<sup>σ</sup>*[2.7:2.8] <sup>v</sup> ) and groups (assuming Tx = 0.7–3.0 keV, Lx <sup>∝</sup> *<sup>σ</sup>*[1.6:2.6] <sup>v</sup> ). Unless otherwise stated, references consider (bolometric) luminosities and temperatures derived within an estimate of R500. Please note that the [143] 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–*σ*<sup>v</sup> relation have used groups selected over contiguous survey regions. One such study was performed by [144]. Groups were selected from regions of the Canadian Network for Observational Cosmology Field Galaxy Redshift Survey 2 (CNOC2, [145]) that were covered by *XMM–Newton* and *Chandra* observations, totaling 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–*σ*<sup>v</sup> of 2.40+0.58 <sup>−</sup>0.60 (including groups with lower quality redshift information yields a slope of 1.35+0.42 <sup>−</sup>0.47). Although initial inspection of the value of the slopes would imply the slope is shallower than the self-similar expectation (as noted in [144]), 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. [146]). Assuming the scaling follows Lx <sup>∝</sup> *<sup>σ</sup>*3+2*<sup>γ</sup>* <sup>v</sup> , then for luminosities in the 0.1–2.4 keV band, the scaling can be given by Lx <sup>∝</sup> *<sup>σ</sup>*[2.0:2.7] <sup>v</sup> (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 [144] is coincident with this scaling. Therefore, if the energy band is considered, the [144] relation is consistent with the self-similar expectation. Hence, it can be assumed that the groups studied in [144] are consistent with the cluster scale (assuming clusters follow the self-similar expectation). Another study using contiguous fields is presented in [143], using groups selected from the 2 deg2 Cosmic Evolution Survey (COSMOS, [147]). This study constructs a catalog of galaxy groups based upon those identified in [148], using *XMM–Newton* and *Chandra* observations of the COSMOS field, reaching an X-ray flux limit of ∼10−<sup>15</sup> erg s−<sup>1</sup> cm−2. Ref. [143] 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, ref. [143] showed that the relation for individual groups appears to follow a shallower relation than clusters (consistent with that found by previous studies, e.g., [137]). However, they note that this trend may be affected by a small number of groups that appear to have anomalously low velocity dispersions (at *σ*<sup>v</sup> - 125 km s−1) for their measured X-ray luminosity (discussed further in Section 3.4). To overcome this, Ref. [143] estimated the median velocity dispersion for 7 bins of groups created from the 74 groups in their sample, finding a slope of Lx ∝ *σ*4.7±0.7 <sup>v</sup> . 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–*σ*<sup>v</sup> 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 [149], using 14 groups with at least 5 galaxy members selected from the 9 deg2 X-Boötes survey [150]. They find that the group scale relation is consistent with the self-similar expectation. To test the effects of Malmquist bias on the observed relations, ref. [149] 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., [104]). The study by [144], 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–*σ*<sup>v</sup> relation of 1.78+0.60 <sup>−</sup>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 affected 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 *σ*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, *σ*<sup>v</sup> ∝T1/2 <sup>x</sup> . In the context of clusters, various studies have found that the *σ*v–Tx relation has a slope steeper than the self-similar expectation (e.g., [138,151]), with others finding a steeper slope but with errors too large to confirm a deviation (e.g., [140]). Unfortunately, the study of the *σ*v–Tx relation for groups is somewhat limited in the literature. An early investigation presented in [94] showed evidence for steepening of the relation at the group scale (where they find a slope of 1.15 ± 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 [94] found evidence for a steepening of the relation, they remarked that a comparison cluster-based relation passes through the center 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 *σ*v–Tx is presented in [151], making use of groups/clusters detected serendipitously in the *XMM* Cluster Survey [152]. Using 19 groups/clusters with redshifts z < 0.5, spanning the temperature range 1.0 <sup>&</sup>lt;<sup>∼</sup> Tx <sup>&</sup>lt;<sup>∼</sup> 5.5 keV (with 50% of clusters with a temperature <sup>&</sup>lt; 3 keV), the *σ*v–Tx relation is found to have a slope of 0.89 ± 0.16. Although again steeper than the self-similar expectation, the result is somewhat shallower (although not significant) than that presented in [94]. Finally, the last relation considered is that given in [153], which investigated the *σ*v–Tx relation for a sample of X-ray selected clusters detected in the XXL survey [124]. Clusters were selected from the 25 deg2 XXL-N region, with spectroscopic data compiled from a range of surveys (see [154], for full details of the spectroscopic coverage). Using a sample of 132 clusters (the majority of which have T*<sup>x</sup>* < 3 keV), ref. [153] found a relation of the form *σ*<sup>v</sup> ∝T0.63±0.05 <sup>x</sup> . Please note that the relation is fitted using an ensemble maximum likelihood method, with fitted slope in tension with the selfsimilar 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., [153]). However, consideration must be given to velocity anisotropies during mass modeling using velocity information, which can vary for loose, compact and virialized groups [155]. However, corrections based upon halo concentration have been developed (e.g., [156]). The study of the *σ*v-Tx relation can also be probed down to the galaxy scale. Ref. [157] used galaxies from the volume-limited MASSIVE survey [158], to study the relation between galaxy kinematics (*σ*e) and X-ray temperature. Ref. [157] found a relation of the form Tx ∝ *σ*1.3−1.8 <sup>e</sup> (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–*σ*v. Although an observational consensus on the magnitude of the deviation from selfsimilarity 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. [159], Farahi et al. [160], Truong et al. [28]). The deviations discussed for the Lx–*σ*<sup>v</sup> and *σ*v–Tx are thought to arise due to the effects of AGN feedback on the ICM (as shown in simulations, e.g., [161]), which has little effect on the galaxy velocity dispersions. Furthermore, Ref. [157] measured a

median value of *β* = 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–*σ*<sup>v</sup> relation, is the presence of low velocity dispersion groups (appearing at *σ*<sup>v</sup> - 200 km s−1) that have a high X-ray luminosity in comparison to their *σ*<sup>v</sup> (conversely, it can be stated that these groups have a low *σ*<sup>v</sup> for their Lx). These low velocity outliers have been noted in various studies in the literature (e.g., [137,162]), attributed as the cause of the flattening of the Lx–*σ*<sup>v</sup> relation at the group scale (e.g., Vajgel et al. [149], Sohn et al. [143]). Although 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 *σ*<sup>v</sup> 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 *σ*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 *σ*<sup>v</sup> outliers were given in [162]. 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. Although 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., [143]) may not be attributed to selection; however various relations involving L*<sup>x</sup>* when using X-ray selected samples characteristically show a flattening when not account for selection (e.g., [163,164]). 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. Ref. [104] 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, [144] have used an optically selected sample of clusters to investigate the form of the Lx–*σ*<sup>v</sup> 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.

#### *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. [37], Ettori et al. [165], Mantz et al. [166], Schellenberger and Reiprich [44]). 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. [167], Gonzalez et al. [73], Giodini et al. [2], Behroozi et al. [168], Zhang et al. [169], Leauthaud et al. [170], Laganá et al. [171], Chiu et al. [172], Decker et al. [173]). 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 [2]. 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 [174]. Mean photometric redshifts were assigned to each candidate and checked against available spectroscopic redshifts from *z*COSMOS [175]. 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 [123] 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. [176], Navarro et al. [177], and constant concentration, *c* = 5). Within R500 the Mstars–M relation was found to follow a form of Mstars ∝ M0.81±0.11 and a stellar mass fraction of the form fstars ∝ M−0.26±0.09 (extending this to higher masses with the inclusion of clusters, the form follows a relation of fstars ∝ M−0.37±0.04). More recent studies have used increased area X-ray surveys. The *XMM Blanco* Cosmology Survey (XMM-BCS, [106]) covers 12 deg<sup>2</sup> of the sky with *XMM–Newton*, and was used by [178] to study the form and evolution of the Mstars–M relation using 46 groups/clusters within a mass and redshift range of (2 - M - <sup>25</sup>) <sup>×</sup> <sup>10</sup><sup>13</sup> <sup>M</sup> and 0.1 *z* - 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 [179]. The fitted relation has the form Mstars ∝ M0.69±0.15(1 + *z*)−0.04±0.47, again consistent with previous results showing a shallower than unity slope of the relation. Furthermore, these results indicate little evolution in the stellar content with stellar mass fraction staying constant out to *z* 1. In Figure 6, we plot the Mstars–M relation for various results obtained in the literature (namely [2,178,180–182]). 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 [2,178] are derived at the group scale, whereas [182] straddles the high-mass groups/low-mass cluster regime (see below) and [180] used primarily high-mass clusters (note that the relation plotted here includes clusters from Gonzalez et al. [76], as detailed in Kravtsov et al. [180]). 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 [181]).

**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 [182] relation is derived from the conversion of LK to Mstars assuming a constant mass-to-light ratio of 0.73 (as used in [182]).

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., [167,183]). The use of LK as a stellar mass proxy was investigated in [182] 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. Ref. [182] found a relation of the form LK <sup>∝</sup> <sup>M</sup>0.85+0.35 −0.27 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, ref. [182] measured a slope of 1.05+0.16 <sup>−</sup>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*σ* uncertainty), which is derived from the LK–MWL relation assuming a constant mass-to-light ratio of 0.73 (as adopted in [182]).

#### **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 cannot be merely understood in isolation as giant self-similar gaseous spheres. Particularly in the last decade, a wide range of evidence has accumulated showing that the SMBHs at the center of each galaxy group are tightly co-evolving with the hot X-ray halo. Such coevolution 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. [184], Prasad et al. [185], Voit et al. [186], Temi et al. [187], Tremblay et al. [188], Gaspari et al. [189], Rose et al. [190], Storchi-Bergmann and Schnorr-Müller [191]). In turn, the SMBH re-ejects back large amount of mass and energy (in particular via jets and outflows; e.g., Tombesi et al. [192], Sa¸dowski and Gaspari [193], Fiore et al. [194]), thus re-heating and re-shaping the IGrM via bubbles, shocks, and turbulence up to the group outskirts (McNamara and Nulsen [195], Fabian [196], Gitti et al. [197], Brighenti et al. [198], Gaspari [199], Liu et al. [200], Yang et al. [201], Wittor and Gaspari [202], Voit et al. [203]). Although the small-scale AGN self-regulation thermodynamics/kinematics is respectively tackled in the companion Eckert/Gastaldello et al. 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 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. [20], 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 M500 ∼ <sup>3</sup> × 1012 − <sup>3</sup> × 1014 M, with most systems in the group regime (Tx ∼ 1 keV) and a few in the poor or cluster tails. The companion Eckert et al. review shows that the M• correlation with Tx is significantly tighter than the classical optical scaling, such as the Magorrian relation (e.g., Kormendy and Ho [204], Saglia et al. [205]), 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. [20]) shows the 1-*σ* intrinsic scatter as light red bands with the dotted lines enveloping the rare 3-*σ* 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 using the R500 scaling to probe the total mass, while smaller extraction radii to probe gas mass (and related properties).

**Figure 7.** Scaling relations between the central (dynamical/direct) SMBH mass and key macro X-ray halo properties adapted from Gaspari et al. [20]. **Top-left** to **bottom-right panels**: gas X-ray luminosity (in the 0.3–7.0 keV band), gas t t l (d i t d b th d k tt t) l t d it C t t d f ti

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 [206], Narayan and Fabian [207]), the larger the thermal entropy of the gas, the stronger the feeding is stifled, as the inflowing gas must 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. [184], Prasad et al. [185], Voit [208], Olivares et al. [209])—would produce major positive and tight correlations with the gas mass and X-ray luminosity (e.g., the cooling rate is ∝ Lx). Therefore, X-ray correlations favor 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. [210], Truong et al. [211]) show this to be effective only at the high-mass end. Moreover, Figure 7 shows that all the mass scaling is 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. [212] 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. [23], McCarthy et al. [14], Kravtsov and Borgani [213], Tremmel et al. [214]). Figure 8 shows the key impact of archetypal feedback models on the evolution of the diffuse hot atmospheres (Gaspari et al. [27]). 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 (Tx < 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. [103] 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*cool (Sun et al. [81], Babyk et al. [215]). 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. [216], Truong et al. [28], Weinberger et al. [217]). For comparisons with further cosmological simulations, we refer the interested reader to the companion reviews by Oppenheimer et al. and Eckert et al. (Section 5).

**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. [103] and Gaspari et al. [27]. The colored individual objects in the top panels are from a wide range of observational works (Helsdon and Ponman [92], Osmond and Ponman [94], Mulchaey et al. [218], Pratt et al. [30], Sun et al. [81], Maughan et al. [219]; luminosities are extracted mostly in the 0.5–2 keV band). The empty circles and solid line show the raw Anderson et al. [103] stacking analysis and the unbiased fit, respectively. The filled black points show evolutionary tracks in large-scale HD simulations (Gaspari et al. [27]) 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.

#### **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 ∼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* (*eROSITA*2, Predehl et al. [220]). 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. [221], and Käfer et al. [222] in particular for a detection and characterization through ICM outskirts that reduces possible biases due the peaked X-ray emission associated with 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 ∼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 M500c <sup>&</sup>gt; 1013M/*<sup>h</sup>* up to redshift <sup>∼</sup>1 (median: *<sup>z</sup>* <sup>∼</sup> 0.3) [223–227]. Almost all groups (and clusters) detected with *eROSITA* will lack sufficient X-ray photons to accurately constrain temperature and mass profiles (Borm et al. [225]). 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*σ* errors of 1%, 1%, 7%, and 25% on *σ*8, Ωm, *w*0, and *w*a, respectively, the critical passages will be: (i) a better knowledge (by a factor of ∼4) of the parameters describing the Lx–M relation to improve the constraints on *σ*<sup>8</sup> and Ωm, 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* (*Athena*3), 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 M500c <sup>&</sup>gt; <sup>5</sup> <sup>×</sup> 1013M and hot gaseous atmospheres at *<sup>z</sup>* <sup>&</sup>gt; 2. For about ten of those, a global gas temperature estimate is expected to be measurable [228]. *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* ∼ 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 [228,229]. 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 [230] and transport processes such as turbulence and diffusion (Cucchetti et al. [231], Roncarelli et al. [232], Mernier et al. [233]).

<sup>2</sup> http://www.mpe.mpg.de/erosita/, accessed on 3 May 2021.

<sup>3</sup> https://www.the-athena-x-ray-observatory.eu/, accessed on 3 May 2021.

Presently, concepts funded for study by NASA for consideration in the 2020 Astrophysics Decadal Survey, *Lynx*<sup>4</sup> (as high-energy flagship mission) and *AXIS*<sup>5</sup> (as probeclass 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 ∼100 with respect to *Chandra* ACIS-I. Their predicted low background level and capability to resolve embedded and background AGN will allow the tracking of group and cluster emissions at very low surface-brightness values. For example, *AXIS* is expected to reach a flux limit of ∼<sup>1</sup> × <sup>10</sup>−<sup>16</sup> erg/s/cm2 (0.5–2 keV) over the 50 deg<sup>2</sup> of the proposed Wide Survey (e.g., [234]), providing the detection of thousands of groups and clusters, and evidence 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* ≈ 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-3G*<sup>6</sup> will extend the work of *SPT-SZ* by covering a nearly identical area of 2500 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 a reduction of the mass limit and extending the redshift coverage with respect to *SPT-SZ*. About 5000 clusters with M500c <sup>&</sup>gt;<sup>∼</sup> <sup>10</sup>14M at a signal-to-noise > 4.5 (corresponding to a 97% purity threshold) are expected by the completion of the survey (2023; [235]). The next-generation ground-based cosmic microwave background experiment *CMB-S4*7, 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 <sup>×</sup>1013 <sup>M</sup> at *<sup>z</sup>* <sup>&</sup>gt; 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 (*Euclid*8-from 2022- and *Nancy Grace Roman Space Telescope*9-formerly *WFIRST*; launch date: 2025) and ground-based missions (*Vera Rubin Observatory*<sup>10</sup> and the 4-metre Multi-Object Spectroscopic Telescope, *4MOST*11) will map the large-scale structures over more than 15,000 deg2, extending the current catalogs of systems with M500c <sup>&</sup>gt; <sup>5</sup> <sup>×</sup> 1013M (see, e.g., results from DES<sup>12</sup> in [48]) by orders of magnitude, in particular at high (*z* > 1) redshifts (e.g., [236]). 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* [237] to provide dynamical mass estimates for ∼10000 clusters at redshift z < 0.6 and masses <sup>&</sup>gt; 1014 <sup>M</sup>. 4*MOST* will also provide spectroscopic confirmation of *eROSITA* detected groups at redshifts <sup>&</sup>lt;0.2 down to a mass limit of 10<sup>13</sup> <sup>M</sup>. Additionally, the wide area vista extragalactic survey (WAVES, [238]) being planned using 4*MOST*, 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 ∼70 (1200) deg2, identifying <sup>∼</sup>50,000 (20,000) dark matter halos down to a mass of 10<sup>14</sup> (1011) M and out to a redshift of *z*phot - 0.2 (0.8). As with X-ray selected objects, optically selected groups are physically heterogenous systems (e.g., see the dynamical analysis by Zheng and Shen [239] 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,

<sup>4</sup> https://www.lynxobservatory.com/, accessed on 3 May 2021.

<sup>5</sup> http://axis.astro.umd.edu, accessed on 3 May 2021.

<sup>6</sup> https://pole.uchicago.edu/, accessed on 3 May 2021.

<sup>7</sup> https://cmb-s4.org/, accessed on 3 May 2021.

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<sup>12</sup> https://www.darkenergysurvey.org/, accessed on 3 May 2021.

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 with 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 a 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 is evidence 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 also provide 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 providing the properties at R2500 (i.e., ∼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. [77]; the choice of the column density, Lovisari and Reiprich [83]; the fitting technique, Balestra et al. [240]) 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.

**Author Contributions:** 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.

**Funding:** 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.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** 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.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Review* **Feedback from Active Galactic Nuclei in Galaxy Groups**

**Dominique Eckert 1,\*,†, Massimo Gaspari 2,3,†, Fabio Gastaldello 4,†, Amandine M. C. Le Brun 5,† and Ewan O'Sullivan 6,†**


**Abstract:** The co-evolution between supermassive black holes and their environment is most directly traced by the hot atmospheres of dark matter halos. The cooling of the hot atmosphere supplies the central regions with fresh gas, igniting active galactic nuclei (AGN) with long duty cycles. Outflows from the central engine tightly couple with the surrounding gaseous medium and provide the dominant heating source preventing runaway cooling by carving cavities and driving shocks across the medium. The AGN feedback loop is a key feature of all modern galaxy evolution models. Here, we review our knowledge of the AGN feedback process in the specific context of galaxy groups. Galaxy groups are uniquely suited to constrain the mechanisms governing the cooling–heating balance. Unlike in more massive halos, the energy that is supplied by the central AGN to the hot intragroup medium can exceed the gravitational binding energy of halo gas particles. We report on the state-of-the-art in observations of the feedback phenomenon and in theoretical models of the heating-cooling balance in galaxy groups. We also describe how our knowledge of the AGN feedback process impacts galaxy evolution models and large-scale baryon distributions. Finally, we discuss how new instrumentation will answer key open questions on the topic.

