Cosmological perturbation theory of primordial compact sources

We construct a position-space cosmological perturbation theory around spatially flat Friedmann-Lema\^itre-Robertson-Walker geometries that allows to model localized primordial sources of gravitational waves. The equations of motion are decoupled using a generalized harmonic gauge, which avoids the use of a scalar-vector-tensor decomposition. We point out that sources cannot generically be defined in a compact domain due to fluctuations of the cosmic perfect fluid. For power law cosmologies, we obtain the exact Green's function necessary to solve for all metric perturbations in terms of a hypergeometric function, which matches with a Green's function derived earlier by Chu. This allows us to derive the closed form expression of the linearized metric perturbation generated by sources up to quadrupolar order in the multipolar expansion.