On Modelling a Relativistic Hierarchical (Fractal) Cosmology by Tolman's Spacetime. I. Theory

TL;DR

This paper develops a relativistic fractal cosmological model using Tolman's spacetime to describe large-scale inhomogeneities, integrating hierarchical clustering with a crossover to homogeneity, and providing a framework for observational comparison.

Contribution

It introduces a relativistic fractal cosmology model based on Tolman's solution, connecting hierarchical clustering with inhomogeneous spacetime and a transition to homogeneity.

Findings

Formulated a Tolman-based fractal cosmological model.

Derived observational relations for the model.

Proposed a numerical approach to find solutions exhibiting fractal behavior.

Abstract

This work examines a relativistic model for the observed inhomogeneities of the large scale structure where the hypothesis that this structure can be described as being a self-similar fractal system is advanced. The concept of hierarchical clustering is identified with a fractal distribution and the problems raised by the use of fractal ideas in a relativistic model are discussed, as well as their relations to the Copernican and Cosmological Principles. Voids, clusters and superclusters of galaxies are assumed to be part of a smoothed-out fractal structure described by a Lemaitre-Tolman solution. The basic concepts of the Newtonian model presented by Pietronero (1987) are reinterpreted and applied to this inhomogeneous curved spacetime. This fractal system is also assumed to have a crossover to homogeneity which leads to a "Swiss cheese" type model, composed by an interior Lemaitre-Tolman metric and an exterior dust Friedmann solution. The Darmois junction conditions between the two spacetimes are calculated, and the observational relations necessary to compare the model with observations are obtained for the interior region. The differential equations of the interior spacetime are set up and a numerical strategy is devised for finding particular Tolman solutions representing a fractal behaviour along the past light cone.