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 charac- terization of galaxy distribution. After reviewing the basics of measuring distances of cosmological sources, various distance definitions are used to calculate the differential density γ and the integral differential density γ ∗ 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 γ and γ ∗ are strongly affected by such a choice; (2) inap- propriate 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 for- mula are unsuitable to probe for a possible fractal pattern as they render γ and γ ∗ constant for all redshifts; (4) a possible galaxy fractal system at scales larger than 100 Mpc (z ≈ 0.03) may only be found if those statistics are calculated with the luminosity or redshift distances, as they are the ones where γ and γ ∗ decrease at higher redshifts; (5) Celerier & Thieberger's (2001) critique of Ribeiro's (1995) earlier study are rendered impaired as their objections were based on misconceptions regarding relativistic distance definitions.