A Fortran Code for Null Geodesic Solutions in the Lemaitre-Tolman-Bondi Spacetime

This paper describes the Fortran 77 code SIMU, version 1.1, designed for numerical simulations of observational relations along the past null geodesic in the Lemaitre-Tolman-Bondi (LTB) spacetime. SIMU aims at finding scale invariant solutions of the average density, but due to its full modularity it can be easily adapted to any application which requires LTB's null geodesic solutions. In version 1.1 the numerical output can be read by the GNUPLOT plotting package to produce a fully graphical output, although other plotting routines can be easily adapted. Details of the code's subroutines are discussed, and an example of its output is shown.Comment: 13 pages, 10 figures, LaTeX. Fortran code included with the LaTeX source code (also available at http://www.if.ufrj.br/~mbr/codes). Accepted for publication in "Computer Physics Communications

Observations in the Einstein--de Sitter Cosmology: Dust Statistics and Limits of Apparent Homogeneity

The two-point correlation function for the dust distribution in the unperturbed Einstein-de Sitter cosmological model is studied along the past light cone. It was found that this function seems unable to represent the theoretical distribution of dust along the backward null cone of this unperturbed model, which has already been determined in a previous paper as being apparently inhomogeneous at ranges usually considered as local. Such result was revisited in order to determine more precisely the quantitative limits where, in theory, we can detect apparent homogeneity, and it was found that this may only happen up to $z \sim 10^{-2}$. A different statistical analysis proposed by Pietronero is used, and it appears to be able to represent more accurately the theoretical distribution of dust in this cosmology. In the light of these results, it is argued that the usual practice of disregarding relativistic effects in studies of distribution of galaxies, by considering them as being placed on local regions, seems to be valid only on much closer scales than it is commonly believed. In the Einstein-de Sitter cosmology with $H_0=75 km s^{-1} Mpc^{-1}$, that may only happen in redshifts as low as $z \approx 0.04$, which means that the local approximation seems to be valid up to zeroth order of approximation only. As at present there are many redshift surveys which have already probed at deeper ranges, it seems that in order to compare the Friedmann models with observations we have to be very careful when ignoring the past light cone problem in observational cosmology, either in theoretical calculations or in data analysis, due to relativistic effects which produce observable inhomogeneity even in spatially homogeneous cosmological models.Comment: standard LaTeX; 23 pages, 2 tables, 3 LaTeX figures; paper published in 1995 is archived her

Cosmological Distances and Fractal Statistics of Galaxy Distribution

This paper studies the effect of the distance choice in radial (non-average) statistical tools used for fractal characterization of galaxy distribution. After reviewing the basics of measuring distances of cosmological sources, various distance definitions are used to calculate the differential density $\gamma$ and the integral differential density $\gamma^\ast$} of the dust distribution in the Einstein-de Sitter cosmology. The main results are as follows: (1) the choice of distance plays a crucial role in determining the scale where relativistic corrections must be taken into account, as both $\gamma$ and $\gamma^\ast$ are strongly affected by such a choice; (2) inappropriate distance choices may lead to failure to find evidence of a galaxy fractal structure when one calculates those quantities, even if such a structure does occur in the galaxy distribution; (3) the comoving distance and the distance given by Mattig's formula are unsuitable to probe for a possible fractal pattern as they render $\gamma$ and $\gamma^\ast$ constant for all redshifts; (4) a possible galaxy fractal system at scales larger than 100Mpc (z \~ 0.03) may only be found if those statistics are calculated with the luminosity or redshift distances, as they are the ones where $\gamma$ and $\gamma^\ast$ decrease at higher redshifts; (5) C\'el\'erier and Thieberger's (2001) critique of Ribeiro's (1995: astro-ph/9910145) earlier study are rendered impaired as their objections were based on misconceptions regarding relativistic distance definitions.Comment: 14 pages, 4 figures, A&A LaTeX macro. Minor linguistic changes to match the version sent to the publisher. Accepted for publication in "Astronomy and Astrophysics

The Apparent Fractal Conjecture

This short communication advances the hypothesis that the observed fractal structure of large-scale distribution of galaxies is due to a geometrical effect, which arises when observational quantities relevant for the characterization of a cosmological fractal structure are calculated along the past light cone. If this hypothesis proves, even partially, correct, most, if not all, objections raised against fractals in cosmology may be solved. For instance, under this view the standard cosmology has zero average density, as predicted by an infinite fractal structure, with, at the same time, the cosmological principle remaining valid. The theoretical results which suggest this conjecture are reviewed, as well as possible ways of checking its validity.Comment: 6 pages, LaTeX. Text unchanged. Two references corrected. Contributed paper presented at the "South Africa Relativistic Cosmology Conference in Honour of George F. R. Ellis 60th Birthday"; University of Cape Town, February 1-5, 199

Scale Invariance in a Perturbed Einstein-de Sitter Cosmology

This paper seeks to check the validity of the "apparent fractal conjecture" (Ribeiro 2001ab: gr-qc/9909093, astro-ph/0104181), 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. Implementing these conditions 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 observationally inhomogeneous, even in the spatially homogeneous spacetime of standard cosmology, change which was already analysed by Ribeiro (1992b, 1993, 1994, 1995: astro-ph/9910145) 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: astro-ph/9810146), 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.Comment: 18 pages, 1 figure, LaTeX. Final version (small changes in the figure plus some references update). Fortran code included with the LaTeX source. To be published in "Fractals

Spatial and observational homogeneities of the galaxy distribution in standard cosmologies

This work discusses the possible empirical verification of the geometrical concept of homogeneity of the standard relativistic cosmology considering its various definitions of distance. We study the physical consequences of the distinction between the usual concept of spatial homogeneity (SH), as defined by the Cosmological Principle, and the concept of observational homogeneity (OH), arguing that OH is in principle falsifiable by means of astronomical observations, whereas verifying SH is only possible indirectly. Simulated counts of cosmological sources are produced by means of a generalized number-distance expression that can be specialized to produce either the counts of the Einstein-de Sitter (EdS) cosmology, which has SH by construction, or other types of counts, which do, or do not, have OH by construction. Expressions for observational volumes and differential densities are derived with the various cosmological distance definitions in the EdS model. Simulated counts that have OH by construction do not always exhibit SH features. The reverse situation is also true. Besides, simulated counts with no OH features at low redshift start showing OH characteristics at high redshift. The comoving distance seems to be the only distance definition where both SH and OH appear simultaneously. The results show that observations indicating possible lack of OH do not necessarily falsify the standard Friedmannian cosmology, meaning that this cosmology will not necessarily always produce observable homogeneous densities. The general conclusion is that the use of different cosmological distances in the characterization of the galaxy distribution lead to significant ambiguities in reaching conclusions about the behavior of the large-scale galaxy distribution in the Universe.Comment: 12 pages, 12 figures, LaTeX. Matches the final version sent to the journal. Accepted for publication in "Astronomy and Astrophysics