The distribution of pairwise peculiar velocities in thenon - linear

The distribution of pairwise relative peculiar velocities, f(u;r), on small non-linear 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 V c / 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 non-linear scales clump{clump motions are unimportant when calculating the distribution of velocity diierences. Comparison with N-body simulations shows that, on small non-linear scales, all these assumptions are accurate. For initially scale-invariant Gaussian density elds, the pairwise velocity distribution evolves in a self-similar manner. This is consistent with other analytic expectations, and with the distribution measured in relevant N-body simulations. For most power-spectra of interest, the resulting line of sight, pairwise relative velocity distribution, f(u r), 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.