**Keywords:** black holes; galaxy groups; elliptical galaxies; intragroup medium/plasma; active nuclei; X-ray observations; hydrodynamical and cosmological simulations

#### **1. Introduction**

Structure formation in the Universe operates as a bottom-up process in which small halos formed at high redshift progressively merge and accrete the surrounding material to form the massive halos we see today [1]. Given the evolution of the halo mass function, the peak of the mass density in the current Universe occurs in halos of <sup>∼</sup>1013*M*—the *galaxy group* regime. At the current epoch, galaxy groups are the building blocks of the structure formation process and, thus, they occupy a key regime in the evolution of galaxies. Typical *L* galaxies exist within groups rather than within isolated halos [2]. The galaxy stellar mass function exhibits a cut-off at *M*- <sup>∼</sup> 1011*M* [3], corresponding to the central galaxies of galaxy groups, brightest group galaxies (BGGs). Abundance matching studies show that the star formation efficiency reaches a maximum at *Mh* <sup>∼</sup> 1012*M* and decreases at both higher and lower masses [4–6]. At the high-mass end, non-gravitational energy input is needed to quench star formation and reproduce the shape of the stellar mass function [7].

Feedback from active supermassive black holes (SMBH) is currently the favored mechanism for regulating the star formation activity in massive galaxies, explaining the observed

**Citation:** Eckert, D.; Gaspari, M.; Gastaldello, F.; Le Brun, A.M.C.; O'Sullivan, E. Feedback from Active Galactic Nuclei in Galaxy Groups. *Universe* **2021**, *7*, 142. https:// doi.org/10.3390/universe7050142

Academic Editor: Francesco Shankar

Received: 31 March 2021 Accepted: 29 April 2021 Published: 11 May 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

shape of the stellar mass function, and quenching catastrophic cooling flows. Outflows and jets from the central active galactic nuclei (AGN) interact with the surrounding hot intragroup medium (IGrM) and release a large amount of energy, which prevents the gas reservoir from cooling efficiently and fueling star formation (for previous reviews, see McNamara and Nulsen [8]; Fabian [9]; Gitti et al. [10]; Gaspari et al. [11]). All of the modern galaxy evolution models include a prescription for AGN feedback to reproduce the shape of the galaxy luminosity function and the halo baryon fraction (see the companion review by Oppenheimer et al.). Earlier attempts at reproducing these observables with supernova feedback resulted in catastrophic cooling and largely overestimated the stellar content of groups (e.g., [12]). State-of-the-art cosmological simulations all implement sub-grid prescriptions for AGN feedback, either in the form of thermal, isotropic feedback, or in the form of mechanical, directional feedback. The adopted feedback scheme strongly affects the gas properties of galaxy groups. Indeed, strong feedback raises the entropy level of the surrounding gas particles, which can lead to a global depletion of baryons in group-scale halos. Cosmological simulations are now facing the important challenge of reproducing at the same time realistic galaxy populations and gas properties.

The imprint of AGN feedback is most easily observed through high-resolution Xray observations of nearby galaxy groups and clusters. Bubbles of expanding energetic material carve cavities in the gas distribution, spatially coinciding with energetic AGN outflows that are traced by their radio emission. Supersonic outflows also drive shock fronts permeating the surrounding IGrM and distributing heat across the environment. In parallel, extended H*α* nebulae demonstrate the existence of efficient gas cooling from the hot phase, thereby feeding the central SMBH. Recent ALMA observations at millimetric wavelengths also provide evidence for large amounts of cool gas at the vicinity of the SMBH. Thus, multi-wavelength observations of the cores of nearby massive structures allow us to investigate, in detail, the balance between gas cooling and AGN heating.

While a great deal of attention has been devoted to studying these phenomena in the most massive nearby clusters, such as Perseus (see [8] for a review), observations of similar quality only exist for a handful of groups, such as NGC 5044 [13] and NGC 5813 [14]. Detailed observations of galaxy groups are crucial for our understanding of feedback processes, as the physical conditions differ from those of galaxy clusters in several important ways. First and foremost, the ratio of feedback energy to gravitational energy is different from that of clusters. The AGN energy input is sometimes parameterized as *E*˙ feed <sup>≈</sup> *<sup>f</sup> rM*˙ BH*c*<sup>2</sup> with *<sup>r</sup>* <sup>∼</sup>10% the energy output of the BH and *<sup>f</sup>* the coupling efficiency between the BH outflows and the surrounding medium. On the other hand, the gravitational binding energy is a strong function of halo mass, *E*bind ∝ *M*<sup>2</sup> *<sup>h</sup>*. If the coupling efficiency only depends on the physical properties of the gas and the feedback loop has a long duty cycle, then the total integrated feedback energy *E*feed becomes comparable to *E*bind or even exceeds it in group-scale halos, whereas it remains substantially lower in the most massive systems. Similarly, the radius inside, which non-gravitational energy dominates over gravitational energy, is comparably larger in group-scale halos. The impact of AGN in groups is spread over much larger volumes and it can even lead to a depletion of baryons within the virial radius. On top of that, the radiative cooling function experiences a transition of regime between the temperature range of clusters and that of groups. For temperatures that are greater than ∼3 keV, the plasma is almost completely ionized and the Bremsstrahlung process dominates. For temperatures of ∼1 keV, line cooling dominates, which makes the radiative losses comparably more important. Therefore, the radiative cooling time can become much shorter than the Hubble time, even at relatively low gas densities, and the supply of gas to the SMBH can be sustained more easily. For all of these reasons, studying the feedback loop across a wide range of halo masses is necessary for informing our theoretical models.

Here, we review the current state of the art in our knowledge of the AGN feedback process in the specific case of galaxy groups. For the purpose of this review, we define galaxy groups as galaxy concentrations with halo masses in the range 1013–1014*M* and with an X-ray bright intragroup medium (IGrM). Such masses correspond to virial temperatures of ∼0.5–2 keV. Most of the processes that are discussed in this review are also relevant in the case of X-ray bright isolated elliptical galaxies and massive spirals with *kT* ∼ 0.3–0.5 keV. Whenever it is appropriate, we will discuss halos of lower masses as well. The paper is organized, as follows. In Section 2, we describe why AGN feedback is presently thought to be a key ingredient in the evolution of galaxy groups and how invoking AGN feedback can solve a number of overarching issues in galaxy evolution. In Section 3, we review the current observational evidence for AGN feedback in nearby galaxy groups, with our main focus on observations at X-ray and radio wavelengths and additional information coming from millimeter and H*α* observations. Section 4 summarizes the theoretical framework that was put into place to interpret the heating/cooling cycle and the main physical processes involved, with a specific focus on the galaxy group regime. Section 5 discusses how AGN feedback is implemented in modern cosmological simulations and its impact on the evolution and large-scale distribution of the baryonic component of the Universe. We conclude our review in Section 6 with a short presentation of the most relevant upcoming experiments and their expected contribution to the field.

#### **2. The Need for AGN Feedback in Galaxy Evolution**

Even though the first indications that AGN feedback could be one of the missing elements in theories of galaxy formation and evolution are already nearly half a century old [15–17], theoretical and observational studies of the process are still in their infancy. In the last fifteen years, AGN feedback has emerged as the most promising solution to a number of overarching problems in galaxy formation and evolution in both semi-analytic models and cosmological hydrodynamical simulations (e.g., [18–22]). In short, the main issues are: cosmic 'downsizing' [23], i.e., the observation that the majority of both star formation and AGN activity took place before redshift *z* ∼ 1 (e.g., [24,25]), the shape of the galaxy stellar mass function at the high-mass end, the gas content of massive galaxies, groups and clusters, the absence of a cooling flow, the deviations from self-similarity of gas scaling relations, and the entropy floor (the latter three are discussed in detail in Section 3.1). AGN feedback progressively appeared to be a promising solution to solve each of these problems separately, eventually leading to the realization that these issues could in fact be seen as different facets of the same problem (e.g., [19,26,27]). In this section, we highlight the main reasons why AGN feedback plays a central role in galaxy evolution models, specifically at the scale of galaxy groups.

#### *2.1. The Shape of the Galaxy Stellar Mass Function*

White and Rees [17] presented one of the first models of galaxy formation in a cosmological context (see also [16,28]). The authors proposed a two-stage process in which galaxies form via radiative cooling of the baryons within halos that had already formed via gravitational collapse of the collisionless dark matter. The authors also argued that an additional non-gravitational process, called feedback, was needed to avoid the overproduction of faint galaxies as compared to observations. Indeed, in that model, the galaxy stellar mass function would simply be a scaled-down version of the dark matter halo mass function (see the discussion in, e.g., [26] illustrated by their Figure 1 and Figure 1 of this review). Later studies pointed out that the scaled halo mass function model overpredicts the abundance of the most massive galaxies when compared to observations (e.g., [26,29–35]). In short, galaxy formation is fundamentally an inefficient process as only a small fraction of the Universe's baryons are in the form of stars (about 10 per cent; e.g., [36–38]) and the star formation efficiency strongly depends on halo mass. Thus, the mass of the dark matter halo plays a fundamental role in shaping the galaxies that it contains. Observational evidence for this halo-mass dependency of the efficiency of galaxy formation was first obtained using galaxy group catalogues [39,40].

**Figure 1.** Galaxy stellar mass function (GSMF) Φ-(*M*-) = *dN dM*∗*dV* in observations and simulations. The dashed purple curve shows the double-Schechter fit to the local GSMF measured in the COSMOS survey [3]. The solid curves show the predictions of the cosmo-OWLS/BAHAMAS model [12,41] in the case with only stellar feedback (REF, yellow), cosmo-OWLS with AGN feedback (AGN-8.0, maroon), and the latest BAHAMAS model (green), in which the feedback model was tuned to reproduce jointly the GSMF and the gas fraction. For comparison, the dotted gray curve shows the Tinker et al. [42] halo mass function in *Planck* cosmology scaled by *fb* = Ω*b*/Ω*m*, which highlights what one expects to see in the case in which each halo is populated by a galaxy with *M*-= *fbMh*.

To illustrate the first point, in Figure 1 we show the local galaxy stellar mass function (GSMF) Φ-(*M*-) that is measured within the COSMOS survey [3] modeled using a double Schechter function. The dashed line shows the Tinker et al. [42] halo mass function with the masses scaled by the universal baryon fraction *fb* = Ω*b*/Ω*m*, which produces the GSMF one would expect to see if every halo was populated by a single galaxy and all the available baryon content had been converted into stars. The observed GSMF vastly differs from the scaled halo mass function both in shape and normalization. The lower normalization implies that the star formation efficiency is much less than 100%, whereas the steep decline at high masses shows that the growth of galaxies does not follow the structure formation process. Attempts at reproducing the shape of the GSMF with feedback from supernovae and star formation were unsuccessful, as the injected energy was insufficient to offset cooling and regulate the star formation efficiency [26]. In Figure 1, we compare the observed GSMF with the predictions of hydrodynamical cosmological simulations from the cosmo-OWLS and BAHAMAS suites [12,41] implementing several prescriptions for baryonic physics. The cosmo-OWLS run, including cooling, star formation, and stellar feedback (labelled REF) suffers from overcooling, and, thus, it overpredicts the observed abundance of massive galaxies (*M*- 1012*M*) by several orders of magnitude. Conversely, including AGN feedback allows the model to closely reproduce the abundance of massive galaxies. AGN feedback is implemented by releasing a fraction of the rest-mass energy of cooling gas particles within the surrounding environment, which reheats the gas and regulates the star formation efficiency (see Section 5.1 for details). The effect of AGN

feedback kicks in around *M*- <sup>∼</sup> 2–3 <sup>×</sup> 1011*M* corresponding to the stellar masses of BGGs, which highlights the crucial role that is played by the galaxy group regime.

Following the early works highlighting the discrepancy between the observed GSMF and the structure formation theory, it took over 30 years to pinpoint the most likely source of feedback. Early models of non-gravitational heating invoked a 'pre-heating' of the baryonic content before the epoch of formation of massive haloes, but did not specify the source of the energy injection (e.g., [31,43,44]). Because these models cannot predict the impact on the galaxy population, but only on the intra-group medium, they are not discussed any further here (but, see Section 3.1 for a detailed discussion). Models implementing gas cooling without feedback were able to reproduce the breakdown of self-similarity that was observed in X-ray selected samples, but this came at the price of greatly overpredicting the abundance of galaxies (e.g., [45–50] for a review). In the late 1990s and early 2000s, teams working with both semi-analytic models and hydrodynamical simulations, who had started to model the effects of cooling, star formation, and stellar feedback, came to the realization that supernova heating, while being a suitable explanation for the inefficiency of galaxy formation at the low-mass end, could neither solve the remaining problem with the high-mass end of the galaxy stellar mass function (if anything it made it worse; see e.g., Menci and Cavaliere [51]; Bower et al. [52]; Benson et al. [26]) nor reproduce the properties of the gas in massive galaxies, groups, and clusters (e.g., [53–55]). Indeed, the amount of energy that is injected by supernovae is insufficient to eject gas from the potential wells of groups and clusters, even if one assumes that the feedback is one hundred per cent efficient (e.g., [56–58]), and such efficiencies were in conflict with contemporary observations of galactic outflows (e.g., [59]). Heating from thermal conduction was also investigated (e.g., [26,60–62]) and eventually ruled out, as it required that the conduction coefficient should exceed the Spitzer rate expected for a fully ionized plasma. By contrast, the energy that is injected by the supermassive black holes at the center of galaxies is, in principle, sufficient to eject gas from the potential wells of groups and clusters (e.g., [18,26,63]). Additionally, the existence of a feedback loop between supermassive black holes and galaxy formation provides an attractive solution to explain the observed correlation between galaxy and black hole properties (see Sections 2.2 and 5.3 for details). Galaxy groups are the best astrophysical laboratories for studying the impact of various feedback mechanisms, since they have managed to retain enough hot gas to allow for a study of the impact of feedback on the IGrM, while, at the same time, representing a transitional regime for the stellar properties of galaxies. We will discuss this point in more detail in the remainder of this review.

#### *2.2. Co-Evolution between Black Hole Mass and Galaxy Properties*

Since the late 1990s, it has become clear that central SMBH co-evolves with the properties of their host galaxies. Thanks to the increasingly precise measurements of SMBH masses obtained through spatially resolved dynamical measurements of stars and gas at the BH's vicinity, it is now well established that the vast majority of galaxies host a central SMBH with a mass that correlates with the properties of its host galaxy (see [64] for an extensive review). The SMBH mass was found to correlate with a galaxy's near-infrared luminosity *LK*, i.e., with the galaxy's integrated stellar content, and with the velocity dispersion *σ<sup>v</sup>* of the stars in the bulge, which is often used as a proxy for the halo mass [64–70]. SMBH masses scale with galaxy properties as *M*BH ∝ *σ*4.5 <sup>e</sup> and *M*BH ∝ *L*1.1 *<sup>K</sup>* . While it is still unclear whether the relations between SMBH mass and galaxy properties are fundamental or derive from correlations between hidden variables, the existence of these scaling relations implies that the processes leading to the growth of BH are tightly linked with the evolution of the host halo. Widespread AGN feedback provides a natural explanation for the existence of SMBH scaling relations [7,18,71]. The radiative and mechanical energy output of AGN outbursts can, in principle, be sufficient to expel cold gas away from galaxy bulges, thereby, at the same time, regulating the AGN accretion rate and the star formation activity of the host galaxy. Feedback from AGN has the potential to directly link the properties of supermassive black holes and their host galaxies. The coupling of the energy released by the formation of the supermassive black hole to the surrounding forming galaxy should lead to a relationship that is close to the observed *M*BH − *σ*<sup>e</sup> relation [63,72]. The generality of these arguments imply that this result may be reasonably independent of the specific details of the feedback model.

While the *M*BH − *LK* and *M*BH − *σ*<sup>e</sup> relations are now well established, the typical intrinsic scatter of optical/stellar relations remains substantial (even in early-type galaxies), ∼ 0.4–0.5 dex, especially for stellar luminosities and masses (e.g., Saglia et al. [73]). Moreover, optical extraction radii are necessarily limited to galactic half-light radii *R*<sup>e</sup> (∼2–5 kpc), neglecting the key role of the host halo of the group. It has been suggested that SMBH properties are fundamentally linked with the mass of the host halo [74,75], and that the *M*BH − *LK* and *M*BH − *σ<sup>v</sup>* relations arise as a byproduct from the scaling relations between halo mass and optical galaxy properties. Several recent studies seem to confirm that the mass of the central SMBH is more tightly related to the temperature of the host gaseous halo, i.e., the global gravitational potential and hot-halo processes [76–78]. We discuss this point, in detail, in Section 5.3.

#### **3. Observational Evidence**

#### *3.1. X-ray Observations*

#### 3.1.1. Feedback-Induced Hydrodynamical Features

The X-ray observatories in orbit for the past 20 years, *Chandra* and XMM-*Newton*, have revolutionized our understanding of the cores of relaxed galaxies, groups, and clusters, which show a highly peaked X-ray emission from a hot interstellar medium whose radiative cooling time is often less than 1 Gyr. Soon after the launch of XMM-*Newton* and *Chandra* it was realized that the gas in the central regions of nearby groups and clusters does not efficiently cool from the X-ray phase, condense, and flow toward the center, as expected from the original 'cooling flow' model [79]. Spectroscopic observations with *Chandra* and XMM-*Newton* have established that there is little evidence for emission from gas cooling below ∼*Tvir*/3 [80–82]. Precisely where the gas should be cooling most rapidly, it appears not to be cooling at all. This effect is known as the 'cooling flow problem' (e.g., [83]).

