Self-similarity and the Pair Velocity Dispersion

Abstract

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. We investigate under what conditions the non-linear corrections, calculated using the Zel'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 $\gamma <2$) do the non-linear corrections have the same self-similar behaviour as the linear correlations. For $(\gamma >2)$ 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'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.

13 pages Latex file with 1 PS figure, To be published in ApJ

References in corpus (1)

• The Evolution of Correlation Functions in the Zel'dovich Approximation and its Implications for the Validity of Perturbation Theory

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