  In this paper, considering the linearized Einstein equation with a
two-parameter family of linear covariant gauges in de Sitter spacetime, we
examine possible vacuum states for the gravitons field with respect to
invariance under the de Sitter group $SO_0(1,4)$. Our calculations explicitly
reveal that there exists no natural de Sitter-invariant vacuum state (the
Euclidean state) for the gravitons field. Indeed, on the foundation of a
rigorous group theoretical reasoning, we prove that if one insists on full
covariance as well as causality for the theory, has to give up the positivity
requirement of the inner product. However, one may still look for states with
as much symmetry as possible, more precisely, a restrictive version of
covariance by considering the gravitons field and the associated vacuum state
which are, respectively, covariant and invariant with respect to some maximal
subgroup of the full de Sitter group. In this regard, we treat $SO(4)$ case,
and find a family of $SO(4)$-invariant states. The associated $SO(4)$-covariant
quantum field is given, as well.