Therefore, a compensating heat source must resupply the radiative losses, and many possibilities have been proposed, including thermal conduction (e.g., [84]), energy released by mergers (e.g., [85,86]), or by supernovae (e.g., [87]); see Section 2.1. However, feedback from the central AGN was rapidly established as the most appealing solution to the problem. There is, in fact, clear observational evidence for AGN heating as the majority of brightest cluster galaxies of cool-core clusters and groups host a radio loud AGN (e.g., [88,89]) and, following the launch of *Chandra*, disturbances, such as shocks, ripples, and cavities, have been found in the central atmospheres of many clusters, groups, and elliptical galaxies (e.g., [90–98]). The cavities, which appear as X-ray surface brightness depressions, have been interpreted as bubbles of low-density relativistic plasma inflated by radio jets, displacing the thermal gas and causing *PdV* heating (e.g., [99]). Weak shocks that are associated with outbursts, long expected in models of jet-fed radio lobes [100], were also finally detected in deep *Chandra* observations, for example, in M87 [93], Hydra A [101], and MS 0735+7241 [102]. The energies available from the AGN were found to be not only comparable to those that are needed to stop gas from cooling, but the mean power of the outbursts was well correlated with the radiative losses from the IGrM [103].

Given the lower surface brightness of groups, in the early days of the *Chandra* era, the study of AGN feedback in these systems did not progress at the same pace as for more massive clusters [8]. However, the situation has improved in more recent times, with a number of studies addressing sizable samples of groups and characterizing the cavities in their IGrM [104–107]. Table A1 in Appendix A presents a list of groups with known cavities being detected using high-resolution X-ray observations. Deep *Chandra* X-ray data are now available for a number of galaxy groups (HCG 62, NGC 5044, NGC 5813). In particular, Randall et al. [14] presented the results of a very deep (650 ks) observation

of NGC 5813, the longest such observation available to date. In Figure 2, we show the *Chandra* image and temperature map of NGC 5813, revealing an impressive number of feedback-induced features. Multiple pairs of X-ray cavities can be observed on both sides of the nucleus, indicating that the system has undergone several consecutive AGN outbursts inflating powerful expanding bubbles. Two pairs of concentric shock fronts were discovered perpendicular to the jet axis. The passage of the shock fronts reheats the IGrM, as evidenced by the higher temperatures that were measured in the post-shock regions (see the right-hand panel of Figure 2).

**Figure 2.** Cavities and shocks in the NGC 5813 galaxy group. The **left panel** shows an adaptively smoothed *Chandra* 0.5–2 keV image of the group, with cavities marked by dashed ellipses and two pairs of shock fronts by solid curved lines. The **right panel** shows a temperature map (in units of keV) with the cavities and outer shock fronts marked. Note the shock-heated gas (red and yellow) behind the outer shock fronts and in the shocked rims of the innermost set of cavities. Images drawn from the *Chandra* Early-type Galaxy Atlas [108].

Because cavities are the most commonly detected feedback-related structures, AGN energy input is usually gauged from their properties, as briefly summarized below. The energy that is required to inflate radio bubbles creating cavities in the X-ray emitting gas is usually expressed as the enthalpy, i.e., the sum of the work done to carve out the cavity and the internal energy of the radio lobes:

$$H = E\_{\rm int} + pV = \frac{\gamma}{\gamma - 1}pV \tag{1}$$

where *p* is the pressure of the surrounding IGrM, *V* is the volume of the cavity, and *γ* is the ratio of the specific heats of the plasma filling the cavities. If the plasma is relativistic, *γ* = 4/3 and *H* = 4 *pV*; if it is non-relativistic, *γ* = 5/3 and *H* = 2.5 *pV*. The exact composition of the cavities is still unknown, even though X-ray, radio ([109] and references therein), and SZ observations [110] suggest that it is likely a mixture of the two species. The age of the cavity is the other key physical quantity that can be estimated from observations. As summarized by Bîrzan et al. [92], several age estimators have been suggested:

(a) the *sonic time*, i.e., the time that is required by the cavity to reach its projected distance *R* at the speed of sound,

$$t\_s = \mathbb{R}/\mathfrak{c}\_s\tag{2}$$

with *cs* = (*γkT*/*μmH*)1/2, *μ* the mean atomic weight of the plasma, and *mH* the proton mass;

(b) the *refill time* that is required by the gas to refill the displaced volume as the cavity rises upward,

$$t\_{\text{ref}} \approx \sqrt{r/g} \tag{3}$$

where *r* is the radius of the cavity and *g* = *GM*(< *R*)/*R*<sup>2</sup> is the gravitational acceleration at the cavity position;

(c) the *buoyancy time*, i.e., the time that iis required for the cavity to rise buoyantly at its terminal velocity,

$$t\_{\text{buoy}} = \mathcal{R}/v\_t \approx \mathcal{R}\sqrt{\mathcal{SC}/2\mathcal{g}V} \tag{4}$$

where *V* and *S* are the volume and the cross-section of the cavity, respectively, and *C* = 0.75 is the drag coefficient [111]

Cavity ages that are estimated using these three methods usually agree within a factor of two, with the buoyancy times typically in between the shorter sonic time and the longer refill time ([10] and references therein). Dividing the enthalpy *H* by the characteristic timescale provides an observational estimate of the cavity power *Pcav*. The cavity power is a lower limit to the mechanical power of the AGN, given the paucity of detected shocks and other possible sources of energy feedback, such as sound waves.

Operationally, estimating *Pcav* requires a measurement of the geometry and size of the cavity and the pressure of the surrounding ICM. The total cavity power can be compared with the gas luminosity inside the cooling radius, *L*cool, which needs to be balanced by the AGN mechanical feedback. *L*cool is usually defined as the total luminosity inside the regions where the cooling time is less than 7.7 × 109 yrs [112], although different thresholds exist in the literature (e.g., 3 Gyr, [113]). *L*cool is estimated by deprojecting the X-ray temperature and emissivity profiles and computing the corresponding bolometric luminosity [96]. A number of possible biases and systematic errors can affect this apparently straightforward observational approach, as the detectability of cavities depends on the depth of the observation, the position of the cavity with respect to the plane of the sky, and uncertainties in the assumed geometry ([8,10,109] and the references therein). The exact impact of these observational uncertainties on the statistics of cavities in clusters and groups is yet to be quantified.

The *Pcav* − *Lcool* relation has been investigated through the years in an increasing number of objects, ranging from ellipticals to groups and clusters ([10,109] and references therein). The general consensus is that the cavity power is enough to offset cooling given an average 4 *pV* injected energy per cavity and that the jet mechanical power correlates well with the cooling luminosity. In Figure 3, we show the relation between cooling luminosity and cavity power from a compilation of literature measurements [104,106,112]; see Table A1. Here, the total cavity power for each system was computed by summing up the power of each individual cavity. While at the high-mass end, the data are broadly consistent with an enthalpy *H* = 4*pV*, being typical of heating by a relativistic plasma, in the group regime, the cavity power is substantially higher. To quantify this effect, we fitted the *L*cool − *P*cav relation with a power law using PyMC3 [114]. The blue curve and shaded area show the best-fit relation, which reads

$$\log\left(\frac{P\_{\text{cav}}}{10^{43}\,\text{erg/s}}\right) = (0.41 \pm 0.09) + (0.70 \pm 0.05)\log\left(\frac{L\_{\text{cool}}}{10^{43}\,\text{erg/s}}\right) \tag{5}$$

with an intrinsic scatter of 0.51 ± 0.07 dex. The slope of the fitted relation is significantly shallower than unity, which would be the expected slope if the feedback efficiency is independent of the halo mass. At face value, this result implies that the feedback efficiency is higher in groups than in clusters, with the cavities injecting enough energy to overheat the cores and deplete the central regions from their gas content. We discuss this point, in detail, in Section 3.1.4.

**Figure 3.** The relation between the luminosity within the cooling radius (*Lcool*) and the power injected by the cavities assuming *H* = 4 *pV* (*Pcav*) for several literature samples. The orange points were either recomputed for this work or collected from papers on individual objects; we refer to Table A1 for detailed references. The blue line shows a fit to the data using a power law with intrinsic scatter. The uncertainty on the fitted relation is indicated by the blue shaded area, whereas the cyan range indicates the intrinsic scatter around the relation.

The presence of shock fronts is another clear observational hydrodynamical feature that is caused by AGN feedback. The passage of a shock front compresses and heats the gas, raising its entropy and providing an effective heat input

$$
\Delta Q \approx T \Delta S = T \Delta \ln K \tag{6}
$$

with *K* the entropy index usually quoted as entropy by X-ray astronomers (see Section 3.1.2). Because of their transient nature, single weak shocks fail to compensate for the radiative losses in a cool core, but the cumulative effect of multiple shocks can be relatively important (see, e.g., the discussion in [109]). This is again highlighted by the exemplar case of NGC 5813 (see Figure 2), where each set of three cavities has been associated with an elliptical shock measured at 1 kpc, 10 kpc and 30 kpc, respectively, with Mach numbers M in the range 1.17–1.78. Generally speaking, the detected shock fronts are weak, i.e., their Mach number falls in the range M ∼1–2 [115]. This range can be understood given the typical evolution of the Mach number as a function of the fundamental parameters, total energy, and duration, of the AGN outbursts (see, for example, the discussion in [116] and Section 4.2). The cumulative heating effect of these successive shock fronts is sufficient to offset cooling within the inner 30 kpc. The shock energy can be estimated as

$$E\_s = p\_1 V\_s (p\_2 / p\_1 - 1) \tag{7}$$

where *p*<sup>1</sup> and *p*<sup>2</sup> are the pre- and post-shock pressures, respectively, and *Vs* is the volume that is enclosed by the shock (e.g., [14]). Liu et al. [115] made an exhaustive search for groups and clusters with detected shocks and studied the dependence of the shock energies and related Mach numbers on the cavity enthalpies (see Figure 4). The shock energies span almost seven orders of magnitude from 4 × <sup>10</sup><sup>61</sup> erg s−<sup>1</sup> in the cluster MS 0735 + 7421 [102] to 1055 erg s−<sup>1</sup> in NGC 4552 [117], with group-scale objects in the range 1056–1059 erg s−<sup>1</sup> and Mach numbers all in the range 1–2 with the exception of Centaurus A. The shock energy is similar to the cavity energy (see the right-hand panel of Figure 4), suggesting that shocks and cavities may play a comparable role in supplying mechanical energy that is provided by the AGN (with the balance between the two mainly driven by the duration of the outburst, e.g., [116,118]). Other heating mechanisms that are discussed in Section 4.2, such as turbulent heating, have yet to be explored observationally at the group scale.

**Figure 4.** Shock energy versus Mach number (**left panel**) and shock energy versus cavity enthalpy (**right panel**) for groups and clusters with available shock energy in the literature. Figure reproduced from Liu et al. [115]. Objects with a red circle have *kT* < 1.3 keV, green square: 1.3 < *kT* < 3.0 keV and blue triangles *kT* > 3.0 keV. The dashed line in the left panel marks M = 1.5 and the dotted, solid, dashed lines in the right panel represent *Eshock*/*Ecav*=10, 1, 0.1 respectively. The objects considered in the plots are: Hydra A [101,119], MS 0735+7421 [102], Centaurus A [120], Cygnus A [92,121], 3C 444 [122], M87 [118], Abell 2052 [123], 3C 310 [124], NGC 4552 [117], NGC 4636 [125], HCG 62 [126], and NGC 5813 [14].

#### 3.1.2. Non-Gravitational Feedback Energy and Entropy Profiles

One of the earliest pieces of evidence for non-gravitational feedback energy came from observed deviations from the self-similar scaling relations that are driven only by gravity [127], in particular the deviation of the observed luminosity-temperature relation from the predicted *L* ∝ *T*<sup>2</sup> [128–130]; see the companion review by Lovisari et al. on the scaling relations of galaxy groups. Advocating a minimum entropy in the pre-collapse intergalactic medium to break self-similarity was one of the first attempts at a solution [43,44], with the result of bending the relation from self-similar at the scale of massive clusters to a steeper slope at the scale of groups. It was recognized that a given entropy level could be reached through different thermodynamic histories, and that the key insight would have been given by the sequence of adiabats through which baryons evolve: the excess entropy could have been achieved prior accretion to the collapsed halo (the external scenario) or in the higher density medium after accretion (internal scenario (e.g., [131,132]). A third option was initially considered while realizing that cooling alone could remove low entropy gas from the centers of halos producing a similar effect to non-gravitational heating (e.g., [46,133]).

Ponman et al. [134] provided the first observational evidence by measuring the entropy (*S* = *T*/*n*2/3 *<sup>e</sup>* ) at a fixed scaled radius (0.1 *rvirial*), and showing that it does not follow the expected linear trend with temperature (see the left-hand panel of Figure 5). *ROSAT* surface brightness profiles were combined with *Ginga* mean temperatures under the assumption of isothermality to derive entropy profiles for 25 objects, six objects with *T* < 2 keV (NGC4261, NGC2300, NGC533, HCG97, HCG94, and HCG62), and objects, like MKW4, MKW3, AWM4, and AWM7, all in the range *T* = 2–4 keV. The authors concluded that the observational data were consistent with a scenario involving an external preheating mechanism through supernova winds, thereby raising the central entropy and enriching the medium in heavy elements. They also established that preheating would have a broader impact on the general picture of structure formation, as, for example, the level and timing of the heating required could not violate the constraints from the Lyman-*α* forest (see also [135]).

**Figure 5. Left**: The gas entropy at the fiducial radius of 0.1*Rvirial* as a function of the temperature for the 25 systems in the sample of Ponman et al. [134]. The solid line shows the relation obtained from numerical non-radiative simulations [136]. Figure reproduced from Ponman et al. [134], arXiv author's version. **Right**: The same relation at different radii for the groups analyzed in the sample of Sun et al. [137], together with the sample of clusters in Vikhlinin et al. [138], both being analyzed with *Chandra* data. Figure reproduced from Sun et al. [137].

A key improvement with respect to this early result was the ability to go beyond the assumption of isothermality by constraining the temperature profiles exploiting the combination of *ROSAT* and *ASCA* data [139,140]. These studies confirmed that low-mass systems exhibit higher scaled entropy profiles. However, they did not show the large isentropic cores that were predicted by simple preheating models (e.g., [141]) and the high entropy excess in galaxy groups was found to extend to large radii (as also shown by the *ASCA* analysis of [142]).

The *Chandra* and XMM-*Newton* results have made the observational picture clearer and more solid. In the comprehensive work that was done by [137], an archival sample of 43 groups observed with *Chandra* with a temperature range of *kT*<sup>500</sup> = 0.7–2.7 keV was analyzed, deriving detailed entropy profiles thanks to the superb spatial resolution of the satellite. The derived entropy-temperature scaling relations at six characteristic radii (30 kpc, 0.15 *R*500, *R*2500, *R*1500, *R*1000, and *R*500) show a large intrinsic scatter at small radii, but already at *R*2500, the scatter reaches a value of 10% and remains the same beyond this point. When combined with similar observations in the galaxy cluster regime (the sample of [138]), the slope of the relation is found to gradually approach the self similar value, steepening from 0.740 ± 0.027 at *R*<sup>2500</sup> to 0.994 ± 0.054 at *R*<sup>500</sup> (see the right-hand panel of Figure 5). The entropy ratios that were calculated with respect to the baseline entropy

profile expected from purely gravitational processes [143] confirm an excess entropy, which is a function of mass and radius, with groups having higher ratios at small radii. The weighted mean ratio for groups decreases from 2.2 at *R*<sup>2500</sup> to 1.6 at *R*500. In general, the entropy profiles of groups have slopes (0.7–0.8) that are flatter than the self-similar expectation from pure gravitational processes of 1.1. Deep observations of nearby poor clusters with *Suzaku* (RX J1159, Humphrey et al. [144]; Virgo, Simionescu et al. [145]; UGC 03957, Thölken et al. [146]) have shown that the entropy excess can extend all the way to the system's virial radius. Similar observations on a larger sample of groups are needed to determine whether the high entropy of galaxy groups is a general feature that is linked to the AGN feedback phenomenon.

The XMM-*Newton* study that wsa performed by [147] on a sample of 29 groups based on the two-dimensional XMM-*Newton* Group Survey, 2DXGS [148,149] supplemented by groups from the sample of Mahdavi et al. [150] divided the objects into cool core (CC) and non-cool core (NCC) objects on the basis of the presence of a temperature gradient in the core, the first time this had been done for galaxy groups. The slope of the scaling relations of the entropy with temperature, incorporating the cluster sample of Sanderson et al. [151], at 0.1 *R*<sup>500</sup> has a slope of 0.79 ± 0.06 consistent with the results of Sun et al. [137]. The entropy profiles of NCC groups show greater scatter than the CC sub-sample, and they have higher central entropies, in qualitative agreement with the results at the cluster scale (e.g., [152]). The excess entropy with respect to the baseline expected from gravitational processes cannot be reproduced by simple theoretical models of entropy modification, such as pure pre-heating or pure cooling, and the required mechanism should provide increasingly large entropy shifts for higher entropy gas.

Another significant advance in the study of entropy profiles of group-scale objects has been provided by the work of Panagoulia et al. [153], which analyzed the entropy profiles of 66 nearby groups and clusters drawn from a volume-limited sample of 101 objects that were assembled from the NORAS and REFLEX catalogues. The study pointed out that the flattening of the entropy profiles at small radii found in previous studies [152] was mainly a matter of resolution and it could be affected by the presence of multi-temperature gas. In particular, for nearby groups, a broken power law model provides the best description of the entropy profiles, with an inner slope of 0.64 within the central 20 kpc. This finding was confirmed in the recent studies of Hogan et al. [154] and Babyk et al. [155], who found that the behavior of the entropy profiles in the inner regions of relaxed clusters and groups can be well described by a broken power law with *K*(*r*) ∝ *R*2/3, a break around 100 kpc (∼0.1*R*2500) and an outer slope of ∼1.1 matching the predictions of gravitational collapse models [143]. Because the cooling time is ∝ *K*3/2, the non-existence of an entropy floor affects the interpretation of the SMBH feeding and feedback processes, as we will discuss in Section 3.1.3 and Section 4.

#### 3.1.3. Thermal Instability Timescale Profiles

In the past decade, a substantial amount of work has been dedicated to understanding the triggering of AGN feedback in galaxy groups and clusters. The energy injected by the central AGN and the cooling of the IGrM appear to be closely balanced over the longterm, as we will review in Section 4. This reflects a tight relation between cooling/feeding (Section 4.1) and heating/feedback (Section 4.2) processes. Initial works suggested that the onset of thermal instability in hot halos and the triggering of runaway cooling can be expressed in terms of the ratio of the cooling time to the free-fall time [156–160]. The cooling time is usually expressed as the ratio of the thermal energy to radiative cooling rate,

$$t\_{\rm cool} = \frac{(3/2)nk\_{\rm B}T}{n\_{\rm e}n\_{\rm i}\Delta} \approx \frac{3k\_{\rm B}T}{n\_{\rm e}\Delta} \tag{8}$$

with Λ the cooling function (see Figure 10) and *n*<sup>i</sup> ≈ *n*<sup>e</sup> the IGrM ion number density. The free-fall time describes the timescale that is necessary for a gas particle to directly fall to the bottom of the potential well,

$$t\_{\rm ff} = \left[2R/\mathcal{g}(\mathcal{R})\right]^{1/2} \tag{9}$$

with *g*(*R*) the local gravitational acceleration. If *t*cool *t*ff, the gas particle is losing internal energy too slowly and radial oscillations are eventually damped. Conversely, as *t*cool becomes comparable to *t*ff, the gas rapidly loses pressure support, developing runaway thermal instability, and eventually sinking radially onto the central galaxy and SMBH. This 'classical' thermal instability (and related TI-ratio) has been also referred to as 'raining' (Gaspari et al. [156]) or 'precipitation' [159], being analogous to early physics studies that are based on analytical approximations (e.g., Field [161]).

