Hot Gas in Clusters of Galaxies: the Punctuated Equilibria Model
  We develop a model to describe the evolution of the intra-cluster X-ray emitting baryons, as they are included in the dark matter potential wells of galaxy clusters evolving through subsequent merging events in the framework of hierarchical clustering. The gas is assumed to re-adjusts to a new hydrostatic equilibrium after each merging event. Before merging it is gravitationally heated at the local virial temperature when bound in subclusters; at early $z$ the gas is preheated by supernova activity following star formation. In detail, we compute analytically the following steps: the dynamic histories of dark matter halos with their merging events; the associated infall of gas into a halo, with compressions and shocks estabilishing the conditions at the cluster boundary; the updated disposition of the gas in the potential well matching such conditions; the statistical convolution of the key quantities over the merging histories. The model predicts the density and surface brightness profiles with no free parameters; the so-called $\beta$ parameter is itself an outcome of the model, and the polytropic index $\gamma$ is internally constrained to a narrow range. We obtain declining temperature profiles, and profiles for the density and for the surface brightness shallower in groups compared with clusters. Our model groups also contain a lower baryonic fraction on average, but with a scatter considerably larger than at cluster scales. Various statistics are obtained analytically upon averaging over the merging histories. In particular, we predict in different cosmologies the statistical correlation $L-T$ of luminosity with temperature; similarly we derivethe correlation $R_X-T$ for the size of the X-ray emitting region. The intrinsic scatter in both correlations is also predicted. 
A. Cavaliere (1), N. Menci (2), P. Tozzi (2,3)