Fractal Dimension as a measure of the scale of Homogeneity

TL;DR

This paper proposes a new method to estimate the scale of homogeneity in large-scale matter distribution using fractal dimensions, and applies it to the LCDM cosmological model, estimating an upper limit near 260 Mpc/h.

Contribution

It introduces a dispersion-based definition of the homogeneity scale and relates it to the correlation function, providing a robust estimate applicable to galaxy surveys.

Findings

Estimated upper limit of homogeneity scale: ~260 Mpc/h for LCDM.

The homogeneity scale is weakly dependent on the tracer used.

The scale remains consistent across different epochs when non-linear effects are negligible.

Abstract

In the multi-fractal analysis of large scale matter distribution, the scale of transition to homogeneity is defined as the scale above which the fractal dimension of underlying point distribution is equal to the ambient dimension of the space in which points are distributed. With finite sized weakly clustered distribution of tracers obtained from galaxy redshift surveys it is difficult to achieve this equality. Recently we have defined the scale of homogeneity to be the scale above which the deviation of fractal dimension from the ambient dimension becomes smaller than the statistical dispersion. In this paper we use the relation between the fractal dimensions and the correlation function to compute the dispersion for any given model in the limit of weak clustering amplitude. We compare the deviation and dispersion for the LCDM model and discuss the implication of this comparison for the expected scale of homogeneity in the concordant model of cosmology. We estimate the upper limit to the scale of homogeneity to be close to 260 Mpc/h for the LCDM model. Actual estimates of the scale of homogeneity should be smaller than this as we have considered only statistical contribution to the dispersion in fractal dimension and we have ignored cosmic variance and contributions due to survey geometry and the selection function. We find that as long as non linear correction are insignificant, scale of homogeneity as defined above does not change with epoch. The scale of homogeneity depends very weakly on the choice of tracer of the density field. Thus the suggested definition of the scale of homogeneity is fairly robust.