Pair correlations in an expanding universe for a multicomponent system

Fall and Saslaw have derived an equation for the growth of pair correlations in an expanding universe of identical self-gravitating point masses which is correlation-free at some initial time. Their equation is rigorously true for the earliest stages of growth, assuming only that the system is spatially homogeneous and isotropic, and that it is characterized in the ''comoving frame'' by a Maxwellian distribution of velocities. This paper generalizes their analysis to the case of a multicomponent system of particles with different masses, each species of which is characterized by a Maxwellian distribution at the same temperature. Here there are two types of pair correlations to consider, namely among members of the same species and among members of different species. The general behavior may be understood most readily by considering the covariance functions, which assume very simple forms. Thus one finds that the ''strength'' of the covariance scales, for sufficiently small radial separations, as the product of the masses, whereas the ''range'' of the covariance varies inversely as the square root of the reduced mass of the two constituents. This implies that, for two very different masses, the ''range'' will be set by the lighter constituent. Knowledge of the covariances also permits the calculation of such objects as the correlational energy densities of the various interactions. Consider, for example, a two-component system. Here one finds that even a very small contamination of heavy masses, which would have a negligible effect upon the total mass or kinetic energy densities, can increase the total correlational energy density, and hence decrease the time scale for the evolution of interesting structure, by orders of magnitude