Self‐Similarity and the Pair Velocity Dispersion

We have considered linear two point correlations of the form $1/{x^{\gamma}}$which are known to have a self-similar behaviour in a $\Omega=1$ universe. Weinvestigate under what conditions the non-linear corrections, calculated usingthe Zel'dovich approximation, have the same self-similar behaviour. We findthat the scaling properties of the non-linear corrections are decided by thespatial behaviour of the linear pair velocity dispersion and it is only for thecases where this quantity keeps on increasing as a power law (i.e. for $\gamma< 2$) do the non-linear corrections have the same self-similar behaviour as thelinear correlations. For $(\gamma > 2)$ we find that the pair velocitydispersion reaches a constant value and the self-similarity is broken by thenon-linear corrections. We find that the scaling properties calculated usingthe Zel'dovich approximation are very similar to those obtained at the lowestorder of non-linearity in gravitational dynamics and we propose that thescaling properties of the non-linear corrections in perturbative gravitationaldynamics also are decided by the spatial behaviour of the linear pair velocitydispersion.Comment: 13 pages Latex file with 1 PS figure, To be published in Ap