Perturbative Growth of Cosmological Clustering II: The Two Point Correlation

We use the BBGKY hierarchy equations to calculate, perturbatively, the lowest order nonlinear correction to the two point correlation and the pair velocity for Gaussian initial conditions in a critical density matter dominated cosmological model. We compare our results with the results obtained using the hydrodynamic equations which neglect pressure and we find that the two match, indicating that thare are no effects of multistreaming at this order of perturbation. We analytically study the effect of small scales on the large scales by calculating the nonlinear correction for a Dirac delta function initial two point correlation. We find that the induced two point correlation has a $x^{-6}$ behaviour at large separations. We have considered a class of initial conditions where the initial power spectrum at small $k$ has the form $k^n$ with $0 < n \le 3$ and have numerically calculated the nonlinear correction to the two point correlation, its average over a sphere and the pair velocity over a large dynamical range. We find that at small separations the effect of the nonlinear term is to enhance the clustering whereas at intermediate scales it can act to either increase or decrease the clustering. At large scales we find a simple formula which gives a very good fit for the nonlinear correction in terms of the initial function. This formula explicitly exhibits the influence of small scales on large scales and because of this coupling the perturbative treatment breaks down at large scales much before one would expect it to if the nonlinearity were local in real space. We physically interpret this formula in terms of a simple diffusion process. We have also investigated the case $n=0$ and we find that it differs from the other cases in certain respects. We investigate a recently proposed scaling property ofComment: 41 pages,(including 13 figure) uuencoded, to appear in Ap

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

We investigate whether it is possible to study perturbatively the transition in cosmological clustering between a single streamed flow to a multi streamed flow. We do this by considereing a system whose dynamics is governed by the Zel'dovich approximation (ZA) and calculating the evolution of the two point correlation function using two methods: 1.Distribution functions 2.Hydrodynamic equations without pressure and vorticity. The latter method breaks down once multistreaming occurs whereas the former does not. We find that the two methods give the same results to all orders in a perturbative expansion of the two point correlation function. We thus conclude that we cannot study the transition from a single stream flow to a multi-stream flow in a perturbative expansion. We expect this conclusion to hold even if we use the full gravitational dynamics (GD) instead of ZA. We use ZA to look at the evolution of the two point correlation function at large spatial separations and we find that until the onset of multi-streaming the evolution can be described by a diffusion process where the linear evolution at large scales gets modified by the rearrangement of matter on small scales. We compare these results with the lowest order nonlinear results from GD. We find that the difference is only in the numerical value of the diffusion coefficient and we interpret this physically. We also use ZA to study the induced three point correlation function. At the lowest order we find that, as in the case of GD, the three point correlation does not necessarily have the hierarchical form. We also find that at large separations the effect of the higher order terms for the three point correlatin function is very similar to that for the the two point correlation and in this case too the evolution can be be described in terms ofComment: 28 pages including 6 figures, Latex, Aastex macros, Accepted in Astrophysical Journa

Modeling non-linear effects in the redshift space two-point correlation function and its implications for the pairwise velocity dispersion

The anisotropies in the galaxy two-point correlation function measured from redshift surveys exhibits deviations from the predictions of the linear theory of redshift space distortion on scales as large 20 Mpc/h where we expect linear theory to hold in real space. Any attempt at analyzing the anisotropies in the redshift correlation function and determining the linear distortion parameter \beta requires these deviations to be correctly modeled and taken into account. These deviations are usually attributed to galaxy random motions and these are incorporated in the analysis through a phenomenological model where the linear redshift correlation is convolved with the random pairwise velocity distribution function along the line of sight. We show that a substantial part of the deviations arise from non-linear effects in the mapping from real to redshift space caused by the coherent flows. Models which incorporate this effect provide a better fit to N-body results as compared to the phenomenological model which has only the effect of random motions. We find that the pairwise velocity dispersion predicted by all the models that we have considered are in excess of the values determined directly from the N-body simulations. This indicates a shortcoming in our understanding of the statistical properties of peculiar velocities and their relation to redshift distortion.Comment: Minor Revisions, Accepted to MNRA