Cosmological Perturbations from a Group Theoretical Point of View

TL;DR

This paper introduces a novel group-theoretical framework for cosmological perturbations using Lorentz group representations, re-deriving standard formalism and providing a new metric expansion aligned with local spacetime symmetries.

Contribution

It develops a new expansion of the perturbed FLRW metric based on Lorentz group irreducible representations, connecting to standard perturbation classifications.

Findings

The Lorentz group basis aligns with local spacetime symmetries.

The new equations do not simplify standard perturbation theory.

The Lorentz algebra's structure limits further simplification.

Abstract

We present a new approach to cosmological perturbations based on the theory of Lie groups and their representations. After re-deriving the standard covariant formalism from SO(3) considerations, we provide a new expansion of the perturbed Friedmann-Lemaitre-Robertson-Walker (FLRW) metric in terms of irreducible representations of the Lorentz group. The resulting decomposition splits into (scalar, scalar), (scalar, vector) and (vector, vector) terms. These equations directly correspond to the standard Lifshitz classification of cosmological perturbations using scalar, vector and tensor modes which arise from the irreducible SO(3) representation of the spatial part of the metric. While the Lorentz group basis matches the underlying local symmetries of the FLRW spacetime better than the SO(3), the new equations do not provide further simplification compared to the standard cosmological perturbation theory. We conjecture that this is due to the fact that the so(3,1) ~ su(2) x su(2) Lorentz algebra has no pair of commuting generators commuting with any of the translation group generators.