  Gravitational waves are considered as metric perturbations about a curved
background metric, rather than the flat Minkowski metric since several
situations of physical interest can be discussed by this generalization. In
this case, when the de Donder gauge is imposed, its preservation under
infinitesimal spacetime diffeomorphisms is guaranteed if and only if the
associated covector is ruled by a second-order hyperbolic operator which is the
classical counterpart of the ghost operator in quantum gravity. In such a wave
equation, the Ricci term has opposite sign with respect to the wave equation
for Maxwell theory in the Lorenz gauge. We are, nevertheless, able to relate
the solutions of the two problems, and the algorithm is applied to the case
when the curved background geometry is the de Sitter spacetime. Such vector
wave equations are studied in two different ways: i) an integral
representation, ii) through a solution by factorization of the hyperbolic
equation. The latter method is extended to the wave equation of metric
perturbations in the de Sitter spacetime. This approach is a step towards a
general discussion of gravitational waves in the de Sitter spacetime and might
assume relevance in cosmology in order to study the stochastic background
emerging from inflation.
