Cosmological Principle and the debate about Large Scale Structures distribution

Abstract

The basic hypothesis of a post-Copernican Cosmological theory is that {\em all the points} of the Universe have to be essentially equivalent: this hypothesis is required in order to avoid any privileged {\em observer}. This assumption has been implemented by Einstein in the so-called Cosmological Principles (CP): {\em all the positions} in the Universe have to be essentially equivalent, so that the Universe is homogeneous. This situation implies also the condition of spherical symmetry about every point, so that the Universe in also Isotropic. There is a hidden assumption in the formulation of the CP in regard to the hypothesis that all the points are equivalent In fact, the condition that all the occupied points are statistically equivalent with respect to their environment correspond to the property of of Local Isotropy. It is generally believed that the Universe cannot be isotropic about every point without being also homogeneous [1]. Actually Local Isotropy does not necessarily implies homogeneity; in fact a topology theorem states that homogeneity is implied by the condition of local isotropy together with {\em the assumption of the analyticity or regularity} for the distribution of matter. Up to the seventies analyticity was an obvious implicit assumption in any physical problem. Recently however we have learned about intrinsically irregular structures for which analyticity should be considered as a property to be tested with appropriate analysis of experiment.