Generalized Statistical Models of Voids and Hierarchical Structure in Cosmology

Abstract

Generalized statistical models of voids and hierarchical structure in cosmology are developed. The often quoted negative binomial model and the frequently used thermodynamic model are shown to be special cases of a more general distribution that contains a parameter a. This parameter is related to the Lévy index α and the Fisher critical exponent τ, the latter of which describes the power-law falloff of clumps of matter around a phase transition. The parameter a, exponent τ, or index α can be obtained from properties of a void scaling function. A stochastic probability variable p is introduced into a statistical model, which represents the adhesive growth of galaxy structure. The galaxy count distribution decays exponentially quickly with size for p < 1/2. For p > 1/2, adhesive growth can go on indefinitely, thereby forming an infinite supercluster. At p = 1/2, a scale-free power-law distribution for the galaxy count distribution is present. The stochastic description also leads to consequences that have some parallels with cosmic string results, percolation theory, and phase transitions.