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Omega_0 and Substructure in Galaxy Clusters

(Abridged) We examine the theoretical relationship between Omega_0 and substructure in galaxy clusters which are formed by the collapse of high density peaks in a gaussian random field. The radial mass distributions of the clusters are computed from the spherical accretion model using the adiabatic approximation following Ryden & Gunn. For a cluster of mass, M(r,t), we compute the quantity dM/M_bar at a cosmic time t and within a radius r, where dM is the accreted mass and M_bar is the average mass of the cluster during the previous relaxation time, which is computed individually for each cluster. For a real cluster in three dimensions we argue that dM/M_bar should be strongly correlated with the low order multipole ratios, Phi^{int}_l/Phi^{int}_0, of the potential due to matter interior to r. It is shown that the expected correlation between dM/M_bar and Phi^{int}_l/Phi^{int}_0 extends to the two-dimensional multipole ratios, Psi^{int}_m/Psi^{int}_0, which are well defined observables of the cluster density distribution. The strongest dependence of dM/M_bar on Omega_0 (lambda_0=0) occurs at z=0 where dM/M_bar propto Omega_0^{1/2} for relaxation times ~1-2 crossing times and only very weakly depends on mass and radius. The fractional accreted mass in CDM models with Omega_0+lambda_0=1 depends very weakly on Omega_0 and has a magnitude similar to the Omega_0=1 value. dM/M_bar evolves more rapidly with redshift in low-density universes and decreases significantly with radius for Omega_0=1 models for z > ~0.5. We discuss how to optimize constraints on Omega_0 and lambda_0 using cluster morphologies.