F. Sylos Labini

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.

Fractal Correlations in the CfA2-South Redshift Survey

We report our analysis of the properties of galaxy clustering for a new redshift sample of galaxies, the CfA2-South catalog, using statistical methods which do not rely on the assumption of homogeneity. We find that, up to ~ 20 Mpc/h, which is the largest scale to which correlation properties can be reliably inferred, the galaxy clustering is scale-invariant and characterized by a fractal dimension D=1.9 pm 0.1. Further there is no statistical evidence for homogeneity at any of the larger scales (up to ~150 Mpc/h) probed more weakly by the catalog. These results means that characteristic ``correlation lengths for the clustering of galaxies derived using standards methods of analysis are not meaningful. Further the results are very consistent with those obtained from many other catalogs using the methods adopted here, which show the D =2 fractal continuing to beyond 100 Mpc/h. The incompleteness of the relevant data conjectured by various authors to give rise to such behaviour is therefore proved to have no significant effect (up to 20 Mpc/h) on the measured correlations.

On the problem of initial conditions in cosmological n-body simulations

Cosmological N-body simulations aim to calculate the non-linear gravitational growth of structures via particle dynamics. A crucial problem concerns the setting-up of the initial particle distribution, as standard theories of galaxy formation predict the properties of the initial continuous density field with small-amplitude correlated Gaussian fluctuations. The discretisation of such a field is a complex issue and particle fluctuations are especially relevant at small scales where non-linear dynamics firstly takes place. In general, most of the procedures which may discretise a continuous field give rise to Poisson noise, which would then dominate the non-linear small-scale dynamics due to nearest-neighbours interactions. In order to avoid such a noise, and to consider the dynamics as due only to large-scale (smooth) fluctuations, an ad hoc method (lattice or glassy system plus correlated displacements) has been introduced and used in cosmological simulations. We show that such a method gives rise to a particle distribution which does not have any of the correlation properties of the theoretical continuous density field. This is because discreteness effects, different from Poisson noise but nevertheless very important, determine particle fluctuations at any scale, making it completely different from the original continuous field. We conclude that discreteness effects play a central role in the non-linear evolution of N-body simulations.

Galaxy number counts and fractal correlations

We report the correlation analysis of various redshift surveys which shows that the available data are consistent with each other and manifest fractal correlations (with dimension D 2) up to the present observational limits ( ≈ 150 h−1Mpc) without any tendency towards homogenization. This result points to a new interpretation of the number counts that represents the main subject of this letter. We show that an analysis of the small-scale fluctuations allows us to reconcile the correlation analysis and the number counts in a new perspective which has a number of important implications.

Angular Correlations of Galaxy Distribution

We study the angular correlations of various galaxy catalogs (the CfA1, SSRS1, Perseus-Pisces, APM Bright Galaxies, and Zwicky catalogs). We find that the angular correlation exponent is γa = 0.1 ± 0.1 rather than γa = 0.7, as usually found by the standard correlation function ω(θ). We identify the problem in the artificial decay of ω(θ). Moreover, we find that no characteristic angular scale is present in any of the analyzed catalogs. Finally, we show that all the available data are consistent with each other, and the angular distribution of galaxies is quite naturally compatible with a fractal structure with D ≈ 2.

Statistical properties of galaxy cluster distribution

We analyze subsamples of Abell and ACO cluster catalogs, in order to study the spatial properties of the large-scale matter distribution. The subsamples analyzed are estimated to be nearly complete and are the standard ones used in the correlation analysis by various authors. The statistical analysis of cluster correlations is performed without any assumptions on the nature of the distribution itself. The cluster samples show fractal correlations up to sample limits (≈70h−1 Mpc) with fractal dimension D ≈ 2, without any tendency towards homogenization. Our analysis shows that the standard correlation methods are incorrect, since they give a finite correlation length for a distribution that does not possess one. In particular, the so-called correlation length r0 is shown to be simply a fraction of the sample size. Moreover we conclude that galaxies and clusters are two different representations of the same self-similar structure and that the correlations of clusters are the continuation of those of galaxies to larger scales.

Cosmological principle and the debate about large scale structures distribution

The basic hypothesis of a post-Copernican Cosmological theory is that {em all the points} of the Universe have to be essentially equivalent: this hypothesis is required in order to avoid any privileged {em observer}. This assumption has been implemented by Einstein in the so-called Cosmological Principles (CP): {em all the positions} in the Universe have to be essentially equivalent, so that the Universe is homogeneous. This situation implies also the condition of spherical symmetry about every point, so that the Universe in also Isotropic. There is a hidden assumption in the formulation of the CP in regard to the hypothesis that all the points are equivalent In fact, the condition that all the occupied points are statistically equivalent with respect to their environment correspond to the property of of Local Isotropy. It is generally believed that the Universe cannot be isotropic about every point without being also homogeneous [1]. Actually Local Isotropy does not necessarily implies homogeneity; in fact a topology theorem states that homogeneity is implied by the condition of local isotropy together with {em the assumption of the analyticity or regularity} for the distribution of matter. Up to the seventies analyticity was an obvious implicit assumption in any physical problem. Recently however we have learned about intrinsically irregular structures for which analyticity should be considered as a property to be tested with appropriate analysis of experiment.

Correlation properties of the large scale matter distribution and galaxy number counts

We introduce the basic techniques used for the analysis of three dimensional and two dimensional galaxy samples. We report the correlation analysis of various red-shift surveys which shows that the available data are consistent with each other and manifest fractal correlations (with dimension D≃ 2) up to the present observational limits without any tendency towards homogenization. This result points to a new interpretation of the galaxy number counts. We show that an analysis of the small scale fluctuations allows us to reconcile the correlation analysis and the number counts in a new perspective which has a number of important implications.

Introduction to Correlation Properties of Galaxy Distribution

Statistical analysis of spatial galaxy distribution is usually performed through the two point function ξ(r). This analysis allows one to determine a correlation length r 0 (ξ(r 0) = 1), which separates a correlated regime (r r 0). Some years ago we criticized this approach and proposed a new one based on the concepts and methods of modern Statistical Physics. Here we present an introduction to these methods and report the results of the analysis to all the available three dimensional catalogues of galaxies and clusters, ie CfA, Perseus-Pisces, SSRS, IRAS, Stromlo-APM, LEDA, Las Campanas and ESP for galaxies and Abell and ACO for clusters. All the data analyzed are consistent with each other and show fractal correlations (with dimension D ≃ 2) up to the deepest scales probed until now (1000h –1Mpc) and even more as indicated from the new interpretation of the number counts. The very first consequence of this result is that the usual statistical methods (as for example ξ(r)), based on the assumption of homogeneity, are therefore inconsistent for all the length scales probed until now. In the range of self-similarity theories should shift from “amplitudes” to “exponents”. 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.