M. Montuori

Scale-invariance of galaxy clustering

Abstract 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 identify the 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. In particular we discuss the properties of the following catalogs: CfA, Perseus-Pisces, SSRS, IRAS, LEDA, APM-Stromlo, Las Campanas and ESP for galaxies and Abell and ACO for galaxy clusters. 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 ∼- 2) up to the deepest scales probed so far (1000 h −1 Mpc) 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 h −1 Mpc due to the statistical robustness of the data but becomes progressively weaker (statistically) at larger distances due to the limited data. In addition, the luminosity distribution is correlated with the space distribution in a specific way. 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.

Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey

We discuss some of the basic implications of recent results on galaxy correlations published by the SDSS collaboration. In particular we focus on the evidence which has been recently presented for the scale and nature of the transition to homogeneity in the galaxy distribution, and results which describe the dependence of clustering on luminosity. The two questions are in fact strictly entangled, as the stability of the measure of the amplitude of the correlation function depends on the scale at which the mean density becomes well defined. We note that the recent results which indicate the convergence to well defined homogeneity in a volume equivalent to that of a sphere of radius 70 Mpc/h, place in doubt previous detections of luminosity bias from measures of the amplitude of the correlation function. We emphasize that the way to resolve these issues is to first use, in volume limited samples corresponding to different ranges of luminosity, the unnormalized two point statistics to establish the scale (and value) at which the mean density becomes well defined. We note also that the recent SDSS results for these statistics are in good agreement with those obtained by us through analyses of many previous samples, confirming in particular that the galaxy distribution is well described by a fractal dimension D 2 up to a scale of at least 20 Mpc/h. We discuss critically the agreement of this new data with current theoretical models.

Facts and ideas in modern cosmology

Abstract A review of the principles of observational testing of cosmological theories is given with a special emphasis on the distinction between observational facts and theoretical hypotheses. A classification of modern cosmological theories and possible observational tests for these theories is presented. The main rival cosmological models are analyzed from the point of view of observational testing of their initial hypothesis. A comparison of modern observational data with theoretical predictions is presented. In particular we discuss in detail the validity of the two basic assumptions of modern cosmology that are the Cosmological Principle and the Expanding Space Paradigm. It is found that classical paradigms need to be reanalyzed and that it is necessary to develop crucial cosmological tests to discriminate alternative theories.

Percolation in real wildfires

This paper focuses on the statistical properties of wild-land fires and, in particular, investigates if spread dynamics relates to simple invasion model. The fractal dimension and lacunarity of three fire scars classified from satellite imagery are analysed. Results indicate that the burned clusters behave similarly to percolation clusters on boundaries and look denser in their core. We show that Dynamical Percolation reproduces this behaviour and can help to describe the fire evolution. By mapping fire dynamics onto the percolation models, the strategies for fire control might be improved.

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 α = D/5 ≈ 0.4 compatible with the fractal dimension D ≈ 2 derived from the full correlation analysis. Instead the observation of an exponent α ≈ 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 ≈ 2 in the entire range 12 ⪅ m ⪅ 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 behaviour. We consider also the distribution of Radio-galaxies, Quasars and γ-ray bursts, and we show their compatibility with a fractal structure with D ≈ 1.6–1.8. Finally we have established a quantitative criterion that allows us to define and predict the statistical validity of a galaxy catalog (angular or three-dimensional).

On the Fractal Structure of Galaxy Distribution and its Implications for Cosmology

Two fundamental empirical laws have been established in the analysis of galaxy space distribution. First, recent analyses have revealed that the three dimensional distribution of galaxies and clusters is characterized by large scale structures and huge voids: such a distribution shows fractal correlations up to the limits of the available samples. This has confirmed the earlier de Vaucouleurs power-law density - distance relation, now corresponding to a fractal structure with dimension

Topological approach to neural complexity.

D \approx 2

Angular projections of fractal sets

, at least, in the range of scales

Statistical Properties of the LEDA Redshift Database

\sim 1 ÷200 Mpc

Galaxy number counts and fractal correlations

(