Second order Boltzmann equation : gauge dependence and gauge invariance

Abstract

In the context of cosmological perturbation theory, we derive the second order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: i) the polarisation of light is incorporated in this formalism by using a tensor-valued distribution function; ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; iii) we perform a separation between temperature and spectral distortion, both for the intensity and for polarisation for the first time; iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gauge dependence for the observed temperature.

25 pages, 1 figure

References in corpus (5)

• Relativistic cosmology and large-scale structure
• The CMB bispectrum from recombination
• Gauge invariant Boltzmann equation and the fluid limit
• Covariant generalization of cosmological perturbation theory
• Quantization of perturbations during inflation in the 1+3 covariant formalism

Cited by in corpus (22)

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• Magnetic field spectrum at cosmological recombination revisited
• Parameterization of temperature and spectral distortions in future CMB experiments
• The full CMB temperature bispectrum from single-field inflation
• Geodesic "curve"-of-sight formulae for the cosmic microwave background: a unified treatment of redshift, time delay, and lensing
• Spectral distortions in the cosmic microwave background polarization
• A precise numerical estimation of the magnetic field generated around recombination
• A New Solution for the Observed Isotropic Cosmic Birefringence Angle and its Implications for the Anisotropic Counterpart through a Boltzmann Approach
• A new line-of-sight approach to the non-linear Cosmic Microwave Background
• Radiative transport of relativistic species in cosmology
• Tetrad formalism for exact cosmological observables
• CMB spectral distortions as solutions to the Boltzmann equations
• Unified approach to secondary effects on the CMB B-mode polarization
• Scalar induced gravitational waves review
• Secondary isocurvature perturbations from acoustic reheating
• CMB spectroscopy at third-order in cosmological perturbations
• Observables for the Effect of Gravity on Electromagnetic Polarization
• Observable Gravitational Wave Strain at Second Order