Finite size effects on the galaxy number counts: evidence for fractal behavior up to the deepest scale

We introduce and study two new concepts which are essential for the quantitative analysis of the statistical quality of the available galaxy samples. These are the dilution effect and the small scale fluctuations. We show that the various data that are considered as pointing to a homogenous distribution are all affected by these spurious effects and their interpretation should be completely changed. In particular, we show that finite size effects strongly affect the determination of the galaxy number counts, namely the number versus magnitude relation ($N(<m)$) as computed from the origin. When one computes $N(<m)$ averaged over all the points of a redshift survey one observes an exponent $\alpha = D/5 \approx 0.4$ compatible with the fractal dimension $D \approx 2$ derived from the full correlation analysis. Instead the observation of an exponent $\alpha \approx 0.6$ at relatively small scales, where the distribution is certainly not homogeneous, is shown to be related to finite size effects. We conclude therefore that the observed counts correspond to a fractal distribution with dimension $D \approx 2$ in the entire range $12 \ltapprox m \ltapprox 28$, that is to say the largest scales ever probed for luminous matter. In addition our results permit to clarify various problems of the angular catalogs, and to show their compatibility with the fractal behavior. We consider also the distribution of Radio-galaxies, Quasars and $\gamma$ ray burst, and we show their compatibility with a fractal structure with $D \approx 1.6 \div 1.8$. Finally we have established a quantitative criterion that allows us to define and {\em predict} the statistical validity of a galaxy catalog (angular or three dimensional).