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Cosmological perturbations

Karim A. Malik, David Wands

TL;DR

This paper surveys cosmological perturbation theory around a flat FRW background, with a focus on gauge freedom and gauge-invariant constructions up to second order. It systematically develops the perturbation framework, including the decomposition into scalar, vector, and tensor modes, the geometry of spatial hypersurfaces, and the energy-momentum description for fluids and scalar fields, both single and multi-component. It then details gauge transformations and the creation of gauge-invariant variables across multiple gauges, followed by the dynamical equations governing perturbations in the linear regime, including adiabatic and entropy modes and their evolution in multi-fluid and multi-field contexts. The non-linear section discusses the δN formalism and the emergence of non-Gaussianity, outlining how second-order effects couple modes and how this informs expectations for observational signatures from inflation and the early universe.

Abstract

We review the study of inhomogeneous perturbations about a homogeneous and isotropic background cosmology. We adopt a coordinate based approach, but give geometrical interpretations of metric perturbations in terms of the expansion, shear and curvature of constant-time hypersurfaces and the orthogonal timelike vector field. We give the gauge transformation rules for metric and matter variables at first and second order. We show how gauge invariant variables are constructed by identifying geometric or matter variables in physically-defined coordinate systems, and give the relations between many commonly used gauge-invariant variables. In particular we show how the Einstein equations or energy-momentum conservation can be used to obtain simple evolution equations at linear order, and discuss extensions to non-linear order. We present evolution equations for systems with multiple interacting fluids and scalar fields, identifying adiabatic and entropy perturbations. As an application we consider the origin of primordial curvature and isocurvature perturbations from field perturbations during inflation in the very early universe.

Cosmological perturbations

TL;DR

This paper surveys cosmological perturbation theory around a flat FRW background, with a focus on gauge freedom and gauge-invariant constructions up to second order. It systematically develops the perturbation framework, including the decomposition into scalar, vector, and tensor modes, the geometry of spatial hypersurfaces, and the energy-momentum description for fluids and scalar fields, both single and multi-component. It then details gauge transformations and the creation of gauge-invariant variables across multiple gauges, followed by the dynamical equations governing perturbations in the linear regime, including adiabatic and entropy modes and their evolution in multi-fluid and multi-field contexts. The non-linear section discusses the δN formalism and the emergence of non-Gaussianity, outlining how second-order effects couple modes and how this informs expectations for observational signatures from inflation and the early universe.

Abstract

We review the study of inhomogeneous perturbations about a homogeneous and isotropic background cosmology. We adopt a coordinate based approach, but give geometrical interpretations of metric perturbations in terms of the expansion, shear and curvature of constant-time hypersurfaces and the orthogonal timelike vector field. We give the gauge transformation rules for metric and matter variables at first and second order. We show how gauge invariant variables are constructed by identifying geometric or matter variables in physically-defined coordinate systems, and give the relations between many commonly used gauge-invariant variables. In particular we show how the Einstein equations or energy-momentum conservation can be used to obtain simple evolution equations at linear order, and discuss extensions to non-linear order. We present evolution equations for systems with multiple interacting fluids and scalar fields, identifying adiabatic and entropy perturbations. As an application we consider the origin of primordial curvature and isocurvature perturbations from field perturbations during inflation in the very early universe.

Paper Structure

This paper contains 78 sections, 354 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Perturbations in cosmology
  3. Defining perturbations
  4. Decomposing tensorial quantities
  5. Vectors
  6. Tensors
  7. Geometry of spatial hypersurfaces
  8. Timelike vector fields
  9. Geometrical quantities
  10. Energy-momentum tensor for fluids
  11. Single fluid
  12. Multiple fluids
  13. Energy-momentum tensor for scalar fields
  14. Single field
  15. Multiple fields
  16. ...and 63 more sections

Figures (1)

  • Figure 1: $r$ and $s$ are used to denote the instantaneous adiabatic and entropy fields respectively, along and orthogonal to the curved background solution in field space.