Clustering Properties of Dark Matter Haloes in Hierarchical Models for Structure Formation in the Universe

In chapter 1 we introduce some basic elements of modern cosmology, describing the main features of the standard hot big-bang model and the concept of inflation. Special attention is devoted to show that recent observational data are consistent with the theoretical framework. In chapter 2 we review the theory of gravitational instability in an expanding universe. Both the Eulerian and the Lagrangian perturbative approaches to the evolution of density perturbations are discussed in detail. The spherical top-hat model and a series of dynamical approximations are also presented. Chapter 3 deals with scaling solutions to the problem of clustering growth. After a brief review, we present some original results regarding the evolution of the autocorrelation function of the mass density field. In particular, we test the predictions of some empirically calibrated scaling Ansatze against the analytical solutions obtained by using the Zel'dovich approximation. Chapter 4 is devoted to the issue of hierarchical clustering. First, we describe the Press-Schechter theory for the abundance and mass distribution of dark matter haloes, and its excursion-set extension. A new model for the clustering of dark haloes in Lagrangian space is then discussed. In chapter 5 we present an astrophysical application of the Press-Schechter formalisrn. In particular, we investigate the lensing effect of background supernovae due to mass condensations in three popular CDM cosmologies. Our results suggest that it is not inconceivable that new and existing search teams will soon be able to detect magnified supernovae by conducting deep pencil beam surveys. In chapter 6 we review the theory of biased galaxy formation. Both the original motivations that led to its formulation and recent developments are discussed. In chapter 7 we present a new stochastic approach to the clustering evolution of dark matter halos in Eulerian space. Our results clearly point to a characterization of the halo-to-mass biasing as a highly non-linear and non-local process. This chapter contains some of the most important results of this thesis. We devote chapter 8 to compare the predictions of the models introduced in chapters 4, 6 and 7 with N-body simulations. In appendix A we introduce the basic concepts of the theory of random fields, while in appendix B we briefly summarize the main results of classical kinetic theory. The definition of the n-point correlation functions for a population of discrete objects (e.g. galaxies) is also given.