The galaxy and matter distribution in the non-linear regime

Non-linear structures are seen to be present in the Universe at nearly any scale: galaxies at the sub-Mpc scale, clusters and voids at scales ranging from 1 to 10, possibly 100 Mpc. These structures arise from initial conditions in the early universe that we are having difficulties in finding out. They arise also from the non-linear behaviour of gravitationally interacting matter inhomogeneities. It is of primordial importance to be able to disentangle the presently observed properties of the galaxy distribution that are due to non-linearity from those that are genuine tracers of the initial conditions. The matter (or galaxy) distribution may be described in terms of correlation functions, not only the usual 2-body correlations, but those of arbitrary order. We start from the suggestion of Peebles that the observed power-law behaviour of the 2-body correlation function may be explained by some virialization process. This in turn implies that the N-body correlation function also are powers-laws. We have explored the consequences of this behaviour, and obtained predictions for the probability of holes, the galaxy and cluster luminosity functions, and more generally on the probability distribution of galaxy counts in a cell of any size, on the fractal distribution of galaxies and matter as well as on the cluster distribution in the past leading to an estimate of their X-ray emission and of the fluctuations they introduce in the microwave background