Conserved Quantities in Perturbed Inflationary Universes

Given that observations seem to favour a \index{density parameter} ${\mathrm{\Omega }}_0<1$, corresponding to an open universe, we consider gauge\hs invariant perturbations of non\hs flat Robertson\hs Walker universes filled with a general imperfect fluid which can also be taken to represent a scalar field. Our aim is to set up the equations that govern the evolution of the density perturbations $\mathrm{\Delta }$ so that it can be determined through a {\it first order differential equation} with a quantity $\text{kk}$ which is conserved at any length scale, even in non\hs flat universe models, acting as a source term. The quantity $\text{kk}$ generalizes other variables that are conserved in specific cases (for example at large scales in a flat universe) and is useful to connect different epochs in the evolution of density perturbations via a transfer function. We show that the problem of finding a conserved $\text{kk}$ can be reduced to determining two auxiliary variables $X$ and $Y$, and illustrate the method with two simple examples.