Non - Linear Properties and Time Evolutionof Gravitational Galaxy

We develop methods for comparing the quasi-equilibrium theory of gravitational galaxy clustering with other theories in the non-linear regime. The volume integrals, 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 N scale as [N=2]N 1[ 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 N constant. Our results provide a simple proof that all the 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 N and b, and the adiabatic evolution of b, we derive the non-linear evolution of the 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 speci c heat, provides a physical measure of the transition from the linear to the non-linear regime. These results are extended to give the non-linear 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 uctuations of galaxy counts on large scales. In this limit they are Gaussian, consistent with recent COBE results. The dispersion, 2, is greater than Ngalaxies by a factor (1 b) 2. This may increase the amplitude of the uctuations in the cosmic microwave background signi cantly. We also explicitly nd the lowest order departures from Gaussian behavior which can be compared to observations of greater sensitivity and resolution than are presently available. subject headings: cosmology: theory { galaxies: clustering { galaxies: formation { gravitation { methods: numerical 2