Scale-invariance of galaxy clustering

Some years ago we proposed a new approach to 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 catalogs. The usual statistical methods, which are based on the assumption of homogeneity, are therefore inconsistent for all the length scales probed so far, and a new, more general, conceptual framework is necessary to identifythe real physical properties of these structures. In the last few years the 3-d catalogs have been significatively improved and we have extended our methods to the analysis of number counts and angular catalogs. This has led to a complete analysis of all the available data that we present in this review. The result is that galaxy structures are highly irregular and self-similar: all the available data are consistent with each other and show fractal correlations (with dimension $D \simeq 2$) up to the deepest scales probed so far (1000 \hmp) and even more as indicated from the new interpretation of the number counts. The evidence for scale-invariance of galaxy clustering is very strong up to 150 \hmp due to the statistical robustness of the data but becomes progressively weaker (statistically) at larger distances due to the limited data. In 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.Comment: Latex file 165 pages, 106 postscript figures. This paper is also available at http://www.phys.uniroma1.it/DOCS/PIL/pil.html To appear in Physics Report (Dec. 1997

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).Comment: 42 Latex pages. Figures and macro are avaialable under request at [email protected]