Nonlinear properties and time evolution of gravitational galaxy clustering
William C. Saslaw, Ravi K. Sheth · The Astrophysical Journal · 1993
We develop methods for comparing the quasi-equilibrium theory of gravitational galaxy clustering with other theories in the nonlinear regime. The volume integrals, ξ^bar^_N_ of the gravitational N-point correlation functions are expressed as functions of the quasi-equilibrium parameter b in a closed form which is exact to all orders of b. This shows that the ξ^bar^_N_ scale as (N/2)^N-1^(ξ^bar^_2_)^N_1^ for small b, but break out of this scaling as b evolves. We show how to derive the quasi-equilibrium distribution functions from the void distribution by explicitly keeping the ξ^bar^_N_ constant. Our results provide a simple proof that all the ξ^bar^_N_ are invariant to random binomial selection of subsamples. They also clarify properties of the skewness and kurtosis of the quasi-equilibrium distribution. From the relations between ξ^bar^_N_ and b, and the adiabatic evolution of b, we derive the nonlinear evolution of the ξ^bar^_N_ as the universe expands. This is compared with earlier linear theory for the evolution of these integrals. A new criterion, based on the maximum of the specific heat, provides a physical measure of the transition from the linear to the nonlinear regime. These results are extended to give the nonlinear evolution of the galaxy velocity dispersions, the internal energy, and the entropy. The average gravitational entropy is shown to always increase as the universe expands. We examine the fluctuations of galaxy counts on large scales. In this limit they are Gaussian, consistent with recent COBE results. The dispersion, σ^2^, is greater than N^bar^_galaxies_ by a factor (1 - b)^-2^. This may increase the amplitude of the fluctuations in the cosmic microwave background significantly. We also explicitly find the lowest order departures from Gaussian behavior, which can be compared with observations of greater sensitivity and resolution than are presently available.