Fractal Dimension as a measure of the scale of Homogeneity
  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.