The observational constraints on thermal instability can be investigated by studying the radial profiles of such IGrM timescales. In the absence of cooling and non-gravitational heating (*t*cool 1 Gyr), the cooling time follows a baseline universal profile set by the structure formation process [143,162]. Deviations from the baseline profile occur in the central regions, where *t*cool 1/*H*<sup>0</sup> and radiative cooling losses can no longer be neglected. The left-hand panel of Figure 6 shows the cooling time profiles for a large sample of galaxy clusters and groups from the ACCEPT database [159,163]. All of the systems exhibiting evidence of multi-phase gas (e.g., H*α*, CO) show a floor set by the threshold *t*cool/*t*ff ∼ 10– 30, such that, on average, the IGrM does not experience runaway cooling. Figure 6 (right panel) includes 40 galaxy groups and massive early-type galaxies with deep *Chandra* observations [155], showing that the ratio of cooling time to free-fall time reaches a floor at *<sup>t</sup>*cool/*t*ff ∼10. In the inner regions where the entropy rises approximately as *<sup>K</sup>* <sup>∝</sup> *<sup>R</sup>*2/3, *t*cool/*t*ff is approximately constant with an average value of *t*cool/*t*ff ∼ 30 (blue line).

**Figure 6.** Observed timescales profiles in the IGrM (mainly related to TI). **Left**: Cooling time profiles for a sample of cool-core clusters (blue) and groups (magenta) hosting multiphase gas (H*α*, CO; figure reproduced from Voit et al. [159], arXiv author's version). The observed profiles are compared with the 'baseline' structure formation profile (maroon) and the typical simulation threshold *t*cool/*t*ff ∼ 10 (pink). **Right**: The ratio of the cooling time to free-fall time in a sample of groups and ellipticals (the figure reproduced from Babyk et al. [155]; the average is shown with a blue line).

On the one hand, the above results show that the TI-ratio is correlated with the presence of multiphase gas. On the other hand, it is clear that the threshold is puzzlingly not unity (as expected in classical thermal instability), and that there is a substantial intrinsic scatter, even in multiphase systems. However, classical thermal instability starts from idealized linear fluctuations and it does not account for key astrophysical processes, such as AGN feedback or mergers, both recurrently injecting substantial gas motions (e.g., turbulence) at small and large radii, respectively (e.g., Lau et al. [164]; see Section 4.2). In

this more realistic IGrM case of 'turbulent' nonlinear TI, the key physical timescale is not the free-fall time, but rather the turbulence eddy turn-over time (Gaspari et al. [165]),

$$t\_{\rm eddy} = \frac{2\pi R^{2/3} L^{1/3}}{\sigma\_{v,L}} \,\prime \tag{10}$$

where *σ<sup>v</sup>* is the turbulence velocity dispersion and *L* is the related injection scale. The velocity dispersion of the *ensemble* warm H*α*-emitting gas should linearly correlate with *σ<sup>v</sup>* of the hot IGrM, allowing for us to convert between the two, in particular by leveraging the higher spectral resolution of optical/IR telescopes. Rough estimates of the injection scale can be also obtained via the size of the ensemble warm gas filaments/nebulae, or via the AGN cavity diameter. In the presence of a turbulent halo, thermal instability develops chaotically and non-linearly in a very rapid way whenever *t*cool/*t*eddy ∼ 1 [166–168]. Future X-ray microcalorimeter missions, like XRISM and Athena (see Section 6), will allow us to measure the eddy time directly and test, in more depth, the above scenarios. In Section 4, we discuss the related processes from a theoretical perspective.

#### 3.1.4. Baryon Content

A key quantity for AGN feedback models in cosmological simulations is the total integrated baryon budget and its dependence on halo mass. While galaxy cluster halos are massive enough to retain all of their baryons (e.g., [169]), energy injection by AGN feedback can lead to an overall *depletion* of baryons all the way out to the virial radius. Observational studies have found that the gas fraction within *R*<sup>500</sup> increases with halo mass [170–177]. Because an estimate of the gas density can be obtained from imaging data only, a lot of attention has been devoted to the study of gas density profiles [175,178,179]. Using a compilation of measurements from the literature, Sun [178] showed that, while the gas density of galaxy group cores is systematically lower than that of more massive systems, at *R*500, the gas density is nearly independent of mass. Eckert et al. [175] studied the gas density profiles of the 100 brightest galaxy clusters and groups in the XMM-XXL survey. While, at galaxy clusters scales, the measured profiles show a well-defined core and a relatively steep decline in the outskirts, the density profiles of the selected groups exhibit a power-law behavior with a very flat index *ne*(*r*) ∝ *r*<sup>−</sup>1.2, indicating that the gas fraction in spherical shells increases steeply with radius. It is believed that most of the gas has been evacuated from the inner regions under the influence of AGN feedback and displaced to larger radii, thereby explaining the observed shallow slopes [12,156].

In Figure 7, we present a compilation of published measurements of the hot gas fraction at *R*<sup>500</sup> as a function of the corresponding halo mass. The gas fractions were derived from X-ray data under the assumption of hydrostatic equilibrium, with the exception of XMM-XXL and SPT-SZ. In the case of XMM-XXL [175], weak lensing measurements for a sub-sample of 35 clusters were used to calibrate the mass-temperature relation. SPT-SZ masses [180] were derived as an ensemble from a joint fit to the cosmological parameters and the relation between SZ observable and halo mass. All of the studies find that the hot gas fraction contained within *R*<sup>500</sup> increases with halo mass roughly as *f*gas ∝ *M*0.2 <sup>500</sup>. Here, we provide a conservative estimate of the *f*gas − *M*<sup>500</sup> relation, which aims at encompassing all state-of-the-art observational studies and their uncertainties. To this aim, we collected the compilation of observational studies from Figure 7 and estimated in each mass bin the median and 90% confidence range of the data points. The resulting relation is shown as the gray band depicted in Figure 7. The gray band can be approximated as a power law, which reads

$$f\_{\rm gas,500} = 0.079^{+0.026}\_{-0.025} \times \left(\frac{M\_{500}}{10^{14}M\_{\odot}}\right)^{0.22^{+0.06}\_{-0.04}}.\tag{11}$$

**Figure 7.** Compilation of existing measurements of the hot gas fraction at *R*<sup>500</sup> in galaxy groups and clusters as a function of halo mass *M*500. The data points show the galaxy group samples of Sun et al. [137] (orange), Lovisari et al. [174] (magenta), Sanderson et al. [181] (cyan), and Nugent et al. [177] (green). The data from the X-COP sample [169] at the high-mass end are shown as the blue points for comparison. The solid lines show the *fgas* − *M* relations that are derived from REXCESS (blue, [173]), XMM-XXL (red, [175]), SPT-SZ (magenta, [180]), and the literature sample of Ettori [176]. The gray shaded area shows the 90% confidence range encompassing the existing observational data and their corresponding uncertainties (see text).

While at the high-mass end, the gas fractions approach the cosmic baryon fraction, on galaxy group scales, the IGrM only contains about half of the baryons that are expected from the self-similar structure formation scenario. On the other hand, the stellar fraction *f*- is a weak function of halo mass and decreases only slightly from 2–3% at 1013*M* to 1–1.5% at 1015*M* [5,6,172,175,180,182,183]. The weak dependence of the stellar fraction on halo mass is insufficient to compensate for the steeper dependence of the gas fraction, which results in a deficit of baryons in galaxy groups with respect to the cosmic baryon fraction. Here, we note that this result is independent of the hydrostatic equilibrium assumption adopted by most authors. Indeed, an additional non-thermal pressure term would lead to a slight underestimation of the mass in these studies (e.g., [184]), which, in turn, would result in the gas fraction being *overestimated* [169]. Thus, a high level of non-thermal pressure would render the lack of baryons in group-scale halos even more severe.

Here, we caution that the measurement of the gas fraction of group-scale halos is a difficult one and it is hampered by numerous systematic uncertainties. While halo mass estimates definitely represent the leading source of systematics, several other sources introduce potential systematic errors. In the temperature range of galaxy groups, line cooling renders the X-ray emissivity highly dependent on gas metallicity, which is difficult to measure away from group cores (see the review conducted by Gastaldello et al. within this issue). This can introduce uncertainties as large as 20% in the recovered gas mass [174]. Sample selection, usually based on *ROSAT* all-sky survey data, may bias the selected samples towards gas-rich systems, especially if the scatter at fixed mass is substantial [185,186]. Finally, most of the studies do not detect the X-ray emission all the way out to *R*<sup>500</sup> (see e.g., Figure 8 of [137]) and must rely on extrapolation. For all of these reasons, the question of what is the exact baryon fraction of galaxy groups within *R*<sup>500</sup> is still very much an open one, let alone within the virial radius.

#### *3.2. Radio Observations*

#### 3.2.1. Interaction between Radio Sources and the IGrM

Radio surveys have made clear that the centers of galaxy groups and clusters are special locations for AGN (e.g., [89,187,188]), with group-central galaxies twice as likely to host radio-mode activity than non-central galaxies of equal mass out to *z* > 1. Deeper observations show that almost all the central galaxies of X-ray luminous groups host some radio emission [189,190], though in the local universe some of these may be contaminated by emission from low-level star formation [191]. The observations of nearby groups show a wide range of radio morphologies (e.g., [192]), with jet-mode feedback dominated by FR-I radio galaxies, as in clusters. Roughly one-third of X-ray luminous groups appear to host currently or recently active jet sources in their central galaxies [106] with typical jet powers in the range 1041–1044 erg s−<sup>1</sup> [190].

While cavities and shocks are the most accurate indicators of the impact of AGN feedback on the IGrM (see Section 3.1), current X-ray instruments have a limited ability to detect these features outside the high surface brightness cores of nearby groups. Radio studies offer an observationally cheaper way to measure feedback, particularly at higher redshifts. Radio galaxies are only periodically active and, once their AGN ceases to power them, their emission fades fastest at high frequencies. Therefore, low-frequency observations can be particularly effective at identifying older, dying radio sources, and measuring their full extent and luminosity (e.g., [193,194]). The radio spectrum can also give an indication of the properties of the source, most notably its age and the lobe pressure, which, for older sources, is usually in equilibrium with the surrounding IGrM. Combining radio and X-ray observations, we can observe multiple cycles of outbursts in individual groups, e.g., NGC 5813 and NGC 5044 (Figures 2 and 8, [14,195]). In particular, in Figure 8, we show the existing high-quality radio, X-ray, and H*α* observations of NGC 5044 [195]. GMRT 235 MHz radio observations trace the oldest outburst via detached lobes and a bent, one-sided radio jet, while *Chandra* detects cavities on ∼5 kpc and ∼150 pc scales. Interestingly, the current radio jets, which are traced by high-resolution VLBA observations (bottom-left panel), are not aligned with the X-ray cavities, possibly indicating the precession of the jet axis with time.

The properties of group-central radio galaxies are closely linked to the IGrM. Both groups and clusters show a correlation between X-ray luminosity and the radio luminosity of the central source [189,196,197]. In clusters, central radio source luminosity is observed to be higher in systems with cooling times <109 yr [198] and, in groups, it appears that radio jets are more common in the central galaxies of groups with short central cooling times, low tcool/tff ratios, and declining central temperature profiles [106]. However, perhaps the most important correlation is that between jet power, as determined from the enthalpy of AGN-inflated cavities, and radio luminosity. This Pcav-Lradio relation was first established for galaxy clusters by Bîrzan et al. [92,112] and later extended to early-type galaxies [104] and galaxy groups [199]. Although there is significant scatter in the relation, it offers a mechanism for determining the energy that is available from AGN feedback in the many systems where direct determination in the X-ray is impossible.

Applying the Pcav-Lradio relation to a large sample of SDSS groups and clusters with radio sources identified from the NVSS and FIRST surveys, Best et al. [89] showed that central radio galaxies dominate the heating of the IGrM within the cooling radius. They also found that the efficiency of AGN heating cannot be constant across the full mass range of groups and clusters; feedback must be less efficient in groups if they are not to be over-heated. Support for this result came from a study of groupsobserved in the COSMOS survey [200], which found that, factoring in the likely duty cycle of the AGN population, group-central radio galaxies can inject energies that are comparable to the binding energy of the IGrM. These results suggest that group-central AGN have the potential to drive gas out of the group core, and perhaps out of the group altogether, unless some mechanism reduces their effectiveness in heating the gas.

**Figure 8.** Multiple cycles of AGN feedback in the NGC 5044 galaxy group. The *upper left* panel shows the 0.5–2 keV *Chandra* image with GMRT 235 MHz contours overlaid. These reveal an old, bent radio jet and detached lobe from a prior AGN outburst, whose structure has been affected by the sloshing front marked with a dashed line. The *upper right* panel zooms in to show more detail of the complex of cavities and cool filaments in the group core. The *lower right* panel zooms in further, with contours showing the MUSE H*α* (cyan) and ACA diffuse CO (blue) in the densest parts of the X-ray filaments and core. The *lower left* panel shows parsec-scale VLBA 6.7 GHz radio emission in the nucleus of the galaxy, evidence of a new cycle of AGN jet activity (adapted from [195]).

#### 3.2.2. Giant Radio Galaxies

Because feedback studies at the group scale are largely limited to the X-ray bright cores of nearby groups where cavities are most easily identified, they have tended to focus on relatively small radio sources, with jet sizes of less than a few tens of kiloparsecs. However, the population of group-central radio galaxies includes much larger objects, some of which extend to very large radii, well beyond the cool core, and even into the outskirts of their groups. Pasini et al. [197] show that radio galaxies larger than 200 kpc are more common in

groups than clusters, and the largest radio galaxies are located in groups, probably because the IGrM is less able to confine their growth than the ICM. There is also evidence from new radio surveys, with greater sensitivity to extended diffuse emission, that giant radio galaxies may be more common, in general, than previously believed [201].

Large radio sources pose a problem for AGN feedback models, in that they may inject a large fraction of their energy into the IGrM at large radii, rather than in the core, where it is needed to balance cooling. Even some medium sized sources appear to have jets that tunnel out of the cool core and inflate cavities outside it (e.g., NGC 4261 [202]). IC 4296 is an extreme example in which at least one cavity is confirmed, which hosts an FR-I radio galaxy whose 160 kpc diameter lobes extend out to a projected radius of ∼230 kpc [203]. This is far beyond the cool core (20–30 kpc radius) and about half of *R*<sup>500</sup> for this ∼1 keV group. In such a system, while some of the energy that is involved in lobe inflation will likely have heated the core, the energy that is bound up in the relativistic particles and magnetic field of the radio lobes will likely be released at large radii, heating gas that is unlikely to contribute to fuelling the AGN. Other nearby examples of group-central giant FR-Is include NGC 315 and NGC 383 [192] and NGC 6251 [204]. As in clusters, group-central FR-II galaxies are uncommon, but not unknown (see, e.g., [196,205]). Their faster, more collimated jets likely provide feedback heating via shocks during expansion (as in, e.g., 3C 88, [115]), and lobe inflation will drive turbulence, but, as with the giant FR-Is, it is less clear how they affect the cooling region once they grow beyond it.

Therefore, giant radio galaxies pose a number of important questions for feedback models of groups. Do they provide feedback that can balance the rapid cooling in group cores, and if so how? The large sizes of these systems, particularly the FR-Is, implies that their jets have been active for very long periods. How do these sources stay active for so long?

#### *3.3. Multiwavelength Observations*

In the cool cores of galaxy clusters, many observations have shown evidence of material cooling from the hot atmosphere, in the form of highly multi-phase filamentary nebulae surrounding the central galaxy and containing gas and dust with temperatures ranging from ∼10<sup>6</sup> K to a few ×10 K. Some cool core galaxy groups show similar structures, although they are generally less luminous and are thus far less thoroughly explored. As of yet, few studies have specifically focused on BGGs, but samples of giant ellipticals provide a window on the group regime.

<sup>H</sup>*<sup>α</sup>* emission from ionized gas with temperatures ∼104 K may be the most accessible tracer of cooled material. Lakhchaura et al. [206] find that, in giant ellipticals, as in galaxy clusters (c.f. [207]), the presence of H*α* emission is associated with high IGrM densities, short cooling times, low values of the thermal instability criterion, t*cool*/tff (see Section 3.1.3), and disturbed X-ray morphologies, with the overlap between galaxies with and without detected H*α*, suggesting that the transition between the two states can happen fairly easily. They also report a weak correlation between the mass of H*α*-emitting gas and Pcav, as expected if the H*α* traces cooling material, some of which will eventually fuel the AGN. Some of the best known X-ray bright groups contain examples of H*α* filaments that are similar to those seen in clusters (e.g., [14,208–210]). As in clusters, the filaments are closely correlated with feedback-related structures, showing signs of having been drawn out behind, or wrapping around, radio lobes and cavities. In some cases they are located in cool X-ray filaments that show signs of being thermally unstable [209]. Figure 8 shows an example of this in the NGC 5044 group, where the H*α* nebula is correlated with the brightest cool X-ray emission and it appears to wrap around the base of the intermediatescale cavities. Spatially resolved spectroscopy shows that, while the inner parts of these H*α* nebulae are generally cospatial with the stellar bodies of the BGGs, they do not rotate with the stars, supporting formation from the IGrM rather than stellar mass loss (e.g., [211–213]). It should be noted that, while BGGs do host some star formation (SF), their H*α* nebulae are not tracing SF. McDonald et al. [214] studied the relation between the star formation

rate inferred from infrared data and the X-ray cooling luminosity (Section 3.1) and found that the inferred star formation rates in BGGs are typically quenched by a factor 10–100 as compared to the pure cooling scenario.

