The Evolution of Correlation Functions in the Zeldovich Approximation and Its Implications for the Validity of Perturbation Theory

We investigate whether it is possible to study perturbatively the transition in cosmological clustering from a single-streamed flow to a multistreamed flow. We do this by considering a system whose dynamics is governed by the Zeldovich approximation (ZA) and calculating the evolution of the two- point correlation function using two methods, (1) distribution functions and (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 yield 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-streamed flow to a multistreamed flow in a perturbative expansion. We expect this conclusion to hold even if full gravitational dynamics (GD) is used 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 multistreaming, the evolution can be described by a diffusion process in which the linear evolution at large scales is 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 of nonlinearity, 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 correlation function is very similar to that for the two-point correlation, and in this case too the evolution can be described in terms of a diffusion process.