Conserved Quantities in Perturbed Inflationary Universes

TL;DR

This paper develops a gauge-invariant framework for analyzing density perturbations in non-flat, imperfect fluid-filled universes, introducing a conserved quantity that links different cosmic epochs.

Contribution

It introduces a generalized conserved quantity  that applies to non-flat universes with imperfect fluids, extending previous conservation laws.

Findings

 remains conserved across different scales and universe geometries.

The evolution of density perturbations can be described by a first-order differential equation involving .

The method is demonstrated with two simple cosmological examples.

Abstract

Given that observations seem to favour a \index{density parameter} $\Omega_0<1$, 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 $\Delta$ so that it can be determined through a {\it first order differential equation} with a quantity $\kk$ which is conserved at any length scale, even in non\hs flat universe models, acting as a source term. The quantity $\kk$ 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 $\kk$ can be reduced to determining two auxiliary variables $X$ and $Y$, and illustrate the method with two simple examples.