  Using the tool of Hodge-Morrey decomposition of forms, we prove a new
decomposition of symmetric rank-2 tensors on Ricci flat manifolds with
boundary. Using this we reconstruct a new cosmological perturbation theory that
allows for the scalar-vector-tensor type separation of the linearized Einstein
equations with general boundary conditions. We discuss gauge transformations,
gauge invariant quantities and as an example how the new decomposition works
out in the single-field inflation scenario. For the scalar modes we get two
copies of Mukhanov-Sasaki equation, one of them with a slight modification.
Additionally we run a Weinberg-like argument for the existence adiabatic modes,
and find some gauge-invariant solutions to the perturbations that exists
whatever the constituents of the universe are.
