Multifractal Analysis of Cluster Distribution in Two Dimensions
Abstract
We perform a multifractal analysis on samples of galaxy clusters, obtained by cutting the Lick catalog at different values of the galaxy surface density, as well as on the Lick galaxy distribution. This kind of analysis is particularly relevant in order to properly test the scaling behavior of the cluster distribution and how it changes when peaks of different height are selected. We find that the highest selected peaks possess welldefined scaling properties up to the angular scale θ~6^deg^  7^deg^, corresponding to ~20 h^1^ Mpc at the characteristic depth of our samples. remarkably similar to the correlation length of rich galaxy systems. This finding suggests a relation between scaleinvariance of the cluster distribution and nonlinear clustering. The resulting spectrum of multifractal dimensions D_q_ indicates that the rich cluster distribution is essentially space filling in the underdense regions (D_q_~ 2 for q <~ 1), while the overdense regions (corresponding to q ~> 1) are characterized by a dimension D_q_~ 1. On the contrary, the distributions of clusters corresponding to lower peaks show no definite scaling properties. In fact, although the dimensionality of these distributions is always near to one at sufficiently small angular scales, they have a dimension around two at larger scales θ ~> 7^deg^ (thus indicating homogeneity) without evidence of scale range where the dimensions take a constant value. A similar result holds also for the distribution of Lick galaxies, which reaches homogeneity already at the scale θ~ 2.5^deg^. We suggest that the absence of scaling for the distributions of clusters identified as lower peaks and of galaxies could well be due to projection effects that homogenize the threedimensional clustering and, thus, are more severe in distorting the spatial scaling properties of less clustered structures. Our results indicate that welldefined scaling properties are associated in our projected samples only with the nonlinear clustering of high peaks and a corresponding dimension D ~ 1 is preferred. In order to properly test the effects of projection on the sky and of luminosity selection, we generate a threedimensional fractal structure with an a priori known multifractal spectrum. We then assign luminosities to the points of this set drawn from a Schechterlike luminosity function, and then project onto a sphere, to generate apparent magnitude limited samples. We find that the resulting multifractal spectrum for these synthetic angular samples turns out to be distorted with respect to the original (threedimensional) one. In particular, overdense regions are smoothed by projection, thus increasing D_q_ for q > 2, while underdense regions are filled, thus lowering the values of the negativeorder dimensions. Despite this variation of the multifractal dimensions, the threedimensional scaleinvariance is however preserved. This result suggests that the break seen in D_q_(θ) (for high values of q) of the highest peak distribution is real and not due to projection or luminosity selection effects. Thus, the presence of a characteristic scale in the angular distribution of rich clusters strongly argues against a pure fractal picture extending with the same dimension over all the considered scales.
 Publication:

The Astrophysical Journal
 Pub Date:
 February 1993
 DOI:
 10.1086/172255
 Bibcode:
 1993ApJ...404...21B
 Keywords:

 Fractals;
 Galactic Clusters;
 Two Dimensional Models;
 Universe;
 Astronomical Catalogs;
 Computational Astrophysics;
 Astrophysics;
 GALAXIES: CLUSTERING;
 COSMOLOGY: LARGESCALE STRUCTURE OF UNIVERSE;
 METHODS: NUMERICAL