The Omega Dependence of the Evolution of the Two-Point Correlation Function

The evolution of the two-point correlation function, $\xi(r,z)$, and the pairwise velocity dispersion, $\sigma(r,z)$, for both the matter, \xirho, and halo population, \xihh, are described. If the evolution of $\xi$ is parameterized by $\xi(r,z)=(1+z)^{-(3+\eps)}\xi(r,0)$, where $\xi(r,0)=(r/r_0)^{-\gamma}$, then $\epsrho = 1.04 \pm 0.09$ for \omeone\ and $\epsrho = 0.18 \pm 0.12$ for \ometwo, as measured by the the evolution of \xirho\ at 1 Mpc (from $z \sim 5$ to the present epoch). For halos, \eps\ depends also on the mean density. A range of \eps\ values is obtained: $-0.2 \simless \epshh \simless 1.0$ for \omeone\ and $-1.4 \simless \epshh \simless -0.4$ for \ometwo. This result could be used to constrain the mean density of the universe. The evolution of the pairwise velocity dispersion for the mass and halo distribution is measured and compared with the evolution predicted by the Cosmic Virial Theorem (CVT). According to the CVT, $\sigma(r,z)^2 \sim G Q \rho(z) r^2 \xi(r,z)$ or $\sigma \propto (1+z)^{-\eps/2}$. The values of $\eps$ measured from our simulated velocities differ from those given by the evolution of $\xi$ and the CVT, keeping $\gamma$ and $Q$ constant: $\eps = 1.78 \pm 0.13$ for \omeone\ or $\eps = 1.40 \pm 0.28$ for \ometwo.