The fractal structure of the universe: Correlations of galaxies and clusters and the average mass density

It is shown that the use of the function ζ(r) for the analysis of statistical correlations for galaxies and clusters is conceptually incorrect. This is due to the fact that ζ(r) provides a correct characterization of correlations only for systems for which an intrinsic and unique average density can be defined independently of the position and size of the sample volume considered. This is the case for liquids but it is manifestly not the case for the large scale structures in the universe. A reinterpretation of the data with the appropriate correlation functions naturally resolves a number of puzzling features of the previous interpretation, like, for example, the amplitude mismatch for ζ(r) of galaxies and clusters. The correct correlation analysis strongly supports the existence of a single fractal (self-similar) structure extending from the galaxy scale up to the present limits of observation. This picture is now fully consistent with the existence of clusters and voids of very large sizes as well as with large scale anomalous velocities. In addition the present observational data do not support the existence of a length-scale above which the distribution of matter becomes homogeneous. This fact poses serious doubts about the current determinations of “a fair sample” from which one deduces the average mass density and the cosmological density parameter Ω.