Power law correlations in galaxy distribution and finite volume effects from the Sloan Digital Sky Survey Data Release Four

We discuss the estimation of galaxy correlation properties in several volume limited samples, in different sky regions, obtained from the Fourth Data Release of the Sloan Digital Sky Survey. The small scale properties are characterized through the determination of the nearest neighbor probability distribution. By using a very conservative statistical analysis, in the range of scales [0.5, ~30] Mpc/h we detect power-law correlations in the conditional density in redshift space, with an exponent $\gamma=1.0~\pm 0.1$. This behavior is stable in all the different samples we considered; thus it does not depend on galaxy luminosity. In the range of scales [~30, ~100] Mpc/h we find evidence for systematic unaveraged fluctuations and we discuss in detail the problems induced by finite volume effects on the determination of the conditional density. We conclude that in such a range of scales there is evidence for a smaller power-law index of the conditional density. However we cannot distinguish between two possibilities: (i) that a crossover to homogeneity (corresponding to $\gamma=0$ in the conditional density) occurs before 100 Mpc/h; (ii) that correlations extend to scales of order 100 Mpc/h (with a smaller exponent $0< \gamma <1$). We emphasize that galaxy distributions in these samples present large fluctuations at the largest scales probed, corresponding to the presence of large scale structures extending up to the boundaries of the present survey. We discuss several differences between the behavior of the conditional density in mock galaxy catalogs built from cosmological N-body simulations and real data. We discuss some theoretical implications of such differences considering also the super-homogeneous features of primordial density fields.