The assumption that the Universe, on sufficiently large scales, is homogeneous and isotropic is crucial to our current understanding of cosmology. In this paper we test if the observed galaxy distribution is actually homogeneous on large scales. We have carried out a multifractal analysis of the galaxy distribution in a volume limited subsample from the SDSS DR6. This considers the scaling properties of different moments of galaxy number counts in spheres of varying radius $r$ centered on galaxies. This analysis gives the spectrum of generalized dimension $D_q(r)$, where $q >0$ quantifies the scaling properties in overdense regions and $q<0$ in underdense regions. We expect $D_q(r)=3$ for a homogeneous, random point distribution. In our analysis we have determined $D_q(r)$ in the range $-4 \le q \le 4$ and $7 \le r \le 98 h^{-1} {\rm Mpc}$. In addition to the SDSS data we have analysed several random samples which are homogeneous by construction. Simulated galaxy samples generated from dark matter N-body simulations and the Millennium Run were also analysed. The SDSS data is considered to be homogeneous if the measured $D_q$ is consistent with that of the random samples. We find that the galaxy distribution becomes homogeneous at a length-scale between 60 and $70 h^{-1} {\rm Mpc}$. The galaxy distribution, we find, is homogeneous at length-scales greater than $70 h^{-1} {\rm Mpc}$. This is consistent with earlier works which find the transition to homogeneity at around $70 h^{-1} {\rm Mpc}$.Comment: 6 pages, 3 figures, Submitted for publicatio

The assumption that the Universe is homogeneous and isotropic on large scales is one of the fundamental postulates of cosmology. We have tested the large-scale homogeneity of the galaxy distribution in the Sloan Digital Sky Survey Data Release One (SDSS-DR1) using volumelimited subsamples extracted from the two equatorial strips that are nearly two-dimensional. The galaxy distribution was projected on the equatorial plane and we carried out a 2D multifractal analysis by counting the number of galaxies inside circles of different radii, r, in the range 5-150 h −1 Mpc centred on galaxies. Different moments of the count-in-cells were analysed to identify a range of length-scales (60-70 h −1 Mpc to 150 h −1 Mpc ), where the moments show a power-law scaling behaviour, and to determine the scaling exponent that gives the spectrum of generalized dimension D q . If the galaxy distribution is homogeneous, D q does not vary with q and is equal to the Euclidean dimension, which in our case is 2. We find that D q varies in the range 1.7-2.2. We also constructed mock data from random, homogeneous point distributions and from lambda cold dark matter ( CDM) N-body simulations with bias b = 1, 1.6 and 2, and analysed these in exactly the same way. The values of D q in the random distribution and the unbiased simulations show much smaller variations and these are not consistent with the actual data. The biased simulations, however, show larger variations in D q and these are consistent with both the random and the actual data. Interpreting the actual data as a realization of a biased CDM universe, we conclude that the galaxy distribution is homogeneous on scales larger than 60-70 h −1 Mpc.

We have tested for luminosity, colour and morphology dependence of the degree of filamentarity in seven nearly two‐dimensional strips from the Sloan Digital Sky Survey Data Release Four (SDSS DR4). The analysis was carried out at various levels of coarse graining allowing us to address different length‐scales. We find that the brighter galaxies have a less filamentary distribution than the fainter ones at all levels of coarse graining. The distribution of red galaxies and ellipticals shows a higher degree of filamentarity compared to blue galaxies and spirals, respectively, at low levels of coarse graining. The behaviour is reversed at higher levels of coarse graining. We propose a picture where the ellipticals are densely distributed in the vicinity of the nodes where the filaments intersect while the spirals are sparsely distributed along the entire extent of the filaments. Our findings indicate that the regions with an excess of ellipticals are larger than galaxy clusters, protruding into the filaments. We have also compared the predictions of a semi‐analytic model of galaxy formation (the Millennium Run galaxy catalogue) against our results for the SDSS. We find the two to be in agreement for the M* galaxies and for the red galaxies, while the model fails to correctly predict the filamentarity of the brighter galaxies and the blue galaxies.

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We propose a method for testing Cosmic homogeneity based on the Shannon entropy in Information theory and test the potentials and limitations of the method on Monte Carlo simulations of some homogeneous and inhomogeneous 3D point process in a finite region of space. We analyze a set of N-body simulations to investigate the prospect of determining the scale of homogeneity with the proposed method and show that the method could serve as an efficient tool for the study of homogeneity.