On the clustering of particles in an expanding Universe

We investigate the clustering of particles in Friedmann models of the Universe using 1000- and 20 000-body numerical simulations. The results of these computations are analysed in terms of the two- and three-point correlation functions, the mean relative peculiar velocity between particle pairs 〈v_{2 1}〉, and the mean square peculiar velocity dispersion between pairs $$\langle\upsilon_{2\enspace 1}^2\rangle$$. In the case of Einstein–de Sitter models we find that on scales corresponding to the transition region $$\xi\sim1,\enspace|\langle\upsilon_{2\enspace 1}\rangle|\gt Hr_{21}$$ and this results in a non-power law form for ξ(r), in rough agreement with simple analytic treatments based on the homogeneous spherical cluster models for the collapse of protoclusters. Our results are in conflict with the kinetic theory calculations of Davis & Peebles who studied the problem in the case of an Einstein–de Sitter Universe and found good agreement with observational data. These authors suggest that clusters develop substantial non-radial motions whilst they are still small density fluctuations, so that when a cluster fragments out of the general Hubble expansion, it is already virialized. This ‘previrialization’ effect does not appear to occur in the numerical models described here. We also examine the effects of particle discreteness and two-body relaxation, which are particularly important in the N-body models but neglected in the approach of Davis & Peebles. Because it is unclear as to whether these effects are important for galaxy clustering in the real Universe, it is difficult to assess the significance of our results. More observational and theoretical work is necessary in order to decide whether our approach is reasonable.