  We solve the equations of motion for a scalar field with domain wall boundary
conditions in a Friedmann-Robertson-Walker (FRW) spacetime. We find (in
agreement with Basu and Vilenkin) that no domain wall solutions exist in de
Sitter spacetime for h = H/m >= 1/2, where H is the Hubble parameter and m is
the scalar mass. In the general FRW case we develop a systematic perturbative
expansion in h to arrive at an approximate solution to the field equations. We
calculate the energy momentum tensor of the domain wall configuration, and show
that the energy density can become negative at the core of the defect for some
values of the non-minimal coupling parameter xi. We develop a translationally
invariant theory for fluctuations of the wall, obtain the effective Lagrangian
for these fluctuations, and quantize them using the Bunch-Davies vacuum in the
de Sitter case. Unlike previous analyses, we find that the fluctuations act as
zero-mass (as opposed to tachyonic) modes. This allows us to calculate the
distortion and the normal-normal correlators for the surface. The normal-normal
correlator decreases logarithmically with the distance between points for large
times and distances, indicating that the interface becomes rougher than in
Minkowski spacetime.
