Parabolic drift towards homogeneity in large-scale structures of galaxies

To describe the progressive transition in large-scale structures of galaxies from a seemingly fractal behavior at small scales to a homogeneous distribution at large scales, we use a new geometrical framework called entropic-skins geometry which is based on a diffusion equation of scale entropy through scale space. In the case of an equipartition of scale entropy losses in scale space, it is shown that fractal dimension (varying from 0 to 3) depends linearly on the logarithm of scale from the average size lc of galaxies until a characteristic length scale l0 beyond which distribution becomes homogeneous. A simple parabolic expression for correlation function can be derived: ln(1+ξi)=(β/2)ln2(lo/li) with β=3/ln(l0/lc)≈0.32 and l0≈55h−1Mpc. This law has been verified using correlation functions measured on several redshift surveys.