Conservation Laws and Cosmological Perturbations in Curved Universes

When working in synchronous gauges, pseudo-tensor conservation laws are often used to set the initial conditions for cosmological scalar perturbations, when those are generated by topological defects which suddenly appear in an up to then perfectly homogeneous and isotropic universe. However those conservation laws are restricted to spatially flat (K=0) Friedmann-Lema\^\i tre spacetimes. In this paper, we first show that in fact they implement a matching condition between the pre- and post- transition eras and, in doing so, we are able to generalize them and set the initial conditions for all $K$. Finally, in the long wavelength limit, we encode them into a vector conservation law having a well-defined geometrical meaning.