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Cosmic microwave background anisotropies in the CDM model: a covariant and gauge-invariant approach

Anthony Challinor, Anthony Lasenby

TL;DR

This work develops a fully covariant, gauge-invariant framework for computing CMB anisotropies in a CDM-dominated universe, avoiding gauge ambiguities by using physically defined variables and a covariant angular decomposition of the photon distribution. It derives a complete set of linearised, frame-independent equations for photons, neutrinos, baryons, and CDM, including the Boltzmann hierarchy and its coupling to metric perturbations across scalar and tensor modes, and provides both integral (line-of-sight) and integral-in-time solutions for the scalar sector. The authors demonstrate adiabatic initial conditions on super-horizon scales, obtain the CMB power spectrum in a standard CDM model, and implement a tight-coupling regime to connect pre- and post-recombination physics, with numerical results that agree with traditional gauge-based analyses. This covariant approach clarifies the physical content of perturbations, facilitates extension to non-linear regimes, and sets the stage for incorporating polarization and hot dark matter in a consistent, gauge-free formalism.

Abstract

We present a fully covariant and gauge-invariant calculation of the evolution of anisotropies in the cosmic microwave background (CMB) radiation. We use the physically appealing covariant approach to cosmological perturbations, which ensures that all variables are gauge-invariant and have a clear physical interpretation. We derive the complete set of frame-independent, linearised equations describing the (Boltzmann) evolution of anisotropy and inhomogeneity in an almost Friedmann-Robertson-Walker (FRW) cold dark matter (CDM) universe. These equations include the contributions of scalar, vector and tensor modes in a unified manner. Frame-independent equations for scalar and tensor perturbations, which are valid for any value of the background curvature, are obtained straightforwardly from the complete set of equations. We discuss the scalar equations in detail, including the integral solution and relation with the line of sight approach, analytic solutions in the early radiation dominated era, and the numerical solution in the standard CDM model. Our results confirm those obtained by other groups, who have worked carefully with non-covariant methods in specific gauges, but are derived here in a completely transparent fashion.

Cosmic microwave background anisotropies in the CDM model: a covariant and gauge-invariant approach

TL;DR

This work develops a fully covariant, gauge-invariant framework for computing CMB anisotropies in a CDM-dominated universe, avoiding gauge ambiguities by using physically defined variables and a covariant angular decomposition of the photon distribution. It derives a complete set of linearised, frame-independent equations for photons, neutrinos, baryons, and CDM, including the Boltzmann hierarchy and its coupling to metric perturbations across scalar and tensor modes, and provides both integral (line-of-sight) and integral-in-time solutions for the scalar sector. The authors demonstrate adiabatic initial conditions on super-horizon scales, obtain the CMB power spectrum in a standard CDM model, and implement a tight-coupling regime to connect pre- and post-recombination physics, with numerical results that agree with traditional gauge-based analyses. This covariant approach clarifies the physical content of perturbations, facilitates extension to non-linear regimes, and sets the stage for incorporating polarization and hot dark matter in a consistent, gauge-free formalism.

Abstract

We present a fully covariant and gauge-invariant calculation of the evolution of anisotropies in the cosmic microwave background (CMB) radiation. We use the physically appealing covariant approach to cosmological perturbations, which ensures that all variables are gauge-invariant and have a clear physical interpretation. We derive the complete set of frame-independent, linearised equations describing the (Boltzmann) evolution of anisotropy and inhomogeneity in an almost Friedmann-Robertson-Walker (FRW) cold dark matter (CDM) universe. These equations include the contributions of scalar, vector and tensor modes in a unified manner. Frame-independent equations for scalar and tensor perturbations, which are valid for any value of the background curvature, are obtained straightforwardly from the complete set of equations. We discuss the scalar equations in detail, including the integral solution and relation with the line of sight approach, analytic solutions in the early radiation dominated era, and the numerical solution in the standard CDM model. Our results confirm those obtained by other groups, who have worked carefully with non-covariant methods in specific gauges, but are derived here in a completely transparent fashion.

Paper Structure

This paper contains 19 sections, 127 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Covariant Approach to Cosmology
  3. Linearised Perturbation Equations for the Total Matter Variables
  4. Equations for Individual Matter Components
  5. Photons
  6. Neutrinos
  7. Baryons
  8. Cold Dark Matter
  9. The CMB Temperature Anisotropy
  10. Scalar Perturbations
  11. The Scalar Equations
  12. The Integral Solution
  13. Initial Conditions on Super-Horizon Scales
  14. Adiabatic Scalar Perturbations in a $K=0$ Universe
  15. The CMB Power Spectrum
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The variation of the harmonic coefficients of the fractional comoving density gradients in the CDM frame with redshift in the standard CDM model: $\Omega_{c}=0.95$, $\Omega_{b}=0.05$, $\Omega_{\Lambda}=0$, $H_{0}=50\hbox{$\mathrm{kms^{-1}Mpc^{-1}}$}$, with Helium fraction $0.24$, for $k=0.01$, $0.1$ and $1.0\hbox{\rm ,Mpc}^{-1}$. The normalisation is chosen so that $\Phi_{k}=1$ for all $k$ at early times.
  • Figure 2: The power spectrum of scalar CMB anisotropies in the standard CDM model: $\Omega_{c}=0.95$, $\Omega_{b}=0.05$, $\Omega_{\Lambda}=0$, $H_{0}=50\hbox{$\mathrm{kms^{-1}Mpc^{-1}}$}$, $n_{s}=1$, with Helium fraction $0.24$. The normalisation of the vertical scale is arbitrary.