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Cosmological models (Cargèse lectures 1998)

George F R Ellis, Henk van Elst

TL;DR

This work develops a comprehensive, coordinate-free framework for relativistic cosmology by advancing the 1+3 covariant formalism and its tetrad completion. It systematically connects geometric and matter evolution through propagation/constraint equations, and uses this to classify exact solutions (FLRW, Bianchi, LTB, Swiss-Cheese) and to study perturbations via gauge-invariant covariant methods. The text then extends to almost-FLRW models, lays out Jeans-type instabilities and phase-plane dynamical systems analyses, and applies a covariant kinetic theory to CBR anisotropies, culminating in an almost-FLRW validation of the standard cosmology and a rigorous treatment of observational relations. The framework unifies exact and perturbative cosmology, clarifies the role of symmetries, and provides tools to link theory with data, while highlighting practical open issues in averaging, entropy of gravitating systems, and the Newtonian limit.

Abstract

The aim of this set of lectures is a systematic presentation of a 1+3 covariant approach to studying the geometry, dynamics, and observational properties of relativistic cosmological models. In giving (i) the basic 1+3 covariant relations for a cosmological fluid, the present lectures cover some of the same ground as a previous set of Cargèse lectures \cite{ell73}, but they then go on to give (ii) the full set of corresponding tetrad equations, (iii) a classification of cosmological models with exact symmetries, (iv) a brief discussion of some of the most useful exact models and their observational properties, and (v) an introduction to the gauge-invariant and 1+3 covariant perturbation theory of almost-Friedmann-Lemaître-Robertson-Walker universes, with a fluid description for the matter and a kinetic theory description of the radiation.

Cosmological models (Cargèse lectures 1998)

TL;DR

This work develops a comprehensive, coordinate-free framework for relativistic cosmology by advancing the 1+3 covariant formalism and its tetrad completion. It systematically connects geometric and matter evolution through propagation/constraint equations, and uses this to classify exact solutions (FLRW, Bianchi, LTB, Swiss-Cheese) and to study perturbations via gauge-invariant covariant methods. The text then extends to almost-FLRW models, lays out Jeans-type instabilities and phase-plane dynamical systems analyses, and applies a covariant kinetic theory to CBR anisotropies, culminating in an almost-FLRW validation of the standard cosmology and a rigorous treatment of observational relations. The framework unifies exact and perturbative cosmology, clarifies the role of symmetries, and provides tools to link theory with data, while highlighting practical open issues in averaging, entropy of gravitating systems, and the Newtonian limit.

Abstract

The aim of this set of lectures is a systematic presentation of a 1+3 covariant approach to studying the geometry, dynamics, and observational properties of relativistic cosmological models. In giving (i) the basic 1+3 covariant relations for a cosmological fluid, the present lectures cover some of the same ground as a previous set of Cargèse lectures \cite{ell73}, but they then go on to give (ii) the full set of corresponding tetrad equations, (iii) a classification of cosmological models with exact symmetries, (iv) a brief discussion of some of the most useful exact models and their observational properties, and (v) an introduction to the gauge-invariant and 1+3 covariant perturbation theory of almost-Friedmann-Lemaître-Robertson-Walker universes, with a fluid description for the matter and a kinetic theory description of the radiation.

Paper Structure

This paper contains 115 sections, 270 equations, 3 figures.

Table of Contents

  1. Basic relations
  2. $1+3$ covariant description
  3. Variables
  4. Average 4-velocity of matter
  5. Kinematical quantities
  6. Matter tensor
  7. Maxwell field strength tensor
  8. Weyl curvature tensor
  9. Auxiliary quantities
  10. $1+3$ covariant propagation and constraint equations
  11. Ricci identities
  12. Twice-contracted Bianchi identities
  13. Other Bianchi identities
  14. Maxwell's field equations
  15. Pressure-free matter ('dust')
  16. ...and 100 more sections

Figures (3)

  • Figure 1: Classification of cosmological models (with $(\mu+p) > 0)$ by isotropy and homogeneity.
  • Figure 2: Canonical structure constants for different Bianchi types. The Class B parameter $h$ is defined as $h = a^2/n_2n_3$ (see, e.g., waiell97).
  • Figure 3: The Bianchi models permitting higher symmetry subcases. The parameter $c$ is zero iff the preferred spatial vector is hypersurface-orthogonal.