Second order Boltzmann equation : gauge dependence and gauge invariance

TL;DR

This paper derives a gauge-invariant second order Boltzmann equation for cosmological radiation, incorporating polarization and spectral distortions, and clarifies the gauge dependence of perturbations affecting observations.

Contribution

It extends the second order Boltzmann equation framework by including polarization, spectral distortions, and explicitly analyzing gauge dependence and invariance.

Findings

Derived gauge-invariant second order Boltzmann equation for radiation.

Incorporated polarization using tensor-valued distribution functions.

Clarified the gauge dependence of perturbations and their impact on observations.

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.