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 \Delta M / \overline { M } at a cosmic time t and within a radius r , where \Delta M is the accreted mass and \overline { M } 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 \Delta M / \overline { M } 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 .
Because our analysis is restricted to considering only the low order moments in the gravitational potential , the uncertainty associated with the survival time of substructure is substantially reduced in relation to previous theoretical studies of the “ frequency of substructure ” in clusters .

We study the dependence of \Delta M / \overline { M } on radius , mass , \Omega _ { 0 } , \lambda _ { 0 } = 1 - \Omega _ { 0 } , redshift , and relaxation timescale in universes with Cold Dark Matter ( CDM ) and power-law power spectra .
The strongest dependence on \Omega _ { 0 } ( \lambda _ { 0 } = 0 ) occurs at z = 0 where \Delta M / \overline { M } \propto \Omega _ { 0 } ^ { 1 / 2 } for relaxation times \sim 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 . \Delta M / \overline { M } evolves more rapidly with redshift in low-density universes and decreases significantly with radius for \Omega _ { 0 } = 1 models for z \ga 0.5 .
We discuss how to optimize constraints on \Omega _ { 0 } and \lambda _ { 0 } using cluster morphologies .

It is shown that the expected correlation between \Delta M / \overline { M } 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 .
We describe how N-body simulations can quantify this correlation and thus allow \Delta M / \overline { M } to be measured directly from observations of cluster morphologies .