Gauge-Invariant Variables on Cosmological Hypersurfaces

TL;DR

The paper addresses gauge ambiguity in cosmological perturbations and shows that gauge-invariant perturbations can be constructed by an unambiguous choice of hypersurface time-slicing defined by a time-like vector field $N^\mu$ orthogonal to the hypersurfaces. In the metric approach, scalar perturbations on 3-D hypersurfaces lead to gauge-invariant combinations $\Phi$ and $\Psi$ that correspond to particular gauges (e.g., longitudinal/zero-shear) and align with Mukhanov's gauge-invariant matter perturbation; in the fluid-flow approach, perturbations built from the fluid velocity yield gauge-invariant quantities like the density perturbation $\delta \tilde{\epsilon}_m$ and the comoving curvature perturbation, with conserved behavior on large scales for adiabatic perturbations. The work clarifies the connection between metric-based and covariant formalisms, demonstrates explicit gauge-invariant reformulations on chosen hypersurfaces, and suggests non-linear extensions and broader gauge choices for robust perturbation analysis. This framework facilitates cross-formalism comparisons and practical calculations of perturbation evolution across different slicing choices.

Abstract

We show how gauge-invariant cosmological perturbations may be constructed by an unambiguous choice of hypersurface-orthogonal time-like vector field (i.e., time-slicing). This may be defined either in terms of the metric quantities such as curvature or shear, or using some matter field. As an example, we show how linear perturbations in the covariant fluid-flow approach can then be presented in an explicitly gauge-invariant form in the coordinate based formalism.