Molecular gas in groups has been observed via multiple tracers. *Herschel* observations revealed [CII] emission from ∼100 K gas with a similar distribution to the H*α*, and [CII]/H*α* flux ratios indicating that both phases are powered by the same source [208]. *Spitzer* IRS spectra show rotational H2 lines in the BGGs of some X-ray bright groups [215], tracing gas at a few ×100 K, and CN has been detected in absorption in a handful of cases via the millimeter-wave band [216]. The forthcoming *James Webb Space Telescope* will open an important observation window on H2, which is likely the dominant mass component of the molecular phase. However, at present, emission from CO is our best tracer for this phase, allowing for us to examine the coolest, densest gas in the cooling regions of groups.

Babyk et al. [217] examine CO in a large sample of local ellipticals (many of which are BGGs) and find that the molecular gas mass M*mol* is correlated with the density of the IGrM and its mass in the central 10 kpc, and that systems with t*cool* < 1 Gyr at 10 kpc are more likely to contain molecular gas. They also find that M*mol* is proportional to Pcav, confirming that the molecular gas is the fuel source for the central AGN. However, cooling from a surrounding hot halo is not the only source of gas for ellipticals. Davis et al. [218] use a combination of the ATLAS3D and MASSIVE samples to show that gas-rich mergers are an important source of molecular gas in these galaxies. The observations of smaller samples of BGGs find some of the same trends, and show that BGGs of X-ray bright, cool core groups are not the CO-richest systems [219,220]. BGGs of X-ray fainter groups can contain more CO (and HI), and it is more often located in disks, rather than filaments. This, again, emphasizes the importance of gas-rich mergers in groups, although IGrM cooling is likely still the more important process in the cool core groups in which AGN feedback is most often observed.

The BGGs of X-ray bright cool core groups generally seem to contain only a few <sup>×</sup>10<sup>6</sup> or <sup>×</sup>107 <sup>M</sup> of molecular gas [220], which makes them challenging targets, even for ALMA. However, being nearer than typical cool core clusters, groups offer an opportunity to study individual molecular cloud associations within the cool core, rather than the overall filamentary structures. Three well-known systems have been studied in detail by ALMA, NGC 4636, NGC 5846 [221], and NGC 5044 [209,222]. The velocity dispersions of the molecular clumps that were observed in these systems suggest that they are not gravitationally bound, and they are likely collections of smaller, denser clouds, with more diffuse gas between them. Atacama Compact Array (ACA) observations of NGC 5044 show that a significant fraction of the molecular gas in the BGG is more diffuse than the clumps observed by ALMA [213], and a similar argument can be made for the other two groups by comparing the CO masses that are derived from ALMA and IRAM 30m observations. The denser CO clumps are generally located within filamentary structures visible at other wavelengths [221], and the extent and velocity distribution of the diffuse CO in NGC 5044 is similar to that of the H*α* and [CII] emission, supporting the idea that all of the observed phases are material cooling from the IGrM. Figure 8 shows the diffuse CO emission in the group core, collocated with the peak of the H*α* and X-ray emission. As with H*α*, the CO in these ALMA-observed systems is cospatial with the stellar component, but shows little sign of rotation or velocity gradients, consistent with formation from the IGrM.

Intriguingly, ALMA studies of giant radio galaxies, some of them group-central systems, show a different CO morphology, with the molecular gas being located in compact disks [223,224]. The difference in cold gas morphology may indicate a difference in the fuelling of the AGN. There are examples of group-dominant giant radio galaxies that appear to be fed by cold gas (e.g., NGC 1167, [225]) and, given the importance of galaxy interactions in groups, the potential for fuelling by gas rich mergers cannot be ignored. With only a handful of group-dominant galaxies mapped thus far in molecular gas, there is a significant opportunity for the exploration of the mechanics of AGN fuelling in these important systems.

#### **4. Theoretical Framework**

Hot halos are a fascinating and crucial element of virialized systems in the Universe, which have been unveiled to be a fundamental engine for the growth and triggering of SMBH, despite the large difference in spatial and temporal scales, which span over nine orders of magnitude (commonly sub-divided in three major scales; see Figure 9: micro meso—macro). Here, we review the AGN feedback process in terms of fundamental physics and why it is expected in the more theoretical framework of accretion out of the hot halo and onto SMBH, with a keen eye on galaxy groups. It is important to appreciate that AGN feedback is only half of the self-regulated cycle, which is bootstrapped via the AGN feeding, the key complementary mechanism on which we will also focus below, as shown in the summary diagram of Figure 9.

**Figure 9.** A schematic representation of the self-regulation loop necessary to fully link feeding and feedback processes over nine orders of magnitude in space (and time) and over the multiphase/multiband cascade, including Xrays (hot halo/outflows), the IR/optical (warm filaments), and radio/mm (molecular clouds) bands—reproduced from Gaspari et al. [11] (arXiv authors' version). In particular, the IGrM experiences relatively stronger top-down condensation rain compared with clusters, due to the lower central cooling times. Tightly related to such an enhanced feeding is the higher frequency of the AGN outflow/jet feedback events, which gently self-regulate each galaxy group for several billion years during the cosmic evolution.

While complex, non-linear thermo-hydrodynamical (THD) mechanisms are at play over the macro (kpc-Mpc), meso (pc-kpc), and micro scales (mpc-pc)—thus requiring expensive numerical 3D Eulerian simulations—it is useful here to understand the whole SMBH-halo system as a unified, co-evolving engine. In essence, such a global THD system can be described via the simple conservation of energy, or analogously via the (Lagrangian) entropy equation (Peterson and Fabian [83], Gaspari [226]):

$$
\mathcal{U}\frac{d}{dt}\ln K = \mathcal{H} - \mathcal{L},
\tag{12}
$$

where *<sup>K</sup>* <sup>=</sup> *<sup>k</sup>*b*T*/*nγ*−<sup>1</sup> is the astrophysical entropy (with *<sup>n</sup>* <sup>=</sup> *<sup>n</sup>*<sup>e</sup> <sup>+</sup> *<sup>n</sup>*<sup>i</sup> <sup>≈</sup> <sup>2</sup> *<sup>n</sup>*<sup>e</sup> the sum of the electron and ion number densities), *U* = *P*/(*γ* − 1) is the internal/thermal gas energy per unit volume (*γ* = 5/3 is the IGrM adiabatic index), H and L are the gas heating and cooling rates per unit volume (erg s−<sup>1</sup> cm<sup>−</sup>3), respectively. A few immediate insights from Equation (12): the internal energy acts as an effective normalization knob (the larger the X-ray temperature times density, the stronger the required heating/AGN feedback, in absolute erg s−<sup>1</sup> values); secondly, the macro entropy evolution is the sole result of the competition of heating and cooling processes, which translates in the competition between AGN feedback and feeding. Let us first discuss the (astro)physics and consequences of the cooling/feeding component of the cycle, i.e., L.

#### *4.1. AGN Feeding & Cooling Processes*

The cooling process is very well understood from basic quantum physics and laboratory plasma/ionized gas experiments, with a radiative cooling loss [227] L = *n*e*n*<sup>i</sup> Λ(*T*, *Z*), where Λ is a cooling function varying with gas temperature and metallicity *Z* (∼0.6–1 *Z* for group cores; Mernier et al. [228], cf. the companion Gastaldello et al. review). The hot IGrM experiences a significantly enhanced Λ due to the influence of line cooling (mostly recombination) taking over from the Bremsstrahlung/free-free emission (Λ ∝ *T*1/2), which instead shapes the more massive galaxy clusters (*T* > 2 keV), as shown in Figure 10.

Figure 10 depicts the three main (quasi)stable phases that arise during the top-down condensation cascade [166], especially during the feeding dominated stage of the AGN cycle. Assuming relatively slow motions over the group macro gravitational potential (quasi pressure equilibrium), Equation (12) can be approximated as (e.g., Pope [229])

$$H - L \approx \frac{c\_s^2}{\gamma - 1} \dot{M}\_{\text{net}} \tag{13}$$

where *c*<sup>2</sup> <sup>s</sup> = *γk*b*T*/*μm*<sup>p</sup> is the (squared) IGrM sound speed (with *μ* ≈ 0.62 the mean atomic weight of the IGrM) and *H*/*L* is the gas heating/cooling power (or luminosity in erg s<sup>−</sup>1). Notably, any net cooling or heating will induce a net mass inflow (*M*˙ cool) or outflow rate (*M*˙ OUT) in the macro-scale halo, being denoted as *M*˙ *net*; here, uppercase subscripts denote macro properties, while lowercase subscripts denote micro properties. Therefore, the thermal evolution of the IGrM is deeply intertwined with feeding/feedback processes, as intuitively anticipated above.

**Figure 10.** Multiwavelength cooling function for the IGrM (adapted from Gaspari et al. [166], unifying the atomic/plasma physics studies by Sutherland and Dopita [227], Dalgarno and McCray [230], Inoue and Inutsuka [231]; *Z* = 1 *Z*), which was specifically used for a typical galaxy group akin to NGC 5044. Above the neutral hydrogen recombination (*<sup>T</sup>* <sup>∼</sup> <sup>10</sup><sup>4</sup> K), the gas is fully ionized and in collisional ionization equilibrium; below this threshold, the gas becomes progressively less ionized (-1%), leading to the formation of neutral filaments and, subsequently, dense molecular clouds. The three magenta ellipses highlight the three key (semi)stable phases of the condensing IGrM, in particular during the feeding dominated part of the AGN cycle that is shown in Figure 9 (bottom insets).

In the absence of any heating, the entire hot atmosphere would rapidly condense and collapse, initiating from the inner denser radial regions (see Section 3.1). As noted in the previous sections, such massive cooling flows are not observed in our Universe, especially in galaxy groups. On the other extreme of (idealized) feeding models, the IGrM halo might experience pure cooling while having significant angular momentum. In this regime, the gas would condense through helical paths onto the equatorial plane, and there form a thin rotating multiphase disk [166]. While extended disks have been found in some BGGs (Hamer et al. [232], Jurá ˇnová et al. [233], Ruffa et al. [223]), such a scenario would induce both large (unobserved) cooling rates in X-ray spectra, as well as drastically reduced accretion rates onto the SMBH due to the preservation of high angular momentum and related centrifugal barrier.

Realistic IGrM atmospheres, instead, often reside in an intermediate THD regime, neither strongly rotating nor in a spherical cooling flow (David et al. [209], Lakhchaura et al. [206], O'Sullivan et al. [106], Temi et al. [221]—Section 3.3). Indeed, hot halos experience significant amount of turbulence, with an irreducible level of 3D turbulent velocity dispersion *σ<sup>v</sup>* ≈ 100– 300 km s<sup>−</sup>1, due to both the previous AGN feedback outbursts and the secular cosmological flows (e.g., Vazza et al. [234], Valentini and Brighenti [235]), as shown by high-resolution HD/cosmological simulations (Lau et al. [164], Gaspari et al. [236], Hillel and Soker [237], Weinberger et al. [238], Wittor and Gaspari [239]) and X-ray spectroscopy (Sanders and Fabian [240], Ogorzalek et al. [241], Hitomi Collaboration et al. [242]). While we review the kinematical features in a companion review (Gastaldello et al.), here we focus on its thermodynamical impact, namely the formation of *chaotic cold accretion* (CCA) and related multiphase rain, a key process driving the bulk of AGN feeding and, hence, the recurrent AGN feedback triggering. In a turbulent hot halo, chaotic multiscale eddies drive local perturbations in relative gas density proportionally to the turbulence sonic Mach number

(*δn*/*n* ∝ M*t*; Gaspari and Churazov [243], Zhuravleva et al. [244]). The relative increase in IGrM density produces in-situ enhanced radiative cooling (L <sup>∝</sup> *<sup>n</sup>*2), thus leading to turbulent non-linear thermal instability (TI; Gaspari et al. [245], Voit [246]). It is important to note that such a chaotic instability is different from classical TI (Field [161], McCourt et al. [157], Pizzolato and Soker [247]), in the sense that direct non-linear fluctuations are seeded by chaotic motions, rather than growing from tiny linear amplitudes. This triggers a quick top-down condensation of localized (soft X-ray) patches to the first quasistable phase at *<sup>T</sup>* ∼104 K, which is best traced via ionized line-emitting (e.g., H*α*+[NII]; Gastaldello et al. [13], McDonald et al. [248], Werner et al. [208]) filaments or nebulae observed in optical/UV (e.g., see the synthetic image in the bottom middle inset of Figure 9). Sustained turbulent perturbations lead to the further condensation cascade onto the last stable and compact gas phase, molecular gas clouds (bottom left inset of Figure 9; see Section 3.3). Such cold clouds will then strongly and frequently collide inelastically within the meso/micro scale, cancelling angular momentum and, thus, feeding the central SMBH (hence, the 'CCA' nomenclature; Gaspari et al. [166]), with the consequent trigger of the next stage of AGN feedback (Section 4.2). CCA feeding recurrently boosts the accretion rates over 100 fold over the feeble and quiescent hot-mode (Bondi [249], Narayan and Fabian [250]) accretion, thereby overcoming the inefficiency of classical hot mode accretion.

On the macro scale, the hot halo can be assessed to reside or predicted to soon enter the CCA raining phase, whenever the ratio of the plasma cooling time and the turbulence eddy gyration/turnover time reaches unity. This reference dimensionless number is called *C*-ratio (from condensation or CCA; Gaspari et al. [165], Olivares et al. [167]),

$$\mathbf{C} \equiv \frac{t\_{\rm cool}}{t\_{\rm eddy}} \sim \mathbf{1},\tag{14}$$

where the cooling and turbulence timescales have been defined in Equations (8) and (10). A correlated ratio and thermal-instability threshold is the TI-ratio <sup>≡</sup> *<sup>t</sup>*cool/*t*ff <sup>&</sup>lt;<sup>∼</sup> 10–30 (e.g., Gaspari et al. [156], Sharma et al. [158], Voit et al. [251]), where the free-fall time is defined in Equation (9). As introduced in Section 3.1.3, all three IGrM timescales can be constrained from X-ray or optical/IR datasets. While both the *C*-ratios and TI-ratios are valuable complementary tools, the simulations show that the *C*-ratio is the more direct physical criterion to apply to probe the onset and extent of *nonlinear* thermal instability (e.g., Gaspari et al. [165]; see also Figure 17). Indeed, unlike in classical linear TI, turbulence acts as an irreducible background of fluctuations over the whole IGrM (e.g., Lau et al. [164]). In particular, AGN bubbles are a key recurrent mechanism for inducing such fluctuations in the IGrM (e.g., McNamara et al. [252], Voit [253]). In this regard, it is not surprising that *t*cool/*t*ff profiles show a large non-trivial deviation above unity, as well as a large intrinsic scatter (e.g., Singh et al. [254]; also see Figure 6). Figure 11 shows the average cooling and turbulence eddy time profiles (with scatter) for the galaxy group regime (Gaspari et al. [165]). Evidently, the crossing of *C* ∼ 1 matches the dotted circle denoting the typical size of the condensed extended multiphase nebulae well (e.g., McDonald et al. [248]).

Together with mild *C*-ratios, a CCA-driven atmosphere—often found in the IGrM cores is described by a low turbulent Taylor number (Jurá ˇnová et al. [168], Gaspari et al. [255]):

$$\text{Ta}\_{\text{t}} \equiv \frac{\upsilon\_{\text{rot}}}{\sigma\_{\text{v}}} \lessapprox 1. \tag{15}$$

Given that the dominant galaxies of hot gas rich galaxy groups tend to be early-type (as do those galaxies that host their own extended hot halos), they have a fairly weak coherent gas rotational velocity *v*rot (e.g., Caon et al. [256], Diehl and Statler [257]), unlike lower-mass/spiral galaxies. Therefore, a median IGrM long-term evolution is to oscillate between stages of strong CCA rain (Tat ∼ 0.3–1, *C* ∼ 0.5–1) and mild rain superposed on a clumpy disk (Tat ∼ 1–3, *C* ∼ 1–2). Evidently, extremes of strong rotation (Tat 1) or overheated quiescence (*C* 1) can lead to periods of disk- or Bondi-driven accretion, both experiencing highly suppressed inflow rates and feedback (albeit such periods must

be short-lived to avert the cooling flow catastrophe). High-resolution HD simulations have shown that the chaotic behaviour of a CCA-driven halo is imprinted not only in the thermodynamical maps and kinematical properties, but also in the time-series spectra. Unlike quiescent and continuous hot modes (Bondi/ADAF), CCA drives a characteristic flicker/'pink' noise power spectrum (logarithmic slope of −1) in the Fourier space of frequencies *f* (Gaspari et al. [166]), thus generating strong self-similar variability on all temporal scales (the integral over *d f* of this spectrum yields constant variance), from several Myr down to years and minutes, as ubiquitously observed in multiwavelength AGN lightcurves (Ulrich et al. [258], Peterson [259]). This triggers the second key part of the self-regulated loop, AGN feedback, the focus of the next section.

**Figure 11.** The average cooling time and turbulence eddy time profiles in the IGrM (with 90% confidence scatter bands), the latter constrained mainly via optical/IR telescopes (figure reproduced from Gaspari et al. [165]; group sub-sample). The dotted circle marks the size of the condensed warm nebular emission, which matches the *C* ≡ *t*cool/*t*eddy ∼ 1 turbulent TI threshold.

#### *4.2. AGN Feedback & Heating Processes*

While pure cooling flows and catastrophic condensation are not detected in IGrM observations, strong overheating is equally ruled out, as virtually all galaxy groups exhibit central cooling times well below 1 Gyr due to the high efficiency of radiative cooling in their characteristic temperature range (see Figure 10). We note that massive galaxy clusters, instead, show a dominant population of non-cool-core systems with central cooling times above the Hubble time (e.g., [260,261]). The system would be in perfect thermal equilibrium in the absence of any heating and cooling terms, as inferred from Equation (12). However, an analogous configuration can be achieved if a heating process (macro AGN feedback) balances the cooling rate (macro condensation). This configuration is the more realistic state of observed galaxy groups, with the characteristic feature that the self-regulation process is intrinsically chaotic (from the macro down to micro scales; Section 4.1), hence only leading to a *statistical* thermal balance H ∼ L . Moreover, while Equation (12) formally allows for the entropy to decrease (pure cooling), fundamental THD physics dictates that entropy shall always increase in real systems (even over the ensemble Universe). With such intuition, we can already expect that the heating rate is as essential—if not eventually more vigorous—than the cooling component (Figure 9).

