Persistent fluctuations and scaling properties in galaxy number counts
  Counts of galaxies as a function of apparent magnitude are among the most time-honored observations in cosmology. In this Letter, we focus on some statistical properties of these counts which are fundamental in order to characterize the large scale correlations in the galaxy spatial distribution. There are, in fact, no longer doubts since two decades of the existence of very large scale structures. The still remaining problem concerns the correct characterization of their statistical properties. We propose to study two properties of galaxy counts data, in order to discriminate between a small scale (~ 5 - 20 Mpc/h) homogeneous distribution, and a fractal structure on large scales (~ 20 - 300 Mpc/h). Firstly, the average slope of the counts which can be associated to an eventual fractal dimension in real space by simple arguments. Secondly, we propose to study fluctuations of counts around the average behavior as a function of apparent magnitude in the whole magnitude range B_j > 11. These fluctuations can discriminate between a genuine fractal distribution and a homogeneous one. In fact, they are related to the very statistical properties of the spatial distribution, independently on cosmological corrections. More specifically, in a fractal distribution one expects to find persistent scale-invariant fluctuations around the average behavior, which do not decay with apparent magnitude. On the other hand, in an homogeneous distribution, on large enough scales, the relative variance of the counts should decrease exponentially with apparent magnitude. Smooth cosmological corrections cannot change such a behavior. 
Francesco Sylos Labini and Andrea Gabrielli