Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Covariant generalization of cosmological perturbation theory

Kari Enqvist, Janne Hogdahl, Sami Nurmi, Filippo Vernizzi

TL;DR

The paper develops a covariant generalization of cosmological perturbation theory by defining nonlinear perturbations through a connecting vector between worldlines in the real inhomogeneous universe. It introduces the covariant family $\zeta_{(n)}$ as the $n$-th Lie-derivative perturbation of the local expansion on uniform density hypersurfaces, and derives an exact evolution equation showing $\zeta_{(n)}$ is conserved on all scales for barotropic fluids. A gauge-invariant construction is provided to connect these covariant quantities with the standard coordinate-based perturbations, reproducing the familiar first- and second-order results and generalizing to arbitrary order. On large scales, the covariant quantities reduce to the curvature perturbation on uniform density hypersurfaces, offering a gauge-independent, physically transparent framework with direct relevance to observations and the separate-universe picture.

Abstract

We present an approach to cosmological perturbations based on a covariant perturbative expansion between two worldlines in the real inhomogeneous universe. As an application, at an arbitrary order we define an exact scalar quantity which describes the inhomogeneities in the number of e-folds on uniform density hypersurfaces and which is conserved on all scales for a barotropic ideal fluid. We derive a compact form for its conservation equation at all orders and assign it a simple physical interpretation. To make a comparison with the standard perturbation theory, we develop a method to construct gauge-invariant quantities in a coordinate system at arbitrary order, which we apply to derive the form of the n-th order perturbation in the number of e-folds on uniform density hypersurfaces and its exact evolution equation. On large scales, this provides the gauge-invariant expression for the curvature perturbation on uniform density hypersurfaces and its evolution equation at any order.

Covariant generalization of cosmological perturbation theory

TL;DR

The paper develops a covariant generalization of cosmological perturbation theory by defining nonlinear perturbations through a connecting vector between worldlines in the real inhomogeneous universe. It introduces the covariant family as the -th Lie-derivative perturbation of the local expansion on uniform density hypersurfaces, and derives an exact evolution equation showing is conserved on all scales for barotropic fluids. A gauge-invariant construction is provided to connect these covariant quantities with the standard coordinate-based perturbations, reproducing the familiar first- and second-order results and generalizing to arbitrary order. On large scales, the covariant quantities reduce to the curvature perturbation on uniform density hypersurfaces, offering a gauge-independent, physically transparent framework with direct relevance to observations and the separate-universe picture.

Abstract

We present an approach to cosmological perturbations based on a covariant perturbative expansion between two worldlines in the real inhomogeneous universe. As an application, at an arbitrary order we define an exact scalar quantity which describes the inhomogeneities in the number of e-folds on uniform density hypersurfaces and which is conserved on all scales for a barotropic ideal fluid. We derive a compact form for its conservation equation at all orders and assign it a simple physical interpretation. To make a comparison with the standard perturbation theory, we develop a method to construct gauge-invariant quantities in a coordinate system at arbitrary order, which we apply to derive the form of the n-th order perturbation in the number of e-folds on uniform density hypersurfaces and its exact evolution equation. On large scales, this provides the gauge-invariant expression for the curvature perturbation on uniform density hypersurfaces and its evolution equation at any order.

Paper Structure

This paper contains 10 sections, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Covariant formalism
  3. Covariant perturbation theory
  4. Covariant perturbations
  5. Evolution of perturbations
  6. Comparison with the coordinate-based approach
  7. Perturbation theory and gauge transformations
  8. Evolution of perturbations
  9. Coordinate-based expansion of the covariant perturbations
  10. Conclusion

Figures (1)

  • Figure 1: Geometric definition of the covariant perturbation $\Delta f_{\lambda_\perp}$