Before tackling the AGN feedback physical sub-processes, here we first discuss the key difference between groups and clusters. In Figure 12, we show an analysis of the potential impact of the stored SMBH energy versus the gravitational binding energy of the hot halo cores (from small groups to clusters), by leveraging the large sample of Gaspari et al. [77] with both direct BH masses and extended hot halos detected. The potentially available SMBH mechanical energy is *E*BH = *ε*M*M*BH*c*2, where *ε*<sup>M</sup> is the macro mechanical efficiency (Gaspari and Sa¸dowski [262]). We test for the now commonly used fixed *<sup>ε</sup>*<sup>M</sup> ∼ <sup>10</sup>−<sup>3</sup> (we will explore variations to this basic modeling further below). The gravitational binding energy is tightly related to the thermal energy via the virial theorem, *E*bind ≈ 2 *E*th ∝ *M*gas*T*x. Here, we consider the integration over a large scale, *R* < 0.15 *R*500. Evidently, the linear regression fit (including the intrinsic scatter band) is significantly shallower than the dashed line of the one-to-one balance. In particular, the mechanical feedback energy that a SMBH can release could potentially overcome the core binding energy if released in a very short period of time—for instance, assuming a quasar-like/Sedov blast scenario. This would drastically overheat and evacuate the gaseous core atmosphere, becoming more serious toward the galaxy group regime and lower mass halos (*E*bind,c < 1059 erg), where the feedback energy might even evacuate the entire gas virial region (Puchwein et al. [22], Gaspari et al. [263], as in Section 5.2). Because observations almost ubiquitously detect hot atmospheres, this indicates that the AGN feedback in groups shall be well self-regulated and relatively gentler than in massive galaxy clusters, which can sustain much stronger and impulsive AGN feedback deviations over the cosmic evolution.

**Figure 12.** Available mechanical feedback energy of the central SMBH versus gravitational binding energy of the hot gas within the core of the host halo (*R* - 0.15 *R*500). The SMBH energy is *E*BH = 10−<sup>3</sup> *M*BH*c*2, while the binding energy is related to the thermal energy via the virial theorem *E*bind ≈ 2 *E*th ∝ *M*gas*T*x. The 85 points are taken from Gaspari et al. [77], which include the observed direct/dynamical SMBH mass with the X-ray halo detected in the host group or cluster. The solid red curve shows a fit to the relation with a power law, with the 16–84 percentile interval being indicated by the red shaded area. The 1-*σ* intrinsic scatter is plotted as a light red band on top of the mean fit. The circle colors reflect the morphological type of the central galaxy: elliptical (blue), lenticular (green), and spiral (cyan). The black dashed line demarks the one-to-one energy equivalence, whereas the magenta arrows highlight the excess BH energy when compared to the binding energy.

Reaching a gentle self-regulation, while avoiding strong overheating, implies two major features of AGN feedback in galaxy groups and related improvements when compared with the above modeling. First, the conversion efficiency of accreted rest mass energy into feedback energy is expected to decline with lower halo mass: equating the macro AGN power to the gas X-ray radiative cooling rate requires to modify the above with

*<sup>ε</sup>*<sup>M</sup> ∼ <sup>10</sup>−3(*T*x/2 keV) (Gaspari and Sa¸dowski [262]; see also Equation (16)). This can be explained by the weaker macro-scale coupling of the AGN jets/outflows with the hot halo, as the IGrM atmospheres are more diffuse than the dense ICM counterparts. Second, in order to avoid the above evacuation outburst, such self-regulated AGN feedback has to be not only gentler, but significantly more frequent, i.e. with larger duty cycle (ratio of on/off activity). This is also naturally explained by the relatively lower cooling times (tens of Myr) in the inner IGrM regions (as compared with the ICM counterparts), due to the lower *T*<sup>x</sup> and the substantially enhanced Λ via line emission (Figure 10). Both such key features have been extensively tested and retrieved by high-resolution HD simulations (Gaspari et al. [156], Sharma et al. [158], Gaspari et al. [264], Prasad et al. [265]), and found in observations (e.g., Best et al. [89], O'Sullivan et al. [106]).

While we have discussed the key characteristics and requirements of AGN heating, the next major question is: how is the AGN feedback energy propagated and dissipated within the IGrM? The problem is challenging, as it entails a wide range of scales and phases, from the milliparsec up to at least the 100 kpc region, as shown in Figure 9 (top insets). Generalrelativistic, radiative-magnetohydrodynamical simulations (GR-rMHD; Sa¸dowski and Gaspari [266]) resolving radial distance of ∼500 *r*<sup>S</sup> (Schwarzschild radii) show that the AGN triggered via CCA is able to transform the inner gravitational energy into wide ultrafast outflows (UFOs) with velocities ∼0.1*c* (top-left inset in Figure 9; Fukumura et al. [267], Tombesi et al. [268]). Under strong magnetic field tower and spin conditions (Tchekhovskoy et al. [269]), the AGN is also able to generate a very collimated relativistic (radio-emitting) jet, perpendicularly to the thick accretion torus. The above GR-rMHD simulations show that kinetic feedback appears to be present over both low and high Eddington ratios (*M*˙ BH/*M*˙ Edd <sup>≡</sup> *<sup>M</sup>*˙ BH/[<sup>23</sup> *<sup>M</sup>* yr−1(MBH/109M)]), with a retrieved *micro* mechanical efficiency *ε*<sup>m</sup> 0.03 ± 0.01. At variance, the radiative efficiency declines dramatically below *ε*<sup>r</sup> 0.01 at *M*˙ BH/*M*˙ Edd < 1%, which is the typical regime of local AGN in massive galaxies (Russell et al. [270]). Further, in order to achieve an efficacious macro self-regulation, the AGN feedback has to satisfy energy conservation (Costa et al. [271]) and related micro- to macro-scale power transfer (Gaspari and Sa¸dowski [262]), such as

$$(P\_{\rm out} \equiv \varepsilon\_{\rm m} \dot{M}\_{\rm BH} c^2) \, = \, (P\_{\rm OUT} \equiv \varepsilon\_{\rm M} \dot{M}\_{\rm cool} c^2) \, \sim \, L\_{\rm cool} \,\tag{16}$$

with *Lcool* the cooling luminosity (see Section 3.1). The discrepancy between the above *macro* and the larger *micro* efficiency is crucial: it implies that most of the accreted matter (*M*˙ out/*M*˙ cool = (<sup>1</sup> <sup>−</sup> *<sup>ε</sup>*M/*ε*m) <sup>&</sup>gt; 90%) is re-ejected back by the SMBH, as discussed above, driven mostly in the kinetic form of UFOs and relativistic jets. Such AGN outflows/jets propagate and percolate deeper into the meso-scale atmosphere and start to entrain progressively more IGrM, loading part of the surrounding gas mass and decreasing their velocity down to several 1000 km s−<sup>1</sup> (e.g., Giovannini [272], Fiore et al. [273]).

The last missing tile of the self-regulated cycle is the macro AGN feedback deposition a strongly debated topic since the launch of *Chandra* and XMM-*Newton* telescopes, as their angular resolution is mostly limited to the macro scale (Figure 9, top-right inset). While numerous physical mechanisms have been proposed to compensate macro cooling flows (e.g., McNamara and Nulsen [109]), heren we focus on the physics of the three major mechanisms that have been firmly established to be present in the majority of hot halos, particularly the IGrM (see the observational evidences in Section 3), namely: buoyant bubbles, shocks, and turbulence. While previous reviews tried to assess what is the dominant or sole driver of the AGN feedback, we show that the macro AGN feedback deposition is a strong nonlinear composition of at least three key processes. We can dissect such non-linearity and sub-processes via the local enstrophy analysis, which we define as the squared magnitude of the flow vorticity = <sup>1</sup> <sup>2</sup> |*ω*| <sup>2</sup> <sup>≡</sup> <sup>1</sup> <sup>2</sup> |*∇* × **v**| 2. Neglecting the

small dissipation term, the Lagrangian (tracer particle) framework leads to the following enstrophy evolution decomposition (Wittor and Gaspari [239]):

$$\frac{d\varepsilon}{dt} = \underbrace{-2\varepsilon(\nabla \cdot \mathbf{v})}\_{F\_{\text{com}}} + \underbrace{2\varepsilon\left(\frac{\omega}{|\omega|} \cdot \nabla\right)\mathbf{v} \cdot \frac{\omega}{|\omega|}}\_{F\_{\text{tr}}} + \underbrace{\frac{\omega}{\rho^2} \cdot (\nabla\rho \times \nabla P)}\_{F\_{\text{bar}}}\tag{17}$$

where the three right-hand-side (positive/negative) terms are compressions/rarefactions, stretching/squeezing motions, and baroclinicity, respectively.

The more that we want to zoom into the detailed processes of the AGN heating/weather, the more we need to rely on nonlinear HD simulations. Figure 13 shows a typical AGN-heating dominated period taken from a self-regulated AGN feedback simulation of meso AGN outflows/jets that consistently balance the macro cooling flow down to 1–10% *M*˙ cool,pure (Gaspari et al. [156]). The 30 million tracer particles injected on top of the Eulerian grid can dissect the major components of the macro AGN feedback. First, the progressively slower and entrained outflow/jet inflates a pair of underdense cavities/bubbles, as its ram pressure is balanced by the surrounding IGrM thermal pressure (e.g., Brighenti et al. [274]). Such bubbles are often—albeit not universally—traced by radio synchrotron emission spilling from the micro/meso jets (see Section 3.2). Within their ellipsoidal volumes *V*<sup>b</sup> (Shin et al. [107]), they contain a substantial amount of enthalpy *E*cav 4*P*b*V*<sup>b</sup> in the purely relativistic case (see Section 3.1); dividing by the buoyancy time (Churazov et al. [111]), the related cavity power/heating rate is *H*cav = *E*cav/*t*buoy. Second, the bubbles are often encased within a cocoon shock, which is the result of the strong compressional motions of the expanding outflow and bubbles (see the thick blue contours in the second panel of Figure 13). At this stage, the shock Mach number has become already weakly transonic, M ∼ 1–2 (see Figure 4); as the AGN outflow recurrently ignites, they generate a series of weak shock ripples in the IGrM (Randall et al. [14], Liu et al. [115]), which heat the gas non-adiabatically via cumulative entropy jumps, with heating rate *H*shock = (*e*th Δ ln *K*)/*t*age (where *e*th is the specific thermal energy and *t* <sup>−</sup><sup>1</sup> age is the frequency of shocks). Figure 13 shows that both processes are indeed present, although the relative heating ratio varies as a function of time, with shock heating being initially more vigorous toward the inner regions, while cavity deposition is more effective at 10–100 kpc radii.

Without the third component—subsonic turbulence (see also Section 4.1)—the final macro AGN feedback deposition would be either highly anisotropic (bubble pairs) or localized (thin shock jumps). Figure 13 (third panel) shows that the AGN feedback induces major turbulence/vorticity in a quasi isotropic manner. While the jet direction is a continuous source of enhanced turbulence, the whole IGrM core experiences a quasi irreducible level of turbulent motions (*σ<sup>v</sup>* <sup>∼</sup> 100–300 km s<sup>−</sup>1; <sup>M</sup>*<sup>t</sup>* <sup>&</sup>lt;<sup>∼</sup> 0.5). At the same time, the simulation shows that the (negative) rarefactions avoid the runaway accumulation of large vorticity by balancing the (positive) stretching term in a volume-filling way. The final panel finally shows that baroclinicity is negligible during the macro AGN feedback deposition, as subsonic turbulence is able to preserve the alignment of density and pressure gradients. It is important to note that, while turbulence provides a key source of isotropic *mixing* (with characteristic scale *t*eddy; Section 4.1), its subsonic nature implies that the heating rate (*H*turb = <sup>1</sup> <sup>2</sup> *<sup>M</sup>*gas*σ*<sup>2</sup> *<sup>v</sup>*/*t*turb ∝ *σ*<sup>3</sup> *<sup>v</sup>* , where *<sup>t</sup>*turb = *<sup>t</sup>*eddy/M<sup>2</sup> *<sup>t</sup>* ; Gaspari et al. [275]) is not only a fraction of the global cooling rate, but it also has a substantially delayed deposition time *t*turb *t*cool [237,276]. Alternatively, reorienting jets similar to the case of NGC 5044 (see Figure 8) may provide an alternative way of heating the gas in a quasi-isotropic way. Cielo et al. [277] presented simulations of AGN/IGrM interaction in the case of precessing jets and claimed that the distributed energy is sufficient for offsetting cooling and reproducing the features seen in real cool-core clusters.

**Figure 13.** The macro AGN feedback transfer and deposition highlighted via the enstrophy decomposition (Equation (17)) into its main positive/negative components: compressions/rarefactions (second panel), stretching/squeezing motions (third), baroclinicity (fourth)—adapted from Wittor and Gaspari [239]. This is achieved via Lagrangian tracer particles on top of an adaptive-mesh-refinement HD simulation of self-regulated AGN outflows/jets in a central massive galaxy (Gaspari et al. [156]). The cycle of CCA rain, AGN outflow injection, bubble inflation, cocoon shock expansion, and turbulence cascade repeats self-similarly over several billion years, recurrently quenching the macro cooling flow.

In closing this theoretical section, we remark a few remaining important differences between galaxy groups and the more massive clusters. While we have discussed above that the IGrM shall be strongly self-regulated to avoid overheating/overcooling, this does not imply that groups are less variable than clusters. Indeed, the tails of the chaotic feeding/feedback loop can generate relatively more disruptive imprints in the less bound IGrM (e.g., Voit et al. [278]). This is reflected in the increased morphological diversity of groups (Sun et al. [137]) and larger intrinsic scatter of the scaling relations toward the lowmass regime, as found in the fundamental *L*<sup>x</sup> − *T*<sup>x</sup> (Goulding et al. [279]; see the companion Lovisari et al. review) and *M*BH − *T*<sup>x</sup> (see Section 5.3) relations. While, in absolute values, the AGN deposition radius is significantly larger in galaxy clusters (up to several 100 kpc), normalized to function of *R*<sup>500</sup> the AGN feedback outliers can pierce through relatively larger regions of the less bound IGrM (e.g., Grossová et al. [203]). Interestingly, many elliptical galaxies (including non-centrals) show the presence of a mini-cool core with a size of ∼ 1 kpc, which could represent the irreducible inner CCA condensation region, enabling the more frequent self-regulated AGN feedback discussed above for galaxy groups.

#### **5. Impact of AGN Feedback on Large Scales**

#### *5.1. AGN Feedback in Cosmological Simulations*

Hydrodynamical cosmological simulations are paramount for self-consistently modelling the highly non-linear formation of large-scale structure. They can simultaneously precisely solve for the gravitational and hydrodynamical aspects of structure formation. Yet, because these simulations have limited spatial and mass resolutions, one needs to implement simplified 'sub-grid' prescriptions for including crucial physical processes, such as cooling, star formation, and the feedback from supernovae and AGN, since these phenomena take place at scales that cannot be resolved by the simulations. For a complete review on numerical simulations of galaxy groups, we refer the reader to Oppenheimer et al. within this issue. Modern simulations of galaxy groups broadly fall into three categories:


These simulations have been run with codes that use different methods for solving the equations of hydrodynamics in a cosmological context. Namely, the Tree Particle Mesh (TreePM) and smoothed particle hydrodynamics (SPH) code GADGET [292] in various versions for the majority (EAGLE, MassiveBlack-II, cosmo-OWLS, BAHAMAS, Magneticum), the moving mesh codes AREPO [293,294], and GIZMO [295] for a smaller number (FABLE, Illustris(TNG), SIMBA), and, finally, the adaptive mesh refinement (AMR) code RAMSES [296] for an even smaller fraction of them (Horizon-AGN, Horizon Run 5, and NewHorizon). Note that ROMULUS was run with the Tree+SPH code CHANGA [297]. All of these simulations include a sophisticated modelization of the non-gravitational processes of galaxy formation, such as metal-dependent radiative cooling, star formation, chemical evolution, accretion onto supermassive black holes, and feedback processes from supernovae, asymptotic giant branch stars, as well as AGN. Some of them have even calibrated the free parameters of these models on observations (e.g., FABLE, EAGLE, Illustris(TNG), and BAHAMAS). Note that the value of these parameters are often at least informed by higher-resolution simulations.

In the majority of cases, cosmological simulations (type 2 and 3) implement some variation of the Booth and Schaye [298] AGN feedback model (hereafter BS09), which is itself largely based upon the Springel et al. [292] model (hereafter S05). In the BS09 model, halos are seeded with BH seeds in their center when their mass, as evaluated by an on-the-fly halo finder, first reaches *M*h,min = 100*mDM*, where *mDM* is the mass of a dark matter particle (as in [21]). At that point, BH seeds are introduced at the bottom of the potential well, with masses *Mseed* = 0.001*mg*, where *mg* is the mass of a gas particle. BH can then grow either by gas accretion or mergers. Specifically, BH accrete from the surrounding gas at a rate that is proportional to that given by the Eddington-limited Bondi–Hoyle–Lyttleton [299,300] formula,

$$
\dot{M}\_{\rm acc} = \alpha \dot{M}\_{\rm Bcondi} = \alpha \frac{4\pi G^2 M\_{\rm BH}^2 \rho}{(c\_s^2 + v^2)^{3/2}} \tag{18}
$$

where *v* is the velocity of the BH relative to the ambient gas. The dimensionless 'boosting' factor *α* was introduced by S05 as a numerical correction factor that attempts to correct for the limitations of numerical simulations (see also Section 4.1). It was independent of density and had a constant value of ∼100 (e.g., [22,292,301–303]). BS09 introduced a density-dependent efficiency that varies as a power law of the density with a power-law index *β* = 2 when the density is above *n*∗ *<sup>H</sup>* = 0.1 cm−<sup>3</sup> and *<sup>β</sup>* = 1 otherwise. Bondi– Hoyle accretion is spatially resolved when the local gas density *n*- *<sup>H</sup>* < 0.1 cm<sup>−</sup>3, which corresponds to the threshold for the formation of a cold (*T* - 104 K) phase, and when the simulations resolve the Jeans length (see BS09 and e.g., [304] for a detailed discussion). The BH growth rate can then be determined from the mass accretion rate by assuming a given radiative efficiency *r*, *<sup>M</sup>*˙ BH <sup>=</sup> *<sup>M</sup>*˙ *acc*(<sup>1</sup> <sup>−</sup> *r*). The total radiative efficiency is always assumed to be 10 per cent, which is the mean value for the radiatively efficient [305]

accretion on to a Schwarzschild BH. Bondi accretion—albeit very simple to implement in subgrid models—is far from a realistic representation of the feeding processes (such as CCA and multiphase precipitation), thus we advocate for fundamental updates of subgrid models in future works, as discussed in Section 4.1.

