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Galaxy groups host the majority of matter and more than half of all the galaxies in the Universe. Their hot (107 K), X-ray emitting intra-group medium (IGrM) reveals emission lines typical of many elements synthesized by stars and supernovae. Because their gravitational potentials are shallower than those of rich galaxy clusters, groups are ideal targets for studying, through X-ray observations , feedback effects, which leave important marks on their gas and metal contents. Here, we review the history and present status of the chemical abundances in the IGrM probed by X-ray spectroscopy. We discuss the limitations of our current knowledge, in particular due to uncertainties in the modeling of the Fe-L shell by plasma codes, and coverage of the volume beyond the central region. We further summarize the constraints on the abundance pattern at the group mass scale and the insight it provides to the history of chemical enrichment. Parallel to the observational efforts, we review the progress made by both cosmological hydrodynamical simulations and controlled high-resolution 3D simulations to reproduce the radial distribution of metals in the IGrM, the dependence on system mass from group to cluster scales, and the role of AGN and SN feedback in producing the observed phenomenology. Finally, we highlight future prospects in this field, where progress will be driven both by a much richer sample of X-ray emitting groups identified with eROSITA, and by a revolution in the study of X-ray spectra expected from micro-calorimeters onboard XRISM and ATHENA.

As building blocks of dust, rocky planets, and even complex life, the chemical elements heavier than hydrogen (H) and helium (He) - called \"metals\" in astronomy - play an essential role in our Universe and its evolution. Up to Fe and Ni, these metals are known to be created by stars and stellar remnants via nuclear fusion and to be ejected into their immediate surroundings to enrich new stellar generations. A spectacular finding, however, is that these processed elements are found even outside stellar systems, in particular in the hot, X-ray atmospheres surrounding early-type galaxies and pervading galaxy clusters and groups. These large-scales structures are thus a remarkable fossil record of the integrated history of the enrichment of our Universe. In this Chapter, we briefly discuss the chemical properties of this intracluster - or intragroup - medium (ICM). After introducing the concept of chemical abundances, and recalling which stellar sources produce which elements, we review the method to derive abundance measurements from observations of the ICM and detail some chemical models implemented in numerical hydrodynamical simulations of cosmic structures. In particular, we explore how synergies between X-ray observations and numerical simulations help us to understand (i) the cosmic epoch at which the bulk of the enrichment occurred, (ii) the physics of these stellar sources responsible for this enrichment, and (iii) the main mechanisms responsible for metal diffusion and mixing outside galaxies.

Theoretical models of structure formation predict the presence of a hot gaseous atmosphere around galaxies. While this hot circum-galactic medium (CGM) has been observationally confirmed through UV absorption lines, the detection of its direct X-ray emission remains scarce. We investigate theoretical predictions of the intrinsic CGM X-ray surface brightness (SB) using simulated galaxies and connect them to their global properties such as gas temperature, hot gas fraction and stellar mass. We select a sample of galaxies from the ultra-high resolution (\\(48\\ \\rm{cMpc\\, h^{-1}}\\)) cosmological volume of the Magneticum Pathfinder set of hydrodynamical cosmological simulations. We classify them as star-forming (SF) or quiescent (QU) based on their specific star-formation rate. For each galaxy we generate X-ray mock data using the X-ray photon simulator PHOX, from which we obtain SB profiles out to the virial radius for different X-ray emitting components, namely gas, active galactic nuclei and X-ray binaries (XRBs). We fit a \\(\\beta\\)-profile to each galaxy and observe trends between its slope and global quantities of the simulated galaxy. We find marginal differences between the average total SB profile of the CGM in SF and QU galaxies, with the contribution from hot gas being the largest (\\(>50\\%\\)) at radii \\(r>0.05\\,R_{\\rm{vir}}\\). The contribution from X-ray binaries (XRBs) equals the gas contribution for small radii and is non-zero for large radii. The galaxy population shows positive correlations between global properties and normalization of the SB profile. The slope of fitted \\(\\beta\\)-profiles correlates strongly with the total gas luminosity, which in turn shows strong connections to the current accretion rate of the central super-massive black hole (SMBH).

We discuss recent advances and prospects in our understanding of the formation of structures on cosmic scales based on surveys of galaxy clusters in the X-ray bands. We highlight the interaction between observations and numerical simulations of the X-ray sky. We show how future surveys will unveil the nature of the dark energy and study its evolution with time.

