In this study we use simulations of 128 ^ { 3 } particles to study the ellipticity and orientation of clusters of galaxies in N-body simulations of differing power-law initial spectra ( P ( k ) \propto k ^ { n } ,n = +1 , 0 , -1 , -2 ) , and density parameters ( \Omega _ { 0 } = 0.2 to 1.0 ) , in a controlled way , based on nearly 3000 simulated clusters .
Furthermore , unlike most theoretical studies we mimic most observers by removing all particles which lie at distances greater than 2 { h ^ { -1 } } Mpc from the cluster center of mass .

We computed the axial ratio and the principal axes using the inertia tensor of each cluster .
The mean ellipticity of clusters increases strongly with increasing n .
We also find that clusters tend to become more spherical at smaller radii .

We compared the orientation of a cluster to the orientation of neighboring clusters as a function of distance ( correlation ) .
In addition , we considered whether a cluster ’ s major axis tends to lie along the line connecting it to a neighboring cluster , as a function of distance ( alignment ) .
Both alignments and correlations were computed in three dimensions and in projection to mimic observational surveys .
Our results show that significant alignments exist for all spectra at small separations ( D < 15 { h ^ { -1 } } Mpc ) but drops off at larger distance in a strongly n - dependent way .
Therefore the most useful study for observers is the variation of alignment with distance .
Correlations exist but are a weaker effect .

We found that differences in \Omega had no measurable effect on mean ellipticity , and a weak effect on cluster alignments and correlations .
Biasing was able to totally hide the effect of greater nonlinearity .
Therefore , we suggest that any effort to probe \Omega in this manner be abandoned unless it can be unambiguously proven to exist on smaller scales .
However , there are systematic effects due to the primordial spectral index , n .
Our results suggest that cluster ellipticity and the scale dependence of cluster alignments probe the primordial power spectrum independently of the parameters of the background cosmology .
Future work should concentrate on these parameters .

\altaffiltext 1Department of Physics and Astronomy , University of Kentucky , Lexington , KY 40506-0055 . \altaffiltext 2Center for Computational Sciences , University of Kentucky , Lexington , KY 40506-0045 . \altaffiltext 3Department of Physics and Astronomy , University of Kansas , Lawrence , KS 66045 .