BH inject a fixed fraction of the rest–mass energy of the gas that they accrete into the surrounding medium. The feedback is only implemented thermally. In that case, the energy is deposited into the surrounding gas by increasing its internal energy, as opposed to kinetic feedback, which deposits energy by kicking the gas. The fraction of the accreted rest-mass energy that is injected is assumed to be independent of both the environment and accretion rate (i.e., no distinction between 'quasar mode' and 'radio mode' feedback as in the models of e.g., [302], which still injected energy thermally in both cases). The amount of energy returned by a BH to its surrounding medium in a time-step Δ*t* is given by

$$E\_{\text{feed}} = \epsilon\_f \epsilon\_r \dot{M}\_{\text{BFI}} c^2 \Delta t \tag{19}$$

where *<sup>f</sup>* is the efficiency with which a BH couples the radiated energy into its surroundings (a free parameter) and *c* is the speed of light. In order to ensure that the thermal feedback from BHs is efficient, and that it is not immediately radiated away, BS09 introduced a minimum heating temperature Δ*T*min. BHs store feedback energy until they have accumulated an energy *E*crit that is large enough to increase the temperature of a number *n*heat of their neighbours by an amount of Δ*T*min, which is,

$$E\_{\rm crit} = \frac{m\_{\rm heat} m\_{\circ} k\_B \Delta T\_{\rm min}}{(\gamma - 1) m\_H}. \tag{20}$$

The internal energy of the heated gas is instantaneously increased by *E*crit. If Δ*T*min is set too low, the cooling time of the heated gas remains very short and the energy is efficiently radiated away. If *n*heatΔ*T*min is set too high, then the energy threshold and the time period between AGN heating events become very large. Thus, *n*heatΔ*T*min is connected to the AGN duty cycle. The energy is then isotropically deposited into the gas.

The recent increase in resolution (from category 3 to category 2) led to improvements of the AGN feedback modeling, as more physical processes could be taken into account. For instance, the EAGLE team modified the BS09 formula for *M*˙ *acc* differently from what had been previously done by BS09 to take the angular momentum of the gas accreted by the BH into account, such that the above accretion rate was multiplied by a factor *α* = min(1, *C*−<sup>1</sup> *visc*(*cs*/*Vφ*)3), where *<sup>V</sup><sup>φ</sup>* is the rotation speed of the gas around the BH [306] and *Cvisc* is a free parameter that s related to the viscosity of the accretion disc. Improvements to the [302] model used by [22] have also been made as part of the Illustris, IllustrisTNG and FABLE projects for the same reasons. Specifically, the authors added a third mode of AGN feedback (i.e., the feedback is now thermal, mechanical, and radiative, as described in [307]) for Illustris. The kinetic AGN feedback model in the low accretion rate regime was updated for IllustrisTNG [308] (see [309] for an exhaustive discussion of the changes between Illustris and IllustrisTNG). In parallel, the FABLE team also modified the Illustris model to alleviate some of its shortcomings, such as the underestimation of the gas fractions of groups and clusters (see Section 5.2, and, in particular, the discussion of Figures 14 and 15 below). The parameters of the feedback model were calibrated on the gas mass fractions using a strategy similar to the one employed for BAHAMAS ([41]; see [282] for detailed discussion). In particular, being inspired by the BS09 model, they introduced a 25 Myr duty cycle for the AGN feedback to reduce artificial overcooling. We note that both Horizon-AGN [283] and NewHorizon [281] use two modes of feedback, as originally introduced by [302], but that the low accretion rate or 'radio' mode is kinetic instead of thermal (the detailed modelling can be found in [310]). Indeed, the observed and realistic astrophysical deposition of heating in hot halos is carried out most of the time via the AGN outflows and jets, as discussed in Section 4.2.

#### *5.2. The Hot Gas Fraction and the AGN Feedback Model*

The total baryon content and its partition between the various gas and stellar phases put fundamental constraints on galaxy formation models and, in particular, on the strength of AGN feedback, as stated in Section 3.1.4. Thus, here we present the gas mass fraction-*M*<sup>500</sup> relation at *z* = 0 for two compilations of massive galaxies, groups and cluster simulations that include different sorts of baryonic physics: *(i)* historical simulations, which is, simulations run before 2014–2015 in Figure 14; and, *(ii)* modern simulations run as from 2014–2015 that calibrated the free parameters of their subgrid models to reproduce at least the galaxy stellar mass function at *z* = 0 in Figure 15. The various simulation sets will be compared with the compilation of observations that we presented in Section 3.1.4 and especially Figure 7, which is shown as a gray band on both figures.

**Figure 14.** Compilation of historical simulation results for the gas fraction within *R*<sup>500</sup> as a function of *M*500. The red, orange, magenta, green and dark blue solid lines correspond to the different sub-grid models of cosmo-OWLS [12], the pink and cyan ones to the 300 clusters run with the GADGET3X [311] and MUSIC [312] codes, respectively, as part of The Three Hundred Project [313], the lime and gold ones correspond to Horizon-AGN and, its counterpart without AGN, Horizon-noAGN [283], the cyan, blue, and crimson ones to the various physical models of the DIANOGA suite [314] and, finally, the purple and brown symbols correspond to the simulations of [22] without and with AGN, respectively. The compilation of observations presented in Figure 7 is shown as a gray band.

In Figure 14, we present the results of a compilation of historical simulations for the gas fraction within *R*<sup>500</sup> as a function of *M*500. The NOCOOL model of cosmo-OWLS and the NR model of DIANOGA both correspond to classical non-radiative simulations, where one includes hydrodynamics, but do not allow the gas to cool through radiative processes. The REF model of cosmo-OWLS, the CSF model of DIANOGA, the 300 clusters run with the MUSIC code, as well as the noAGN models of [22] and the Horizon suite, all include prescriptions for radiative cooling, star formation, and stellar feedback, but not

for AGN feedback. As first noted by [22,315], the inclusion of AGN feedback substantially lowers the gas fractions of both groups and clusters.The intensity and, thus, the duty cycle of the AGN feedback, as parameterized by Δ*T*min in BS09 (see Equation (20)), can be used to eject more or less gas from the potential well, as can be seen by comparing the models AGN 8.0, 8.5, and 8.7 of cosmo-OWLS. Here the number corresponds to the logarithm of the value of Δ*T*min chosen, i.e., 8.0 corresponds to Δ*T*min = 108 K. The REF, CSF, and noAGN models also yield reasonable gas mass fractions, but the relation with mass is flatter than observed, because the SF efficiency does not strongly depend on halo mass. The low gas fractions in these models are achieved by overly efficient star formation (e.g., [12,315] and Figure 1). Note that, while the cosmo-OWLS models that include AGN feedback use the BS09 AGN feedback model summarized in Section 5.1, which is fully thermal, Horizon-AGN, DIANOGA and 300 G3X resort to a mixture of thermal and kinetic feedback, as originally developed by [310] for the former and [316] latter two.

**Figure 15.** Compilation of gas fractions within *R*<sup>500</sup> from modern simulations as a function of *M*500. The blue line corresponds to BAHAMAS [41], the red and green lines to the Reference and dT9 models of EAGLE [284], the green symbols to C-EAGLE and Hydrangea [317,318], as it uses the same sub-grid model as EAGLE-dT9, the olive line to the 300 clusters run with the GIZMO-SIMBA code (Cui et al. in preparation) as part of The Three Hundred Project [313], the salmon line to Horizon-AGN [283], the gold one to FABLE [282], the orange one to SIMBA [287,288], and the pink and deep pink symbols, as well as the plum, orchid, and lime lines correspond to various models from the Illustris and IllustrisTNG suites [285,286]. The compilation of observations presented in Figure 7 is shown as a gray band.

In Figure 15, we present the results of a compilation of modern simulations for the gas fraction within *R*<sup>500</sup> as a function of *M*500. Despite the fact that most modern simulations have been calibrated to reproduce the local galaxy stellar mass function (see Figure 1), the predictions on the hot gas fraction are vastly different. For instance, Illustris (plum line) vastly underpredicts the observed hot gas fractions, whereas the reference EAGLE model

(red line) clearly overpredicts them. Therefore, a setup that broadly reproduces the stellar content of galaxies in the Universe may simultaneously fail at reproducing the properties of the hot gas phase. Note that, in the case of BAHAMAS (blue line) and FABLE (gold line), the free parameters of the stellar and AGN feedback have been adjusted to reproduce both the *z* = 0 galaxy stellar mass function and gas content of groups and clusters (see discussions in [41,282]). IllustrisTNG and EAGLE-dT9 (and the associated simulations) are versions of Illustris and EAGLE in which the AGN feedback parameters were slightly adjusted to reduce the discrepancies with the gas content of massive groups and clusters. It is worth noting that SIMBA uses a fully kinetic AGN feedback model [287], while the simulations from the Illustris series and FABLE include a mix of thermal, kinetic/mechanical, and radiative feedback [282,285,309] in the vein of the one first developed by [302].

Generally speaking, we stress that the hot gas fraction of galaxy groups is an extremely sensitive probe of the feedback scheme implemented in cosmological simulations. Modern simulation suites have little predictive power on the baryon content of groups, even when the properties of the galaxy population are accurately reproduced (see Figure 1). Some of the simulations are actually *calibrated* on the gas mass fractions, i.e., the parameters governing the feedback model were tuned to produce reasonable gas fractions in the group regime. Major observational (Section 3.1.4) and theoretical advances (Section 4) are required to understand the ejection of baryons from halos by AGN feedback and inform the mainstream galaxy evolution models.

#### *5.3. Co-Evolution between the IGrM and the Central AGN*

SMBH masses are known to correlate with the properties of their host galaxy, in particular the integrated K-band luminosity *LK* and the velocity dispersion of the stars in the bulge, *σ*<sup>e</sup> (see [64] for a review), as discussed in Section 2.2. However, it is still unclear whether the optical scaling relations of SMBH are fundamental or derive from correlations with other key quantities. Recent findings have instead unveiled that the SMBH masses are more tightly correlated with the properties of the host X-ray gaseous halos, especially in the IGrM regime [76–78]. In Figure 16, we summarize our current knowledge of the relation between SMBH mass and X-ray temperature within the core of galaxy groups (*<sup>R</sup>* <sup>&</sup>lt;<sup>∼</sup> 0.15 *<sup>R</sup>*500). It is important to note that the SMBH masses shall be directly observed via dynamical measurements to properly unveil intrinsic scaling relations. The largest existing study is provided by Gaspari et al. [77] with 85 systems with measured SMBH masses, most of which with temperatures ∼0.5–1 keV that were typical of galaxy groups. A Bayesian fit to the relation finds slopes *M*BH ∝ *T*2.1 <sup>x</sup> (Figure 16, green) and *M*BH ∝ *L*0.4 <sup>x</sup> . At the high-mass end, Bogdán et al. [76] measure a somewhat flatter slope, *M*BH ∝ *T*1.7 <sup>x</sup> . Notably, the intrinsic scatter goes down to ∼0.2 dex, with a very high correlation coefficient at or above the 0.9 level, when compared to ∼0.5 dex for the K-band luminosity. The correlations hold, regardless of the large diversity of systems, from BGGs and ETGs to non-central lenticular/spiral galaxies. Varying the extraction radius by slightly enlarging or decreasing it (group outskirts or CGM) does not significantly vary such conclusions (see the companion Lovisari et al. review for the complementary *R*<sup>500</sup> scalings, such as *M*BH − *L*<sup>500</sup> and *M*BH − *M*tot). We note that multi-variate X-ray correlations (a.k.a. 'fundamental planes') do not further improve the intrinsic scatter. In sum, by comparing the different X-ray/optical scaling relations, it has emerged that the extended plasma (collisional) atmospheres seem to play a more fundamental role than small-scale (collisionless) stellar properties in the co-evolution of SMBH and groups. This is further supported by zoom-in cosmological simulations [319–321]. On the other hand, the slope (and scatter) of the current cosmological simulations still remain too low when compared with the observations (dotted lines in Figure 16), indicating the need to model more realistic feeding and feedback physics (see Section 4) into the coarse subgrid numerical modules.

**Figure 16.** Relation between BH mass and IGrM X-ray temperature. The data points show dynamical measurements of BH mass plotted against the spectroscopic X-ray temperature of the host halo. The solid curves show the fitted observational relations from Gaspari et al. [77] (green; together with the quoted log-normal scatter), Bogdán et al. [76] (blue), and the "BCG" subsample of Lakhchaura et al. [78] (red). The dashed lines show the predictions of cosmological simulations with AGN feedback (Illustris TNG, Truong et al. [321]; DIANOGA, Bassini et al. [319]). The gray data points are taken from the sample of Gaspari et al. [77], which already included the smaller samples that were used by Bogdán et al. [76] and Lakhchaura et al. [78].

The above SMBH versus X-ray correlations are crucial for testing models of galaxy/group formation and evolution. Accretion/feeding models can be broadly divided into cold and hot accretion modes, as discussed in Section 4.1. Besides the cosmic dawn, hierarchical binary BH mergers are a present, but subdominant growth channel over most of cosmic time [77,319]. In hot accretion (usually Bondi or Advection Dominated Accretion Flow— ADAF; Bondi [249], Narayan and Fabian [250]), the larger the thermal entropy of the gas, the more strongly feeding is stifled, since the inflowing gas has to overcome the outward thermal pressure of the hot halo. This would induce negative correlations with the IGrM properties, which are ruled out by the strongly positive correlations that are shown in Figure 16. Conversely, cold-mode accretion (Gaspari et al. [245], Voit [246]; Section 4.1) typically in chaotic form (due to the turbulent condensation out of the IGrM generating randomly colliding clouds)—is linearly and tightly correlated with the X-ray luminosity and gas mass. Figure 17 (left) shows the CCA final mass growth via theoretical/numerical predictions [77] when compared to direct measurements of SMBH masses. During the Gyr evolution, the turbulent IGrM locally condenses into extended warm filaments and cold molecular clouds via nonlinear thermal instability. The clouds inelastic collisions at the meso/kpc scale boost the micro accretion rate down to the Schwarzschild radius, hence triggering strong AGN feedback heating. Such recurrent SMBH growth drives the *M*BH that is shown in Figure 17, with excellent agreement with the SMBH mass observed in our local universe.

**Figure 17.** X-ray scaling relations are key to constrain different baryonic physics of the IGrM, here in terms of feeding models. *Left:* Direct SMBH masses plotted against the mass derived from theoretical/numerical predictions of chaotic cold accretion (CCA) via the X-ray core properties—adapted from Gaspari et al. [77]. *Right:* Hot-halo condensation radius as a function of BH mass and thus IGrM halo mass (the locus where *C* ≡ *t*cool/*t*eddy = 1; see Section 3.1.3 and Section 4.1). See Figure 12 for the description of the analysis, color coding, and sample. Figure reproduced from Gaspari et al. [77].

Scaling relations allow us to predict other baryonic physics of the IGrM, such as the extent of the IGrM multiphase condensation radius, which is shown in the right panel of Figure 17. As introduced in Section 4.1, such a radius is the locus at which the IGrM cooling time and the turbulent eddy-turnover time match (*C* ≡ *t*cool/*t*eddy = 1), both of which can be retrieved via the X-ray scaling relations as a function of *T*<sup>x</sup> and *L*<sup>x</sup> (Gaspari et al. [77]). Evidently, lower mass groups have condensation radii of less than a few kpc, while massive groups can reach *R*cond of a few 10 kpc (e.g., David et al. [209], Olivares et al. [167], Lakhchaura et al. [206]). Overall, scaling relations between X-ray macro-scale properties translates into scaling relations of micro-scale properties (*M*BH), corroborating a tight co-evolution between multi-scale processes in the IGrM (as depicted in Figure 9).

#### *5.4. Impact on Cosmological Probes*

During the past few years, it has become clear that AGN feedback will play an important role as a leading source of systematic uncertainties for upcoming high-profile cosmology experiments. Indeed, the energy that is injected by the central AGN affects the global distribution of baryons (see Section 5.2), leading to local depletion or excesses of matter with respect to the expectations of models, including dark matter only. This effect is most important in galaxy groups, since these systems correspond to the peak of the local halo mass density and their baryonic properties are highly sensitive to feedback. As a result, the matter power spectrum at *z* = 0 can be substantially altered by baryonic physics and, in particular, AGN feedback (e.g., [322,323]). The shape of the power spectrum is strongly affected by baryonic processes on scales *k* 10−<sup>1</sup> *h*/Mpc. For 10−<sup>1</sup> < *k* < 10 *h*/Mpc, most of the simulations predict a *deficit* of power with respect to the N-body case, although the actual amplitude of the effect is highly uncertain [324]. The evacuation of baryons from the central regions of galaxy groups under the influence of feedback is responsible for the deficit of power on scales of ∼ 1 Mpc, i.e., roughly the typical size of galaxy groups. On smaller scales (*k* 10 *h*/Mpc), cooling and condensation of baryons in the central regions lead to a rapidly increasing power.

Accurately predicting the shape of the matter power spectrum is crucial for the success of future cosmic shear experiments, such as *Euclid*, which aim at determining the growth of structures by measuring the matter power spectrum and its evolution [325]. Semboloni et al. [323] showed that neglecting baryonic effects would imply important systematics on the determination of cosmological parameters. Systematic effects can be mitigated by excluding the small scales (*k* 10−<sup>1</sup> *h*/Mpc) when fitting the measured power spectra, although that comes at the price of greatly increased uncertainties in the resulting cosmological parameters. Constraints on extended cosmologies, such as massive neutrinos, variable dark energy equation of state, or chameleon gravity, require sensitivity on smaller scales, and their effect is strongly degenerate with that of baryonic physics [326,327].

Chisari et al. [324] showed that numerical simulations have not yet converged on the actual impact of feedback on the power spectrum (see their Figure 3). For instance, the very strong feedback that was implemented in the original Illustris simulation, which was sufficient to completely evacuate the gas content from most groups (see Figure 15), leads to a very strong suppression of power (>30%) on scales of a few hundred kpc. Conversely, simulations implementing a more gentle feedback scheme (EAGLE, Horizon-AGN, MassiveBlackII) predict small corrections with respect to the fiducial DM-only case for *k* - 10 *h*/Mpc. These simulations also predict a high gas fraction in galaxy groups (see Section 5.2 and [328] for an extensive discussion). Recently, Schneider et al. [329] used a semi-analytic model to predict the impact of baryons on the matter power spectrum based on the observed gas properties of groups. The authors modified the mass profiles of halos in large N-body simulations to account for star formation and AGN feedback. In particular, the semi-analytic model of Schneider et al. [329] is highly sensitive to the parameter *θej*, which governs the ejection of gas from the central regions of the halo by AGN feedback. In Figure 18, we show how the predicted matter power spectrum depends on *θej*. With increasing feedback, a progressively larger fraction of the gas is ejected from the halo and, thus, the expected power gets more strongly suppressed. Calibrating their semi-analytic model on the observed gas fraction and gas density profiles of group-scale halos, Schneider et al. [329] provided a range of predictions matching the existing observational constraints. High-precision measurements of the gas density profiles in a representative sample of galaxy groups would allow us to precisely determine the expected shape of the power spectrum [327], thereby providing a key input for upcoming cosmology experiments.

