Fractal Dimension as a measure of the scale of Homogeneity Jaswant K. Yadav $^{1}$,$^{2}$, J. S. Bagla2 and Nishikanta Khandai 3 $^{1}$Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul 130722, South Korea $^{2}$Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, INDIA $^{3}$McWilliams Center for Cosmology, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. E-Mail: jaswant@kias.re.kr, jasjeet@hri.res.in, nkhandai@andrew.cmu.edu 29 October 2018 ABSTRACT In the multi-fractal analysis of large scale matter distribution, the scale of transition to ho- mogeneity is defined as the scale above which the fractal dime nsion ( D$_{q}$ ) of underlying point distribution is equal to the ambient dimension (D) of the space in which points are distributed. With finite sized weakly clustered distribution of tracers o btained from galaxy redshift sur- veys it is difficult to achieve this equality. Recently Baglaet al. (2008) have defined the scale of homogeneity to be the scale above which the deviation ( $^{Δ}$D$_{q}$ ) of fractal dimension from the ambient dimension becomes smaller than the statistical dispersion of $^{Δ}$D$_{q}$ , i.e., σ$_{Δ}$$_{D}$$_{q}$ . In this paper we use the relation between the fractal dimensions and the correlation function to compute σ$_{Δ}$$_{D}$$_{q}$ for any given model in the limit of weak clustering amplitude. We compare $^{Δ}$D$_{q}$ and σ$_{Δ}$$_{D}$$_{q}$ for the $^{Λ}$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 $^{Λ}$CDM model. Actual estimates of the scale of homogeneity should be smaller than this as we have considered only statistical contribution to σ$_{Δ}$$_{D}$$_{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$_{Δ}$$_{D}$$_{q}$ and as $^{Δ}$D$_{q}$ decreases with increasing scale, we expect to measure a smaller scale of homogeneity. We find that as long as non linear correction to t he computation of $^{Δ}$D$_{q}$ are insignificant, scale of homogeneity does not change with epo ch. The scale of homogeneity depends very weakly on the choice of tracer of the density fiel d. Thus the suggested definition of the scale of homogeneity is fairly robust. Key words: cosmology : theory, large scale structure of the universe — methods: statistical 1 INTRODUCTION using fractal dimensions over other analyses is that in the former we don’t require the assumption of an average density in the point set (Jones et al. 2004). One of the primary aims of galaxy redshift surveys is to determine the distribution of luminous matter in the Universe (Huchra et al. 1983; Lilly et al. 1995; Shectman et al. 1996; Huchra et al. 1999; York et al. 2000; Colless et al. 2001; Giavalisco et al. 2004; Le F`evre et al. 2005; Scoville et al. 2007). These surveys ha ve re- vealed a large variety of structures starting from groups and clus- ters of galaxies, extending to super-clusters and an interconnected network of filaments which appears to extend across very larg e scales (Bond, Kofman, & Pogosyan 1996; Springel et al. 2005; Faucher-Gigu`ere, Lidz, & Hernquist 2008). We expect the ga laxy distribution to be homogeneous on large scales. In fact, the assump- tion of large scale homogeneity and isotropy of the universe is the basis of most cosmological models (Einstein 1917). In addition to determining the large scale structures, the redshift surveys of galax- ies can also be used to verify whether the galaxy distribution does indeed become homogeneous at some scale(Einasto & Gramann 1993; Martinez et al. 1998; Yadav et al. 2005; Sylos Labini et al. 2009a,b). Fractal dimensions of the galaxy matter distribution can be used to test the conjecture of homogeneity. One advantage of When doing our analysis we often work with volume limited sub-samples extracted from the the full magnitude samples of the galaxies. This is done in order to avoid an explicit use of the selec- tion function. The volume limited sub-samples constructed in this manner from a flux limited parent sample naturally have a much smaller number of galaxies. This was found to be too restrictive for the earliest surveys and corrections for varying selection function were used explicitly in order to determine the scale of homogene- ity (Bharadwaj et al. 1999; Amendola & Palladino 1999). But with modern galaxy redshift surveys, this limitation is less severe. Making a volume limited sub-sample requires assumption of a cosmological model and this may be thought of as an undesirable feature of data analysis. However, if we directly use raw data and do not account for a redshift dependent selection function in any way, it is obvious that the selection function will dominate in any large scale description of the distribution of galaxies. This is only arXiv:1001.0617v2[astro-ph.CO]15 Mar 2010 Mon. Not. R. Astron. Soc. 000 , 000–000 (0000) Printed 29 October 2018 (MN L A T E X style file v2.2)