Fractal dimension as a measure of the scale of homogeneity

Abstract

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 ( D q ) of the fractal dimension from the ambient dimension becomes smaller than the statistical dispersion of D q , i.e. σ D q . In this paper we use the relation between the fractal dimensions and the correlation function to compute σ D q for any given model in the limit of weak clustering amplitude. We compare D q and σ D q for the cold dark matter ( 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 CDM model. Actual estimates of the scale of homogeneity should be smaller than this as we have considered only the statistical contribution to σ D q and we have ignored cosmic variance and contributions due to survey geometry and the selection function. Errors arising due to these factors enhance σ D q and as 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 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.

Antineutrino Star Model of the Big Bang and Dark Energy

“…Thus, sufficient time has elapsed for the formation of large structures in the distant universe. This is qualitatively consistent with the existence of the Hercules-Corona Borealis Great Wall (Horváth et al 2014;Balazs et al 2015); its size of 7-10 Gly challenges the homogeneity assumption of the ΛCDM model, which predicts structures not larger than ∼ 1.21 Gly (Yadav et al 2010;Clowes et al 2013). Second, the Copernican Principle would suggest that we are not exactly at the center of the antineutrino star.…”

First-Order Cosmological Perturbations Engendered by Point-Like Masses

“…Therefore, the current value of λ is λ 0 ≈ 3700 Mpc ≈ 12 Gly. It is very interesting that this Yukawa interaction range and the sizes of the largest known cosmic structures (Clowes et al 2013;Horvath et al 2014;Balazs et al 2015) are of the same order, thereby hinting at the opportunity to resolve the formidable challenge lying in the fact that their sizes essentially exceed the previously reported epoch-independent scale of homogeneity ∼ 370 Mpc (Yadav et al 2010). The authors arrived at this underestimate by comparing the deviation of the fractal dimension, characterizing the distribution of matter, from 3 (dimensionality of space) to its statistical dispersion.…”

A giant ring-like structure at 0.78 < z < 0.86 displayed by GRBs

“…Bagla, Yadav & Seshadri (2008) showed that in the concordance model, the fractal dimension makes a rapid transition to values close to 3 at scales between 40 and 100 Mpc. Sarkar et al (2009) found the galaxy distribution to be homogeneous at length-scales greater than 70 h −1 Mpc, and Yadav, Bagla & Khandai (2010) estimated the upper limit to the scale of homogeneity as being close to 260 h −1 Mpc for the ΛCDM model. Söchting et al (2012) studied the Ultra Deep Catalogue of Galaxy Structures; the cluster catalogue contains 1780 structures covering the redshift range 0.2 < z < 3.0, spanning three orders of magnitude in luminosity (10 8 < L4 < 5 × 10 11 L⊙) and richness from eight to hundreds of galaxies.…”