In addition to the matter power spectrum, AGN feedback on the scales of galaxy groups also affects several other cosmological observables, such as the thermal SZ power spectrum (e.g., [330,331]). Indeed, AGN feedback affects the pressure profiles of halos and thus modifies the amplitude of the power spectrum on small scales ( 1000, [332]). Ramos-Ceja et al. [333] showed that tSZ models based on the universal pressure profile [334] overpredict the power measured by SPT and ACT on small scales. A strong feedback scenario and a low gas fraction on group scales are needed to fit the measured power. The effect of feedback also implies modifications to the cross-correlations between the tSZ and other observables (e.g., [335,336]). Finally, the choice of the feedback scheme affects the shape of the halo mass function (e.g., [337–339]) and the structure of dark-matter halos (e.g., [337,340]). We refer to the companion Oppenheimer et al. review for a more general discussion of the topic.

**Figure 18.** Modification of the local matter power spectrum with respect to pure N-body simulations in the presence of AGN feedback at the scale of galaxy groups in the semi-analytic model of Schneider et al. [329]. The various curves show how the power spectrum depends on the parameter *θej* governing gas ejection from the central regions of groups under the influence of AGN feedback. A strong ejection of gas from the core of halos implies substantial modifications to the matter power spectrum on scales of ∼1 Mpc. The figure reproduced from Schneider et al. [329].

#### **6. Future Observatories**

#### *6.1. eROSITA*

At present, our knowledge of the population of galaxy groups in the local Universe comes largely from the *ROSAT* All-Sky Survey (RASS). Groups that were identified in the RASS (or even the *Einstein* slew survey) form the basis of most studies of the mechanics of AGN feedback at this mass scale, but, unfortunately, groups are at the lower limit of sensitivity for these surveys. RASS is therefore biased toward the detection of relaxed, centrally-concentrated, cool-core systems, with the strength of the bias increasing as mass decreases from poor clusters to groups [185]. Searches tailored to the detection of more extended sources in RASS reveal a population of low surface brightness groups undetected by the original survey [341] confirming the expected bias, while XMM-*Newton* observations of optically-selected groups identify both low luminosity and disturbed systems previously not detected or not recognised as groups in RASS [106].

*Spectrum Roentgen Gamma* (SRG, launched in 2019) hosts the eROSITA instrument, a set of seven co-aligned soft X-ray telescopes covering the 0.2–10 keV band, with a field of view of 1◦ and ∼15 spatial resolution. SRG will spend four years surveying the whole sky once every six months, with eROSITA building up a map ∼20× deeper than RASS in the 0.5–2 keV band [342]. This is sufficient to detect essentially every galaxy group with a virialized halo in the local universe [106]. More massive groups with luminosities ∼1042 erg s−<sup>1</sup> should be detectable to *<sup>z</sup>* ∼ 0.1 in the final eRASS:8 survey. Käfer et al. [343] performed detailed simulations to evaluate the sensitivity of eRASS:8 to galaxy groups. The authors used a wavelet decomposition algorithm sensitive to large-scale diffuse emission. In Figure 19, we show the corresponding sensitivity curves for two possible source detection setups: a decomposition over wavelets of scales 1–4 that were optimized for relatively compact sources, and the other for scales in the range 1–16 sensitive to the most extended nearby sources. Using these setups, the authors predict eRASS:8 will detect all the galaxy groups with *<sup>M</sup>*<sup>500</sup> <sup>&</sup>gt; 1013*M* out to *<sup>z</sup>* <sup>=</sup> 0.05. The most massive groups (∼1014*M*) will be detected out to *<sup>z</sup>* <sup>=</sup> 0.5. While the survey observations will typically only provide luminosity and morphology information for individual halos, the

group samples derived from them will be a solid base from which to investigate the impact of cooling and AGN feedback in groups, particularly when combined with radio surveys.

Pointed observations with eROSITA, possible once the survey phase is complete, may also prove useful for studies of groups. The combination of a large field of view and soft band effective area (roughly double that of the XMM-*Newton* EPIC-pn) is well-suited to observations of the outskirts of nearby groups, and the search for cavities or other structures that are associated with giant group-central radio galaxies. Thanks to its short focal length and very stable background [344], eROSITA is very sensitive to diffuse X-ray emission below 2 keV, which means that it is well suited to study the diffuse X-ray emission of galaxy groups.

**Figure 19.** Expected sensitivity curve of the final eROSITA survey (eRASS:8) compared to the sensitivity of existing and upcoming X-ray and SZ surveys. The eROSITA sensitivity curve was computed from synthetic data using a wavelet decomposition algorithm [343] sensitive to scales of 1–4 (solid blue curve) and 1–16 (dashed blue curve). For comparison, the dashed black curve shows the sensitivity of the ROSAT all-sky survey assuming a fixed soft X-ray flux threshold of 1.8 <sup>×</sup> <sup>10</sup>−<sup>12</sup> erg s−1, which is the typical sensitivity of the REFLEX and NORAS samples (green asterisks, [345]). The red points show the systems selected from the second *Planck* SZ catalogue [346]. The green curve shows the expected sensitivity of the SPT-3G experiment [347].

#### *6.2. XRISM*

The X-ray Imaging and Spectroscopy Mission (XRISM), expected to launch by April 2023, will open a new era of high spectral-resolution observations of galaxy groups. Its X-ray microcalorimeter (*Resolve*, [348]) will have a constant 7 eV energy resolution across its 0.3–12 keV band, resolving the forest of emission lines that characterizes emissions from the IGrM. This offers opportunities in a number of important areas, including measurements of bulk flows and turbulence in the hot gas. At present, grating spectra only provide upper limits on the turbulence of the IGrM [240,349], with possible hints of asymmetries that are associated with sloshing motions [350]. *Hitomi* demonstrated the capabilities of microcalorimeters, but it was only able to make a single turbulence measurement in the Perseus cluster [351]. XRISM should be able to measure turbulent velocities down to tens of km s<sup>−</sup>1, providing a measure of the kinetic energy stored in the IGrM and allowing us to determine how much of the energy of AGN outbursts can be diffused out into the IGrM by these gas motions. The spectra from the *Resolve* microcalorimeter will also provide a detailed view of shock heating and cooling, with individual emission lines accurately tracing gas at different temperatures. Performance verification targets for the mission include NGC 5044 and NGC 4636, and, while the spatial resolution (>1 ) and effective area of the observatory may limit its use to relatively bright nearby groups, its results are likely to be ground-breaking.

#### *6.3. Athena*

The Advanced Telescope for High Energy Astrophysics (Athena), which is expected to launch in the early 2030s, represents the next generation of major X-ray observatories, with 5 spatial resolution and an effective area of 1.4 m<sup>2</sup> at 1 keV (roughly 45× that of XRISM's *Resolve* instrument). It will carry a 40 ×40 active pixel detector (the wide field imager, WFI) providing CCD-like spectral imaging, and a 5 -diameter microcalorimeter array (the X-ray Integral Field Unit, X-IFU) with ≤5 pixels, capable of 2.5 eV spectral resolution (see, e.g., [352]). The combination of the very large collecting area with these two instruments will open up several new fields of study for galaxy groups. The WFI survey, performed over the first four years of operations, is expected to find >10,000 groups and clusters at *<sup>z</sup>* <sup>≥</sup>0.5, including <sup>∼</sup>20 groups with M500 <sup>≥</sup>5×1013 <sup>M</sup> at *<sup>z</sup>* <sup>∼</sup> 2, and measure their temperature to better than 25% accuracy [353]. This will provide a clear view of the evolution of AGN feedback in groups back to the era of peak star formation and black hole growth. The identification of cavities and spectral mapping will be possible for moderating redshift, showing us the impact of feedback over the past few gigayears.

X-IFU offers capabilities that are similar to that of XRISM's *Resolve*, but with greatly improved spatial resolution and the ability to examine even low luminosity systems in the local universe. It will allow the mapping of turbulence and bulk flows in the IGrM, tracing gas motions that are associated with mergers, sloshing, uplift behind rising radio bubbles, or AGN-driven outflows. By mapping the kinetic and thermal energy content of the IGrM on spatial scales similar to those at which energy is injected by radio galaxies, it will allow us to quantify how much energy is injected into the hot gas by outbursts, determine where and when energy is transferred out of the radio jets and lobes, and see how it is then transported out into the surrounding halo [354]. It will also provide a clear view of the location of the coolest gas and allow us to trace the process by which it cools out of the hot phase.

#### *6.4. Lynx*

The Lynx mission concept (https://www.lynxobservatory.com/report accessed on 25 April 2021, [355]) will, if approved, go beyond Athena, with sub-arcsecond resolution over a 22 ×22 field of view and an effective area of 2 m<sup>2</sup> at 1 keV, giving 50× the throughput of *Chandra*. As with Athena, an active pixel array (the high-definition X-ray imager, HDXI) would provide wide-field CCD-like spectral imaging, but with 0.3 pixels to take advantage of the exquisite spatial resolution. The Lynx X-ray Microcalorimeter (LXM) would bring 3 eV spectral resolution on 1 spatial scales over a 5 field of view, with sub-arrays offering 0.5 spatial resolution or 0.3 eV spectral resolution in 1 fields. Much of Lynx's proposed science relates to the detailed physics of accretion and galaxy evolution, and the X-ray universe at high redshift; the survey observations with the HDXI would be capable of detecting groups with masses as low as 2 <sup>×</sup> <sup>10</sup><sup>13</sup> <sup>M</sup> out to a *<sup>z</sup>* <sup>∼</sup> 3. In low-redshift systems, LXM could examine the conditions within individual cooling filaments in group cores, and measure the velocities of the weak shocks and sound waves that are produced during AGN outbursts. As with Athena, the observatory would be sensitive enough to trace the IGrM out to the virial radius in a large sample of groups, but with fine spatial resolution, making the identification of structure easier and the rejection of background sources cleaner. It is notable that Lynx would be the first mission after *Chandra* to be able to provide the finely detailed images that have proved to be so useful in the study of AGN feedback, reaching scales that are comparable to those of optical and radio observations.

#### *6.5. The Square Kilometer Array and Its Precursors*

Radio astronomy has undergone something of a renaissance in recent years, with new and improved capabilities coming online, e.g., the upgraded Jansky Very Large Array (JVLA) and Giant Metrewave Radio Telescope (uGMRT), the Low-Frequency Array (LOFAR), and the Atacama Large Milimeter Array (ALMA). These are providing improved radio continuum surveys in the northern hemisphere and equatorial sky. However, new telescopes in southern Africa and Australia are opening up new opportunities, as they begin to survey the relatively poorly-explored southern sky. These include the Murchison Widefield Array (MWA), the Australian Square Kilometer Array Pathfinder (ASKAP), and MeerKAT in South Africa's Karoo region. The Galactic and Extragalactic All-sky MWA Survey (GLEAM, [356]) provides an early example, covering the entire southern sky below Declination +30◦ at frequencies 72–231 MHz. While its spatial resolution is modest (∼100), its high sensitivity at low frequency and provision of fluxes in multiple bands makes it a powerful tool for studying radio galaxies, particularly old, fading sources. Higher frequency (∼1 GHz), higher spatial resolution (8-30) surveys of continuum emission and HI are becoming available from ASKAP (e.g., RACS, WALLABY, [357,358]) and MeerKAT (e.g., MIGHTEE, [359]), and these observatories are beginning to produce interesting findings on, e.g., group dynamics and evolution [360,361] and the population of giant radio galaxies [197,201].

These telescopes are the precursors of the Square Kilometer Array (SKA), a set of nextgeneration telescopes that are expected to begin operations in the late 2020s, combining wide frequency coverage with unprecedented sensitivity. The SKA will be built in phases, with phase 1 consisting of two components: the SKA1-Low, covering the 50–350 MHz band, with baselines up to 65 km providing spatial resolution of ∼4 at 300 MHz and sensitivity a factor of 5–10 better than LOFAR or GMRT; and the SKA1-Mid, covering 350 MHz to 15 GHz with resolution 0.4 at 1.4 GHz and a sensitivity up to an order of magnitude better than JVLA [362]. The proposed phase 2 SKA would improve sensitivity by another order of magnitude. Much of the SKA's proposed science relates to the early universe, but it will be an extraordinary tool for studies of feedback in groups and clusters, tracing the entire AGN population to high redshift, not merely the radio-loud systems that dominate current samples [363]. SKA surveys are likely to be sensitive to sources down to 1022 W Hz−<sup>1</sup> out to *z* = 3–4 [364]. This offers an opportunity to detect the radio counterparts of most galaxies that are identified in current optical surveys, including essentially all group and cluster-dominant galaxies, with sufficient resolution to allow AGN and star formation emission to be disentangled, and with the wide frequency coverage that is necessary to determine the state and age of jets and lobes. Given the very large numbers of groups and clusters that are likely to be detected in the southern sky by, e.g., eROSITA, such surveys will play an important role in identifying systems with active cooling and feedback. HI observations reaching low column densities may provide another window on cooling from the IGrM. SKA will also open up the study of diffuse radio structures and magnetic fields in groups (e.g., [365]), providing constraints on rates of energy transport and conduction in the IGrM, as well as information on gas motions [366], and perhaps even on turbulence and shocks.

#### *6.6. Upcoming SZ Facilities*

Upcoming surveys of the cosmic microwave background (CMB), such as CCATprime [367], Simons Observatory [368], and CMB-S4 [369], will likely also play a role in advancing our understanding of AGN feedback at group scale by providing complementary information to X-ray surveys. While the thermal SZ effect (hereafter tSZ) is a steep function of mass (*YSZ* ∝ *M*5/3 <sup>500</sup> ), stacking of the tSZ effect over large samples (either X-ray or optically selected) can lead to a detection down to *<sup>M</sup>*<sup>500</sup> <sup>∼</sup>1013*M*. As a pilot study, Planck Collaboration et al. [370] presented the stacked tSZ signal from a large sample of SDSS galaxies selected to be central to their halo, and found that the tSZ-to-mass scaling relation extends with no break all the way down to *<sup>M</sup>*<sup>500</sup> <sup>∼</sup>1012.5*M*. However, the interpretation of

the result is rendered difficult by the large *Planck* beam (∼8 ), which dilutes the signal [371]. While detecting the tSZ signal from individual galaxy groups will be challenging for the new generation of CMB survey instruments, the angular resolution of the foreseen facilities (- 1 ) will be sufficient to study the distribution of the stacked tSZ signal and determine the origin of the signal identified by Planck Collaboration et al. [370]. On the other hand, large single-dish facilities such as AtLAST [372] will be sensitive enough to detect the tSZ effect from galaxy groups [373].

Recently, the kinetic SZ effect (kSZ), i.e., the Doppler shift of the CMB spectrum induced by moving electron clouds, has emerged as a promising tool for studying the baryon content of galaxy groups [374,375]. The kSZ signal is independent of the gas temperature, which makes it, in principle, more suitable than the tSZ for the study of low-mass systems. The kSZ signal cannot be detected directly by stacking CMB observations, given that the average velocity of structures with respect to the CMB rest frame vanishes. However, the kSZ signal can be measured by cross-correlating CMB maps with spectroscopic galaxy surveys, thereby fixing each system's velocity; this technique is known as the *pairwise* kSZ. Several recent studies reported low-significance detections of the kSZ with this technique [376–378]. These early results may indicate that the flat gas density profiles that are inferred from X-ray data (see Section 3.1.4) extend far beyond the halo's virial radius, which, if confirmed, provides a detection of the gas expelled from the central regions of halos by AGN feedback. Cross-correlating the kSZ data from future CMB experiments with large spectroscopic surveys, like DESI, may yield a detection of the pairwise kSZ at high significance [379] and possibly out to high redshifts [380].

**Author Contributions:** D.E.: lead author, Sections 1, 2, 3.1.3, 3.1.4, 5.2, 5.3, 5.4, 6.1 and 6.6; M.G.: Sections 3.1.3, 4 and 5.3; F.G.: Sections 3.1.1 and 3.1.2; A.M.C.L.B.: Sections 2, 5.1 and 5.2; E.O.: Sections 3.2, 3.3 and 6. All authors have read and agreed to the published version of the manuscript.

**Funding:** M.G. acknowledges partial support by NASA Chandra GO8-19104X/GO9-20114X and HST GO-15890.020-A grants. A.M.C.L.B. is supported by a fellowship of PSL University at the Paris Observatory. E.O. acknowledges support from NASA through *XMM-Newton* award number 80NSSC19K1056 and *Chandra* award number GO8-19112A.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The review is based on public data and/or published papers.

**Acknowledgments:** We deeply thank Yannick Bahé, Weigang Cui, Marco De Petris, Federico Sembolini, Yohan Dubois, Scott Kay, Alisson Pellissier, Ewald Puchwein, Elena Rasia, Dylan Robson, Marcel van Daalen, Mark Vogelsberger and Rainer Weinberger for providing data from their simulations for inclusion in Figures 14 and 15, even if some could not be included in the end. We also thank Paul Nulsen for providing cavity parameters for inclusion in Table A1, and the anonymous referees for useful comments. We thank Ming Sun, Mark Voit, Iurii Babyk, Trevor Ponman and Aurel Schneider for granting us permission to reprint some of their figures. E.O. thanks G. Schellenberger and K. Kolokythas for useful conversations. D.E. thanks A. Finoguenov for useful discussions. D.E. and A.M.C.L.B. thank Benjamin Oppenheimer for useful discussions. We are also grateful to Tony Mroczkowski, Joop Schaye and Ming Sun for sending comments on the accepted version while we were checking the proofs, which were taken into account for the published version.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. List of the Properties of Detected AGN Cavities in Galaxy Groups**

**Table A1.** Properties of the cavities detected in groups. Only quantities available from the literature are included, thus for some systems the listing will be incomplete. <sup>1</sup> Projected semi-major axis of the cavity. <sup>2</sup> Projected semi-minor axis of the cavity. <sup>3</sup> Projected distance from the cavity center to the core. <sup>4</sup> Ages are reported as *ts*-*tbuoy*-*trefill*. Where only a single value is reported, this is *ts*. <sup>5</sup> Bolometric luminosity between 0.001 and 100 keV inside *rcool*, where the cooling time is less than 7.7 Gyr. -Additional data provided by Nulsen, P., priv. comm.



**Table A1.** *Cont*.

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