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Clusters of galaxies are expected to form more recently in denser cosmologies, suggesting that $\Omega_o$ may be observationally constrained by the evolution of cluster abundances (as a function of velocity dispersion, for example). Another, related constraint is the amount of substructure observed in nearby clusters; in a low-$\Omega$ universe, clusters have had more time to relax dynamically since formation. N-body simulations have born out the analytical prediction that cluster formation time depends on $\Omega$. \medskip
In this study, we examine the evolution of individual clusters through N-body simulations to see how interior dynamics affect both velocity dispersions and the relaxation of substructure. We have performed simulations for the cosmologies $\Omega_o$=1.0; $\Omega_o$=0.2; $\Omega_o$=0.1; and $\Omega_o$=0.2, $\lambda_o$=0.8 with initial power spectra $P(k) = Ak^n$, $n = -2, -1$, and 0. We use $64^3$ particles to simulate a volume $64h^{-1}$ Mpc across, and save outputs about every 100 million years. This resolution allows us to follow the evolution of a cluster as an object is accreted and subsumed. We focus on two aspects of this evolution: \ (1) the increase in cluster dispersion when an accreted object rushes past a cluster core, producing a scatter in the mass-dispersions relation (and therefore affecting predictions for the abundance function $n(>\sigma)$); and (2) the time required for substructure to relax --- that is, to become fully mixed with the cluster in both position and velocity space. Quantification of these effects is necessary in order to make accurate predictions of cluster properties and thus constrain cosmological models.
