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 nd that the two match, indicating that thare are no eeects of multistreaming at this order of perturbation. We analytically study the eeect of small scales on the large scales by calculating the nonlinear correction for a Dirac delta function initial two point correlation. We nd 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 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 nd that at small separations the eeect 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 nd a simple formula which gives a very good t for the nonlinear correction in terms of the initial function. This formula explicitly exhibits the innuence 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 diiusion process. We have also investigated the case n = 0 and we nd that it diiers from the other cases in certain respects. We investigate a recently proposed scaling property of gravitational clustering and we nd that the lowest order nonlinear terms cause deviations from the scaling relations which are strictly valid in the linear regime. The approximate validity of these relations in the nonlinear regime in N-body simulations cannot be understood at this order of evolution.