The distribution of pairwise peculiar velocities in the non-linear regime

The distribution of pairwise, relative peculiar velocities, f (u;r), on small nonlinear scales, r, is derived from the Press--Schechter approach. This derivation assumes that Press--Schechter clumps are virialized and isothermal. The virialized assumption requires that the circular velocity, Vc \ M^1/3, where M denotes the mass of the clump. The isothermal assumption means that the circular velocity is independent of radius. Further, it is assumed that the velocity distribution within a clump is Maxwellian, that the pairwise relative velocity distribution is isotropic, and that on nonlinear scales clump-clump motions are unimportant when calculating the distribution of velocity differences. Comparison with N-body simulations shows that, on small nonlinear scales, all these assumptions are accurate. For most power spectra of interest, the resulting line of sight, pairwise, relative velocity distribution, f (u\ ₑ), is well approximated by an exponential, rather than a Gaussian distribution. This simple Press--Schechter model is also able to provide a natural explanation for the observed, non-Maxwellian shape of f (v), the distribution of peculiar velocities.