Relativistic Fractal Cosmologies
This article presents a review of an approach for constructing a simple relativistic fractal cosmology, whose main aim is to model the observed inhomogeneities of the distribution of galaxies by means of the Tolman solution of Einstein’s field equations for spherically symmetric dust in comoving coordinates. Such model is based on earlier works developed by L. Pietronero and J. R. Wertz on Newtonian cosmology, and the main points of these models are also discussed. Observational relations in Tolman’s spacetime are presented, together with a strategy for finding numerical solutions which approximate an averaged and smoothed out single fractal structure in the past light cone. Such fractal solutions are actually obtained and one of them is found to be in agreement with basic observational constraints, namely the linearity of the redshift-distance relation for z < 1, the decay of the average density with the distance as a power law (the de Vaucouleurs’ density power law), the fractal dimension within the range 1 ≤ D ≤ 2, and the present range of uncertainty for the Hubble constant. The spatially homogeneous Friedmann model is discussed as a special case of the Tolman solution, and it is found that once we apply the observational relations developed for the fractal model we find that all Friedmann models look inhomogeneous along the backward null cone, with a departure from the observable homogeneous region at relatively close ranges. It is also shown that with these same observational relations the Einstein-de Sitter model can have an interpretation where it has zero global density, a result consistent with the “zero global density postulate” advanced by Wertz for hierarchical cosmologies and conjectured by Pietronero for fractal cosmological models. The article ends with a brief discussion on the possible link between this model and nonlinear and chaotic dynamics.
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