On the Fractal Structure of the Visible Universe
  Some years ago we proposed a new approach for the analysis of galaxy and cluster correlations based on the concepts and methods of modern statistical physics. This led to the surprising result that galaxy correlations are fractal and not homogeneous up to the limits of the available catalogues. In the meantime many more redshifts have been measured and we have extended our methods also to the analysis of number counts and angular catalogues. This has led to a complete analysis of all the available data that we are going to present in detail in this lecture. In particular we will discuss the properties of the following catalogues: CfA, Perseus-Pisces, SSRS, IRAS, LEDA, Las Campanas and ESP for galaxies and Abell and ACO for clusters. The usual statistical methods, that are based on the assumption of homogeneity, are therefore inconsistent for all the length scales probed until now. A new, more general, conceptual framework is necessary to identify the real physical properties of these structures. In the range of self-similarity theories should shift from "amplitudes" to "exponents". The new analysis shows that all the available data are consistent with each other and show fractal correlations (with dimension $D \simeq 2$) up to the deepest scales probed until now ($1000 h^{-1}Mpc$) and even more as indicated from the new interpretation of the number counts. The distribution of luminous matter in the universe is therefore fractal and not homogeneous. The evidence for this being very strong up to $150 h^{-1}Mpc$ due to the statistical roboustness of the data and progressively weaker (statistically) at larger distances due to the limited data. These facts lead to fascinating conceptual implications about our knowledge of the universe and to a new scenario for the theoretical challenge in this field. 
L. Pietronero, M. Montuori and F. Sylos Labini