Fractal dimension as a measure of the scale of homogeneity

In the multifractal analysis of the large-scale matter distribution, the scale of the transition to homogeneity is defined as the scale above which the fractal dimension (D-q) of the underlying point distribution is equal to the ambient dimension (D) of the space in which points are distributed. With the finite sized weakly clustered distribution of tracers obtained from galaxy redshift surveys it is difficult to achieve this equality. Recently Bagla et al. have defined the scale of homogeneity to be the scale above which the deviation (Delta D-q) of the fractal dimension from the ambient dimension becomes smaller than the statistical dispersion of Delta Dq, i.e. sigma(Delta Dq). In this paper we use the relation between the fractal dimensions and the correlation function to compute sigma(Delta Dq) for any given model in the limit of weak clustering amplitude. We compare Delta Dq and sigma(Delta Dq) for the Lambda cold dark matter (Lambda CDM) 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 h(-1) Mpc for the Lambda CDM model. Actual estimates of the scale of homogeneity should be smaller than this as we have considered only the statistical contribution to sigma(Delta Dq) and we have ignored cosmic variance and contributions due to survey geometry and the selection function. Errors arising due to these factors enhance sigma(Delta Dq) and as Delta D-q decreases with increasing scale, we expect to measure a smaller scale of homogeneity. We find that as long as non-linear corrections to the computation of Delta D-q are insignificant, the scale of homogeneity 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.