Testing the Homogeneity of Large-scale Structure with the SDSS Data

One of the basic assumptions in cosmology is that the universe is homogeneous and isotropic on large scales. This assumption is the most important keystone of modern cosmology. In order to verify the homogeneity of galaxy distribution on large scales, we have computed the fractal dimensionality of the galaxy distribution in SDSS-DR4. The fractal dimensionality of the observed spatial geometric bodies is determined with random samples. The redshifts of sample galaxies are in the range 0.01～0.26. When the scale grows continuously to dozens of Mpc, the fractal dimensionality of the galaxy distribution approaches to 3 consistently. All the 6 samples exhibit obviously a transition scale. For scales larger than the transition scale, the fractal dimensionality D_G of the galaxy distribution is very close to 3, the galaxy distribution is homogeneous. This result supports the assumption that the universe is homogeneous on large scales. The transition scale of the sample increases with the luminosity of the sample. This means that the galaxy distribution on small scales is not a of simple fractal distribution, but a multi-fractal distribution. The transition scale of high-luminosity galaxies is very large, the distribution will not become homogeneous till about 100 h^?1Mpc.