Spherically Symmetric Non Linear Structures

Abstract

We present an analytical method to extract observational predictions about non linear evolution of perturbations in a Tolman Universe. We assume no a priori profile for them. We solve perturbatively a Hamilton - Jacobi equation for a timelike geodesic and obtain the null one as a limiting case in two situations: for an observer located in the center of symmetry and for a non - centered one. In the first case we find expressions to evaluate the density contrast and the number count and luminosity distance vs redshift relationships up to second order in the perturbations. In the second situation we calculate the CMBR anisotropies at large angular scales produced by the density contrast and by the asymmetry of the observer's location, up to first order in the perturbations. We develope our argument in such a way that the formulae are valid for any shape of the primordial spectrum.