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

Abstract

This paper presents numerical solutions of particular Tolman models approximating a fractal behaviour along the past light cone. 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 Tolman solution and solved numerically. Extending the results of paper II on the Einstein-de Sitter model, to the K = ±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

Scale-invariance of galaxy clustering

“…Spherical symmetrical inhomogeneity has been used [207,208] to represent isotropic fractal, with the conclusion that there is a strong deflection from linear Hubble law inside a fractal inhomogeneity cell. The Lemaitre-TolmanBondi (LTB) solution of Einstein equations was utilized [209] for modeling a relativistic fractal Cosmology, and it was concluded that it is difficult to get a linear redshift-distance law.…”

Probing the cosmic distance-duality relation with the Sunyaev-Zel’dovich effect, X-ray observations and supernovae Ia

“…Therefore, the DD relation is valid for both homogeneous and inhomogeneous cosmological models, requiring only that source and observer are connected by null geodesics in a Riemannian spacetime and that the number of photons is conserved. The DD relation plays an essential role in modern cosmology, ranging from gravitational lensing studies (Schneider et al 1992) to analyses of galaxy distribution and galaxy clusters observations (Lima et al 2003;Cunha et al 2007;Rangel Lemos & Ribeiro 2008;Ribeiro 1992Ribeiro , 1993Ribeiro , 2005Ribeiro & Stoeger 2003;Albani et al 2007;Mantz et al 2010;Komatsu et al 2011), as well as the plethora of cosmic consequences from primary and secondary temperature anisotropies of the cosmic microwave blackbody radiation (CMBR) observations (Komatsu et al 2011). Other consequences of Etherington's reciprocity relation are the temperature shift equation T o = T e /(1 + z), where T o is the observed temperature and T e is the emitted temperature, a key result for analyzing CMBR observations, as well as the optical theorem that surface brightness of an extended source does not depend on the angular diameter distance of the observer from the source, an important result for understanding lensing brightness (Ellis 2007).…”

Scale Invariance in a Perturbed Einstein-De Sitter Cosmology

“…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 (1992bRibeiro ( , 1993Ribeiro ( , 1994Ribeiro ( , 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 , where the scale factor is expanded in power series to yield perturbative terms.…”

“…Therefore, f 2 → 1, R 2 /f 2 → constant, as r → 0. In other words, we are requiring that metric (1) should obey the central regularity condition (Bonnor 1974;Ribeiro 1993;Humphreys, Matravers and Marteens 1998),…”

“…That was fully explained in Ribeiro (1993Ribeiro ( , 1994Ribeiro ( , 1995, and it implies that if one is simply doing calculations along the null cone it is most probable that one will find no scaling pattern of any kind at all. 3 Thus, to even start considering scale invariance in relativistic cosmology it was necessary to adapt the original analytical tools proposed by Pietronero (1987) into a relativistic framework, and when doing this it became clear that the chosen set of observational quantities had to have their behaviour studied along the past null cone (Ribeiro 1992a).…”