Conserved Quantities in Expanding Gödel Cosmology

TL;DR

This work examines linear perturbations of Expanding Gödel ($EG$) cosmology, a rotating and expanding spacetime, and identifies gauge-invariant perturbations that enable linking early and late-time behavior. It constructs a gauge-invariant spatial metric trace perturbation, $zetaSMTP$, and shows it is conserved on large scales for dust, with a corresponding evolution when non-adiabatic pressure is present. The analysis is extended to a perfect-fluid content, yielding analogous conserved quantities and a similar large-scale conservation law for barotropic fluids. These results provide a robust framework for studying structure formation in rotating, expanding cosmologies and offer practical tools for validating simulations and exploring potential observational signatures of primordial rotation.

Abstract

At linear order we study perturbations to a Gödel background spacetime which includes expansion in addition to rotation. We investigate the transformation behaviour of these perturbations under gauge transformations and construct gauge invariant quantities. Using the perturbed energy conservation equation we find that there are conserved quantities in Expanding Gödel (EG) Cosmology, in particular a spatial metric trace perturbation, ζ SMTP , which is conserved on large scales for pressureless dust. We finally extend our discussion to a perfect fluid matter content to also obtain conserved quantities in this context.