On Modelling a Relativistic Hierarchical (Fractal) Cosmology by Tolman's Spacetime. III. Numerical Results

Abstract

This paper presents numerical solutions of particular Lemaitre-Tolman models approximating a fractal behaviour along the past light cone, as discussed in paper I (0807.0866) of this series. The initial conditions of the numerical problem are discussed and the algorithm used to carry out the numerical integrations is presented. It was found that the numerical solutions are stiff across the flat-curved interface necessary to obtain the initial conditions of the problem. The spatially homogeneous Friedmann models are treated as special cases of the Lemaitre-Tolman solution and solved numerically. Extending the results of paper II (0807.0869) on the Einstein-de Sitter model, to the $K = \pm 1$ models, it was found that the open and closed Friedmann models also do not appear to remain homogeneous along the backward null cone, with a vanishing volume (average) density as one approaches the Big Bang singularity hypersurface. Fractal solutions, that is, solutions representing an averaged and smoothed-out single fractal, were obtained in all three classes of the Lemaitre-Tolman metric, but only the hyperbolic ones were found to be in agreement with observations, meaning that a possible Friedmann background universe would have to be an open one. The best fractal metric obtained through numerical simulations is also analysed in terms of evolution, homothetic self-similarity, comparison with the respective spatially homogeneous case and the fitting problem in cosmology. The paper finishes with a discussion on some objections raised by some authors against a fractal cosmology.