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Galaxy clusters are the largest gravitationally bound objects in the universe. They are also among the rarest objects in the universe. While these two facts about galaxy clusters may seem disparate, they are in fact intimately related. Our current theory of the origin and evolution of galaxy clusters places them within the broader context of cosmological structure formation in which galaxies, galaxy groups, galaxy clusters, and galaxy superclusters all arise from gravitational instability amplifying perturbations in the cold dark matter density field in an expanding universe. At early times the perturbations are linear in amplitude, and are extremely well described by a Gaussian random field with a known power spectrum (the LambdaCDM power spectrum; cf. Fig. 2). At later times, density perturbations become nonlinear and collapse into gravitationally bound systems. The shape of the LambdaCDM power spectrum is such that structure forms from the "bottom up", with galaxies forming first and clusters forming later. It just so happens that we live in a universe in which cluster-scale perturbations collapsed rather recently (since z ~ 1), which accounts for their rarity as well as their sometimes complex substructure.

As cluster-scale perturbations collapse, they bring in all matter within a sphere of comoving radius of about 15 Mpc, which includes galaxies, intergalactic gas, and anything else in that patch of the universe. Because the escape velocity of galaxy clusters is of order 1000 km/s, everything but relativistic particles become trapped in the cluster potential well. For this reason it is often said that clusters represent a fair sample of the universe. This is true from the standpoint of their matter content. However, from the standpoint of cosmological structure this could not be further from the truth. Galaxy clusters form and evolve in the rarest peaks (~ 3sigma) of the density field (Fig. 1). Galaxy formation begins sooner in such regions, and the galaxies evolve due to internal and external processes which are somewhat different from the general field (e.g., ram pressure stripping).

Figure 1

Figure 1. AMR hydrodynamic cosmological simulation of cosmic structure in a 700 Mpc volume of the universe. Up to seven levels of adaptive mesh refinement (AMR) resolve the distribution of baryons within and between galaxy clusters, for an effective resolution of 65,5363. Volume rendering of baryon density. From [70].

Galaxy clusters are interesting objects in their own right, and for decades have been extensively studied in the optical, Xray, and radio wavebands [65]. More recently this has been extended to the microwave, infrared, and extreme UV [66, 67], motivated in part by the fact that galaxy clusters are excellent cosmological probes. Because of their large size and high X-ray luminosities, they can be seen to great distances. As discussed in this volume and in [67], galaxy clusters can also be seen in absorption/emission against the cosmic microwave background (CMB) via the Sunyaev-Zeldovich effect (SZE). As discussed by Rephaeli elsewhere in these proceedings, the SZE is redshift independent, meaning that deep microwave surveys should detect all galaxy clusters in a particular region of the sky regardless of their redshift provided the telescope has enough angular resolution and sensitivity. Indeed such surveys are underway now and results are expected soon.

One of the most challenging measurements in modern observational cosmology is the dark energy equation of state which describes the time rate of change of the vacuum energy density dot{Lambda} responsible for the accelerating expansion of the universe [5]. Galaxy clusters were identified by the US Dark Energy Task Force (DETF) as one of four complementary methods for doing this. However in order to measure dot{Lambda} we must measure the cluster mass distribution function versus redshift to very high accuracy. Accurately measuring the mass of a galaxy cluster is actually quite difficult despite the number of ingenious techniques that have been developed. This motivates attempts to measure the "mass-observable" relationships in observed samples of clusters, and find the ones with the least scatter and least bias.

Numerical simulations are helpful in this regard, as one can in principle calibrate the mass-observable relationships by comparing simulated observations with in situ measurements. Where this has been done, a discouraging result is found: most of the observables of a given simulated cluster depend sensitively on numerical resolution and assumed baryonic physics, including radiative cooling, star formation, galactic and AGN feedback processes. However it has been shown that the integrated SZE signal is rather insensitive to assumed baryonic physics [48, 71], perhaps relaxing requirements on modelers and giving encouragement to SZE cluster observers that the dark energy program may be feasible after all. The point is that the astrophysics of galaxy clusters and their utility as cosmological probes are inextricably linked, and both are worthwhile of study.

I was asked by the organizers to lecture on three topics relavant to the theme of the summer school. First I was asked to review the standard cosmological framework and basic results from the theory of cosmological structure formation within which galaxy clusters can be understood. Second, I was asked to review how galaxy clusters are simulated on a computer, and summarize the basic findings. I do this in the next two sections, which are slightly updated and abbreviated versions of the lecture notes I published in the 2004 Varenna Summer School volume [67]. Finally I was asked to give a lecture of my choosing, which was on recent progress in galaxy cluster modeling focusing on the incorporation of additional baryonic physics and simulating SZE surveys . These topics are presented in Sections 5 and 6 respectively.

In line with the character of the summer school, I have attempted to be pedagogical, emphasizing the key concepts and results that a student needs to know if s/he wants to understand the current literature or do research in this area. Literature citations are kept to a minimum, except for textbooks, reviews, and research papers that I found to be particularly helpful in preparing this article. I am indebted to Dr. Rocky Kolb whose slides much of Section 2 are based upon.

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