Self‐Similarity and the Pair Velocity Dispersion

Abstract

We have considered linear two-point correlations of the form 1/xγ, which are known to have a self-similar behavior in a Ω = 1 universe. We investigate under what conditions the nonlinear corrections, calculated using the Zeldovich approximation, have the same self-similar behavior. We find that the scaling properties of the nonlinear corrections are decided by the spatial behavior of the linear pair velocity dispersion, and it is only for the cases where this quantity keeps on increasing as a power law (i.e., for γ < 2) that the nonlinear corrections have the same self-similar behavior as the linear correlations. For γ > 2 we find that the pair velocity dispersion reaches a constant value and the self-similarity is broken by the nonlinear corrections. We find that the scaling properties calculated using the Zeldovich approximation are very similar to those obtained at the lowest order of nonlinearity in gravitational dynamics, and we propose that the scaling properties of the nonlinear corrections in perturbative gravitational dynamics also are decided by the spatial behavior of the linear pair velocity dispersion.