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.