Scale Invariance in a Perturbed Einstein-de Sitter Cosmology

This paper seeks to check the validity of the apparent fractal conjecture (Ribeiro 2001ab), which states that the observed power-law behaviour for the average density of large-scale distribution of galaxies arises when some observational quantities, selected by their relevance in average density profile determination, are calculated along the past light cone. Since general relativity states that astronomical observations are carried out in this spacetime hypersurface, observables necessary for direct comparison with astronomical data must be calculated along it. Implementing this condition in the proposed set of observational relations profoundly changes the behaviour of many observables in the standard cosmological models. In particular, the average density becomes inhomogeneous, even in the spatially homogeneous spacetime of standard cosmology, change which was already analysed by Ribeiro (1992b, 1993, 1994, 1995) for a non-perturbed model. Here we derive observational relations in a perturbed Einstein-de Sitter cosmology by means of the perturbation scheme proposed by Abdalla and Mohayaee (1999), where the scale factor is expanded in power series to yield perturbative terms. The differential equations derived in this perturbative context, and other observables necessary in our analysis, are solved numerically. The results show that our perturbed Einstein-de Sitter cosmology can be approximately described by a decaying power-law like average density profile, meaning that the dust distribution of this cosmology has a scaling behaviour compatible with the power-law profile of the density-distance correlation observed in the galaxy catalogues. These results show that, in the context of this work, the apparent fractal conjecture is correct.