  We investigate the dependence of ellipticities of clusters of galaxies on
cosmological parameters using large-scale cosmological simulations. We
determine cluster ellipticities out to redshift unity for LCDM models with
different mean densities $\Omega_m$ and amplitudes of mass fluctuation
$\sigma_{8,0}$. The mean ellipticity increases monotonically with redshift for
all models. Larger values of $\sigma_{8,0}$, i.e., earlier cluster formation
time, produce lower ellipticities. The dependence of ellipticity on $\Omega_m$
is relatively weak in the range $0.2 \leq \Omega_m \leq 0.5$ for high mass
clusters. The mean ellipticity $\bar{e}(z)$ decreases linearly with the
amplitude of fluctuations at the cluster redshift $z$, nearly independent of
$\Omega_m$; on average, older clusters are more relaxed and are thus less
elliptical. The distribution of ellipticities about the mean is approximated by
a Gaussian, allowing a simple characterization of the evolution of ellipticity
with redshift as a function of cosmological parameters. At $z=0$, the mean
ellipticity of high mass clusters is approximated by $\bar{e}(z=0) =
0.248-0.069 \sigma_{8,0} + 0.013 \Omega_{m,0}$. This relation opens up the
possibility that, when compared with future observations of large cluster
samples, the mean cluster ellipticity and its evolution could be used as a new,
independent tool to constrain cosmological parameters, especially the amplitude
of mass fluctuations, $\sigma_{8,0}$.
