Self-similarity and the Pair Velocity Dispersion

Abstract

We have considered linear two point correlations of the form $1/{x^γ}$ which are known to have a self-similar behaviour in a $Ω=1$ universe. We investigate under what conditions the non-linear corrections, calculated using the Zel&#39;dovich approximation, have the same self-similar behaviour. We find that the scaling properties of the non-linear corrections are decided by the spatial behaviour 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 $γ< 2$) do the non-linear corrections have the same self-similar behaviour 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 non-linear corrections. We find that the scaling properties calculated using the Zel&#39;dovich approximation are very similar to those obtained at the lowest order of non-linearity in gravitational dynamics and we propose that the scaling properties of the non-linear corrections in perturbative gravitational dynamics also are decided by the spatial behaviour of the linear pair velocity dispersion.