  Analyzing the halo and galaxy catalogs from the Millennium simulations at
redshifts $z=0,\ 0.5,\ 1$, we determine the alignment profiles of cluster
galaxies by measuring the average alignments between the major axes of the
pseudo inertia tensors from all satellites within cluster's virial radius and
from only those satellites within some smaller radius as a function of the
top-hat scale difference. The alignment profiles quantify how well the
satellite galaxies retain the memory of the external tidal fields after merging
into their host clusters and how fast they lose the initial alignment tendency
as the cluster's relaxation proceeds. It is found that the alignment profile
drops faster at higher redshifts and on smaller mass scales. This result is
consistent with the picture that the faster merging of the satellites and
earlier onset of the nonlinear effect inside clusters tend to break the
preferential alignments of the satellites with the external tidal fields.
Modeling the alignment profile of cluster galaxies as a power-law of the
density correlation coefficient that is independent of the power spectrum
normalization ($\sigma_{8}$) and demonstrating that the density correlation
coefficient varies sensitively with the density parameter ($\Omega_{m}$) and
neutrino mass fraction ($f_{\nu}$), we suggest that the alignment profile of
cluster galaxies might be useful for breaking the $\Omega_{m}$-$\sigma_{8}$ and
$f_{\nu}$-$\sigma_{8}$ degeneracies.

Cosmology and Nongalactic Astrophysics