Linear and non-linear contributions to pairwise peculiar velocities

We write the correlation function of dark matter particles, ξ(r), as the sum of two terms -one which accounts for non-linear evolution, and dominates on small scales, and another which is essentially the term from linear theory, and dominates on large scales. We use models of the number and spatial distribution of haloes and halo density profiles to describe the non-linear term and its evolution. The result provides a good description of the evolution of ξ(r) in simulations. We then use this decomposition to provide simple and accurate models of how the single-particle velocity dispersion evolves with time, and how the first and second moments of the pairwise velocity distribution depend on scale. The key idea is to use the simple physics of linear theory on large scales, the simple physics of the virial theorem on small scales and our model for the correlation function to tell us how to weight the two types of contributions (linear and non-linear) to the pairwise velocity statistics. When incorporated into the streaming model, our results will allow a simple accurate description of redshift-space distortions over the entire range of linear to highly non-linear regimes.