We provide a numerical framework with which spatially and spectrally accurate representations of X-ray binary populations can be studied from hydrodynamical cosmological simulations. We construct average spectra accounting for a hot gas component and verify the emergence of observed scaling relations between galaxy wide X-ray luminosity (\\(L_{X}\\)) and stellar mass (\\(M_{\\star}\\)) as well as star-formation rate (SFR). Using simulated galaxy halos extracted from the \\((48\\,h^{-1} \\mathrm{cMpc})^3\\) volume of the Magneticum Pathfinder cosmological simulations at \\(z = 0.07\\) we generate mock spectra with the X-ray photon-simulator Phox. We extend the Phox code to account for the stellar component in the simulation and study the resulting contribution in composite galactic spectra. Average X-ray luminosity functions are perfectly reproduced up to the one-photon luminosity limit. Comparing our resulting \\(L_{X}-\\mathrm{SFR}-M_{\\star}\\) relation for X-ray binaries with recent observations of field galaxies in the Virgo galaxy cluster we find significant overlap. Invoking a metallicity dependent model for high-mass X-ray binaries yields an anti-correlation between mass-weighted stellar metallicity and SFR normalized luminosity. The spatial distribution of high-mass X-ray binaries coincides with star-formation regions of simulated galaxies while low-mass X-ray binaries follow the stellar mass surface density. X-ray binary emission is the dominant contribution in the 2-10 keV band in the absence of an actively accreting central super-massive black hole with 50% contribution in the 0.5-2 keV band rivaling the hot gas component. Our modelling remains consistent with observations despite uncertainties connected to our approach. The predictive power and easily extendable framework hold great value for future investigations of galactic X-ray spectra.

(Abridged) Cluster environment has a strong impact on the star formation rate and AGN activity in cluster galaxies. In this work, we investigate the behaviour of different galaxy populations in galaxy clusters and their vicinity by means of cosmological hydrodynamical simulations. We studied galaxies with stellar mass \\(\\log M_\\ast (M_\\odot) > 10.15\\) in galaxy clusters with mass \\(M_{500} > 10^{13} M_\\odot\\) extracted from box2b (640 comoving Mpc/\\(h\\)) of the Magneticum Pathfinder suite of cosmological hydrodynamical simulations at redshifts 0.25 and 0.90. We examined the influence of stellar mass, distance to the nearest neighbouring galaxy, clustercentric radius, substructure membership and large-scale surroundings on the fraction of galaxies hosting an AGN, star formation rate and the ratio between star-forming and quiescent galaxies. We found that in low-mass galaxies, AGN activity and star formation are similarly affected by the environment and decline towards the cluster centre. In massive galaxies, the impact is different; star-formation level increases in the inner regions and peaks between 0.5 and 1 \\(R_{500}\\) with a rapid decline in the centre, whereas AGN activity declines in the inner regions and rapidly rises below \\(R_{500}\\) towards the centre - likely due to stellar mass stripping and the consequent selection of galaxies with more massive black holes. After disentangling the contributions of neighbouring cluster regions, we found an excess of AGN activity in massive galaxies on the cluster outskirts (\\(\\sim 3 R_{500}\\)). We also found that the local density, substructure membership and stellar mass strongly influence star formation and AGN activity but verified that they cannot fully account for the observed radial trends.

Our aim is to understand how the interplay between AGN feedback and merge processes can effectively turn cool-core galaxy clusters into hot-core clusters in the modern universe. Additionally, we also aim to clarify which parameters of the AGN feedback model used in simulations can cause an excess of feedback at the scale of galaxy groups while not efficiently suppressing star formation at the scale of galaxy clusters. To obtain robust statistics of the cool-core population, we compare the modern Universe snapshot (z=0.25) of the largest Magneticum simulation (Box2b/hr) with the eROSITA eFEDS survey and Planck SZ-selected clusters observed with XMM-Newton. Additionally, we compare the AGN feedback injected by the simulation in radio mode with Chandra observations of X-ray cavities, and LOFAR observations of radio emission. We confirm a decreasing trend in cool-core fractions towards the most massive galaxy clusters, which is well reproduced by the Magneticum simulations. This evolution is connected with an increased merge activity that injects high-energy particles into the core region, but it also requires thermalization and conductivity to enhance mixing through the ICM core, where both factors are increasingly efficient towards the high mass end. On the other hand, AGN feedback remains as the dominant factor at the scale of galaxy groups, while its relative impact decreases towards the most massive clusters. The problems suppressing star formation in simulations are not caused by low AGN feedback efficiencies. They root in the definition of the black hole sphere of influence used to distribute the feedback, which decreases as density and accretion rate increase. Actually, a decreasing AGN feedback efficiency towards low-mass galaxy groups is required to prevent overheating.

