X iv : g r - qc / 0 50 50 83 v 1 1 7 M ay 2 00 5 Multifractal Analysis of Packed Swiss Cheese Cosmologies

The multifractal spectrum of various three-dimensional representations of Packed Swiss Cheese cosmologies in open, closed, and flat spaces are measured, and it is determined that the curvature of the space does not alter the associated fractal structure. These results are compared to observational data and simulated models of large scale galaxy clustering, to assess the viability of the PSC as a candidate for such structure formation. It is found that the PSC dimension spectra do not match those of observation, and possible solutions to this discrepancy are offered, including accounting for potential luminosity biasing effects. Various random and uniform sets are also analyzed to provide insight into the meaning of the multifractal spectrum as it relates to the observed scaling behaviors. The notion of a hierarchically-structured world is a recurrent theme in our understanding of Nature [1]. According to the Cosmological Principle, the Universe must be homogeneous and isotropic. This oft cited-as-fact stipulation is the basis for the Friedmann-Robertson-Walker (FRW) solutions to Einstein's Field Equations, from which the expansion dynamics of the Universe are derived [2]. However, the breakdown of homogeneity (at least on smaller distance scales in the Universe) is quickly becoming an accepted ideal in cosmological circles. That is, while homogeneity requires matter to scale uniformly in space (i.e. D F = 3, equal probability scaling behavior in all spatial directions), actual measurements of the distributions suggest otherwise. The existence of fractally-clustered matter was further emphasized by Peebles (see [3]), whose various two-point correlation analyses of three-dimensional catalogs yielded the exponent γ ∼ 1.7, the co-dimension of which was taken to be the fractal dimension D F = 3 − γ ∼ 1.3. Some successive works confirmed this value, showing a study of the CfA redshift survey to agree statistically [4, 5]. However, these same studies cite model simulations with higher dimensionality, in particular D ∼ 2 1. Over the course of a decade, increasing evidence has been put forth to suggest that the dimensionality of a wide range of redshift survey catalogs yield such clustering dimensions. In the comprehensive publication [6], the results of fractal dimension analysis of numerous galactic catalogs are reviewed , with the general consensus that each data set reveals a unanimous D F ∼ 2 (see Table 1). Since each catalog is limited in size and spatial extent (i.e. volume), the associated dimensions can only be statistically viable up to some effective radius …