Fractal Approach to Large–Scale Galaxy Distribution

We present a review of the history and the present state of the fractal approach to the large-scale distribution of galaxies. The roots of the modern idea of cosmic fractality go to the beginning of the 20th century, when hierarchical world models were proposed by Fournier and Charlier. Fundamental aspects of hierarchical matter distribution was discussed by Einstein and Selety in the correspondence concerning inhomogeneous cosmological models. The Great Debate on the nature of spiral nebulae went over to a struggle on the structure and extension of the galaxy clustering. Already during the epoch of galaxy angular catalogues astronomers detected superclusters of galaxies on the sky. However, early counts of bright galaxies were close to the 0.6m-law, pointing to homogeneity. Also the debated variable extinction seemed to be a reasonable explanation of apparent galaxy clustering. Angular correlation function analysis gave small a homogeneity scale Rhom ≈ 10 Mpc within which the correlation exponent corresponds to a fractal dimension D ≈ 1.2 and outside which the galaxy distribution is homogeneous. It was realized later that a normalization condition for the reduced correlation function estimator results in distorted values for both Rhom and D. Moreover, according to a theorem on projections of fractals, galaxy angular catalogues can not be used for detecting a structure with the fractal dimension D ≥ 2. For this 3-d maps are required, and indeed modern extensive redshift-based 3-d maps have revealed the “hidden” fractal dimension of about 2, and have confirmed superclustering at scales even up to 500 Mpc (the Sloan Great Wall). On scales, where the fractal analysis is possible in completely embedded spheres, a power–law density field has been found. Two new fundamental cosmic numbers have appeared, the fractal dimension D and the crossover scale to homogeneity Rhom. Their values have been debated, and D = 1.2 indirectly deduced from angular catalogues has been replaced by D = 2.2 ± 0.2 directly obtained from 3-d maps and Rhom has expanded from 10 Mpc to scales approaching 100 Mpc. In concordance with the 3-d map results, modern all sky galaxy counts in the interval 10÷15 give a 0.44m-law which corresponds to D = 2.2 within a radius of 100h 100 Mpc. The narrow cones of the existing deep galaxy surveys and poorly known peculiar velocities at small scales are still the main limiting factors hampering precise estimates of D and Rhom. We emphasize that the fractal mass–radius law of galaxy clustering has become a key phenomenon in observational cosmology. It creates novel challenges for theoretical understanding of the origin and evolution of the galaxy distribution, including the role of dark matter and dark energy.