Cosmological perturbations out of the box I

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.