Theoretical models of structure formation predict the presence of a hot gaseous atmosphere around galaxies. While this hot circum-galactic medium (CGM) has been observationally confirmed through UV absorption lines, the detection of its direct X-ray emission remains scarce. We investigate theoretical predictions of the intrinsic CGM X-ray surface brightness (SB) using simulated galaxies and connect them to their global properties such as gas temperature, hot gas fraction and stellar mass. We select a sample of galaxies from the ultra-high resolution (\\(48\\ \\rm{cMpc\\, h^{-1}}\\)) cosmological volume of the Magneticum Pathfinder set of hydrodynamical cosmological simulations. We classify them as star-forming (SF) or quiescent (QU) based on their specific star-formation rate. For each galaxy we generate X-ray mock data using the X-ray photon simulator PHOX, from which we obtain SB profiles out to the virial radius for different X-ray emitting components, namely gas, active galactic nuclei and X-ray binaries (XRBs). We fit a \\(\\beta\\)-profile to each galaxy and observe trends between its slope and global quantities of the simulated galaxy. We find marginal differences between the average total SB profile of the CGM in SF and QU galaxies, with the contribution from hot gas being the largest (\\(>50\\%\\)) at radii \\(r>0.05\\,R_{\\rm{vir}}\\). The contribution from X-ray binaries (XRBs) equals the gas contribution for small radii and is non-zero for large radii. The galaxy population shows positive correlations between global properties and normalization of the SB profile. The slope of fitted \\(\\beta\\)-profiles correlates strongly with the total gas luminosity, which in turn shows strong connections to the current accretion rate of the central super-massive black hole (SMBH).

The baryon fraction of galaxy clusters is a powerful tool to inform on the cosmological parameters while the hot-gas fraction provides indications on the physics of the intracluster plasma and its interplay with the processes driving galaxy formation. Using cosmological hydrodynamical simulations from The Three Hundred collaboration of about 300 simulated massive galaxy clusters with median mass \\(M_{500}\\approx7 \\times 10^{14}\\)M\\(_{\\odot}\\) at \\(z=0\\), we model the relations between total mass and either baryon fraction or the hot gas fractions at overdensities \\(\\Delta = 2500\\), \\(500\\), and \\(200\\) with respect to the cosmic critical density, and their evolution from \\(z\\sim 0\\) to \\(z\\sim 1.3\\). We fit the simulation results for such scaling relations against three analytic forms (linear, quadratic, and logarithmic in a logarithmic plane) and three forms for the redshift dependence, considering as a variable both the inverse of cosmic scale factor, \\((1+z)\\), and the Hubble expansion rate, \\(E(z)\\). We show that power-law dependencies on cluster mass poorly describe the investigated relations. A power-law fails to simultaneously capture the flattening of the total baryon and gas fractions at high masses, their drop at the low masses, and the transition between these two regimes. The other two functional forms provide a more accurate description of the curvature in mass scaling. The fractions measured within smaller radii exhibit a stronger evolution than those measured within larger radii. From the analysis of these simulations, we conclude that as long as we include systems in the mass range herein investigated, the baryon or gas fraction can be accurately related to the total mass through either a parabola or a logarithm in the logarithmic plane. The trends are common to all modern hydro simulations, although the amplitude of the drop at low masses might differ [Abridged].

To assume hydrostatic equilibrium between the intracluster medium and the gravitational potential of galaxy clusters is an extensively used method to investigate their total masses. We want to test hydrostatic masses obtained with an observational code in the context of the SRG/eROSITA survey. We use the hydrostatic modeling code MBProj2 to fit surface-brightness profiles to simulated clusters with idealized properties as well as to a sample of 93 clusters taken from the Magneticum Pathfinder simulations. We investigate the latter under the assumption of idealized observational conditions and also for realistic eROSITA data quality. The comparison of the fitted cumulative total mass profiles and the true mass profiles provided by the simulations allows to gain knowledge about the reliability of our approach. Furthermore, we use the true profiles for gas density and pressure to compute hydrostatic mass profiles based on theory for every cluster. For an idealized cluster that was simulated to fulfill perfect hydrostatic equilibrium, we find that the cumulative total mass at the true \\(r_{500}\\) and \\(r_{200}\\) can be reproduced with deviations of less than 7%. For the clusters from the Magneticum Pathfinder simulations under idealized observational conditions, the median values of the fitted cumulative total masses at the true \\(r_{500}\\) and \\(r_{200}\\) are in agreement with our expectations, taking into account the hydrostatic mass bias. Nevertheless, we find a tendency towards a too high steepness of the cumulative total mass profiles in the outskirts. For realistic eROSITA data quality, this steepness problem intensifies for clusters with high redshifts and thus leads to too high cumulative total masses at \\(r_{200}\\). For the hydrostatic masses based on the true profiles known from the simulations, we find a good agreement with our expectations concerning the hydrostatic mass.