  The Wightman function and the vacuum expectation values of the field squared
and of the energy-momentum tensor are obtained, for a massive scalar field with
an arbitrary curvature coupling parameter, in the region between two infinite
parallel plates, on the background of de Sitter spacetime. The field is
prepared in the Bunch-Davies vacuum state and is constrained to satisfy Robin
boundary conditions on the plates. For the calculation, a mode-summation method
is used, supplemented with a variant of the generalized Abel-Plana formula.
This allows to explicitly extract the contributions to the expectation values
which come from each single boundary, and to expand the second-plate-induced
part in terms of exponentially convergent integrals. Several limiting cases of
interest are then studied. Moreover, the Casimir forces acting on the plates
are evaluated, and it is shown that the curvature of the background spacetime
decisively influences the behavior of these forces at separations larger than
the curvature scale of de Sitter spacetime. In terms of the curvature coupling
parameter and the mass of the field, two very different regimes are realized,
which exhibit monotonic and oscillatory behavior of the vacuum expectation
values, respectively. The decay of the Casimir force at large plate separation
is shown to be power-law (monotonic or oscillating), with independence of the
value of the field mass.
