6. SUMMARY AND OUTLOOK

All of the main elements of the overall narrative of how clusters form and evolve discussed in this review have been established over the past four decades. The remarkable progress in our understanding of cluster formation has been accompanied by great progress in multi-wavelength observations of clusters and our knowledge of the properties of the main mass constituents of clusters: stars, hot intracluster gas, and gravitationally dominant DM.

Formation of galaxy clusters is a complicated, non-linear process accompanied by a host of physical phenomena on a wide range of scales. Yet, some aspects of clusters exhibit remarkable regularity, and their internal structure, abundance, and spatial distribution carry an indelible memory of the initial linear density perturbation field and the cosmic expansion history. This is manifested both by tight scaling relations between cluster properties and the total mass, as well as by the approximate universality of the cluster mass function and bias, when expressed as a function of the peak height .

Likewise, there is abundant observational evidence that complex processes - in the form of a non-linear, self-regulating cycle of gas cooling and accretion onto the SMBHs and associated feedback - have been operating in the central regions of clusters. In addition, the ICM is stirred by continuing accretion of the intergalactic gas, motion of cluster galaxies, and AGN bubbles. Studies of cluster cores provide a unique window into the interplay between the evolution of the most massive galaxies, taking place under extreme environmental conditions, and the physics of the diffuse hot baryons. At the same time, processes accompanying galaxy formation also leave a mark on the ICM properties at larger radii. In these regions, the gas entropy measured from observations is considerably higher than predicted by simple models that do not include such processes, and the ICM is also significantly enriched by heavy elements. This highlights that the ICM properties are the end product of the past interaction between the galaxy evolution processes and the intergalactic medium. Nevertheless, at intermediate radii, r₂₅₀₀ r r₅₀₀, the scaling of the radial profiles of gas density, temperature, and pressure with the total mass is close to simple, self-similar expectations for clusters of sufficiently large mass (corresponding to kT 2-3 keV). This implies that the baryon processes affecting the ICM during cluster formation do not introduce a new mass scale. Such regular behaviour of the ICM profiles provides a basis for the definition of integrated quantities, such as the core-excised X-ray luminosity and temperature, gas mass, or integrated pressure, which are tightly correlated with each other and with the total cluster mass.

The low-scatter scaling relations are used to interpret abundance and spatial distribution of clusters and derive cosmological constraints (see Allen, Evrard & Mantz 2011 and Weinberg et al. 2012 for recent reviews). Currently, cluster counts measured at high redshifts provide interesting constraints on cosmological parameters complementary to other methods (e.g., Vikhlinin et al. 2009c, Mantz et al. 2010b, Rozo et al. 2010) and a crucial test of the entire class of CDM and quintessence models (e.g., Jee et al. 2011, Benson et al. 2011, Mortonson, Hu & Huterer 2011). Although the statistical power of large future cluster surveys will put increasingly more stringent requirements on the theoretical uncertainties associated with cluster scaling relations and mass function (Cunha & Evrard 2010, Wu, Zentner & Wechsler 2010), future cluster samples can provide competitive constraints on the non-Gaussianity in the initial density field and deviations from GR gravity.

A combination of cluster abundance and large-scale clustering measurements can be used to derive stringent constraints on cosmological parameters and possible deviations from the standard CDM paradigm. As an example, Figure 16 shows the constraints on the normalization of the power spectrum and the f_NL parameter, (from Sartoris et al. 2010) expected for a future high-sensitivity X-ray cluster survey. It shows that future cluster surveys can achieve a precision of _fNL 5-10 (see also Cunha, Huterer & Doré 2010, Pillepich, Porciani & Reiprich 2012), thus complementing at smaller scales constraints on non-Gaussianity, which are to be provided on larger scales by observations of CMB anisotropies from the Planck satellite.

Figure 16. The potential of future cluster X-ray surveys to constrain deviations from Gaussian density perturbations (adapted from Sartoris et al. 2010). The figure shows constraints on the power-spectrum normalization, 8, and non-Gaussianity parameter, fNL, expected from surveys of galaxy clusters to be carried out with the next-generation Wide Field X-ray Telescope. Dot-dashed blue curve and dashed green curve show the 68% confidence regions provided by the evolution of power spectrum (PS) of the cluster distribution and cluster number counts (NC), respectively. The solid red ellipse shows the constraints obtained by combining number counts and power spectrum information. Cocmic Microwave Background Planck priors for Gaussian perturbations have been included in the analysis.

Although a variety of methods will provide constraints on the equation of state of DE and other cosmological parameters (e.g., Weinberg et al. 2012), clusters will remain one of the most powerful ways to probe deviations from the GR gravity (e.g., Lombriser et al. 2009). Even now, the strongest constraints on deviations from the GR on the Hubble horizon scales are derived from the combination of the measured redshift evolution of cluster number counts and geometrical probes of cosmic expansion (Schmidt, Vikhlinin & Hu 2009). Figure 17 illustrates the potential constraints on the linear rate of perturbation growth that can be derived from a future high-sensitivity X-ray cluster survey using similar analysis. The figure shows that a sample of about 2000 clusters at z < 2 with well-calibrated mass measurements would allow one to distinguish the standard CDM model from a braneworld-modified gravity model with the identical expansion history at a high confidence level.

Figure 17. The potential of future cluster surveys to constrain deviations from General Relativity (from Vikhlinin et al. (2009d). The linear growth factor of density perturbations, G(z) = D(z) (not normalized to unity at z = 0), recovered from 2000 clusters, distributed in 20 redshift bins, each containing 100 massive clusters, identified in a high-sensitivity X-ray cluster survey. The solid black line indicates the evolution of the linear growth factor for a CDM model, whereas the dashed blue curve is the prediction of a modified gravity model (the brane world model by Dvali, Gabadadze & Porrati 2000), having the same expansion history of the CDM model.

We are grateful to Brad Benson, Klaus Dolag, Surhud More, Piero Rosati, Elena Rasia, Ming Sun, Paolo Tozzi, Alexey Vikhlinin, Mark Voit, and Mark Wyman for useful discussions and comments, and to John Carlstrom for a careful reading of the manuscript. We thank Dunja Fabjan and Barbara Sartoris for their help in producing Fig. 15 and Fig. 16, respectively. The authors wish to thank the Kavli Institute for Theoretical Physics (KITP) in Santa Barbara for hospitality during the early phase of preparation of this review and participants of the KITP workshop "Galaxy clusters: crossroads of astrophysics and cosmology" for many stimulating discussions. AK was supported by NSF grants AST-0507596 and AST-0807444, NASA grant NAG5-13274, and by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0551142 and PHY-1125897. SB acknowledges partial support by the European Commissions FP7 Marie Curie Initial Training Network CosmoComp (PITN-GA-2009-238356), by the PRIN-INAF09 project "Towards an Italian Network for Computational Cosmology", by the PRIN-MIUR09 "Tracing the growth of structures in the Universe" and by the PD51 INFN grant.