Around a flat background, a plane wave propagating in the $z$ direction is given by $h_{\mu\nu} = \epsilon_{\mu\nu} \cos(\omega t -kz)$.
What is the generalisation of this to a de Sitter background?
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$\begingroup$ @J.G.: That is for a scalar field. Presumably OP is interested in gravitational waves. $\endgroup$– A.V.S.Commented Aug 12, 2018 at 6:53
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$\begingroup$ Yes, I am interested in gravitational wave propagation. $\endgroup$– supercoolphysicistCommented Aug 12, 2018 at 8:32
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To lowest order in $\Lambda$, i.e. taking the metric to be Minkowski plus a perturbation from $\Lambda$, plus a perturbation from the plane wave, the plane wave is given in the following papers
The fate of a gravitational wave in de Sitter spacetime About the propagation of the Gravitational Waves in an asymptotically de-Sitter space
answered Aug 12, 2018 at 15:18