Causal perturbation theory in general FRW cosmologies I: energy momentum conservation and matching conditions
We describe energy--momentum conservation in relativistic perturbation theory in general FRW backgrounds with causal source terms, such as the presence of cosmic defect networks. We provide a prescription for a linear energy--momentum pseudo-tensor in a curved FRW universe, decomposing it using eigenfunctions of the Helmholtz equation. We also construct conserved vector densities for the conformal geometry of these spacetimes and relate these to our pseudo-tensor, demonstrating the equivalence of these two approaches. We also relate these techniques to the role played by residual gauge freedom in establishing matching conditions at early phase transitions, which we can express in terms of components of our pseudo-tensor. This formalism is concise and geometrically sound on both sub- and superhorizon scales, thus extending existing work to a physically (and numerically) useful context.