Gauge invariant Boltzmann equation and the fluid limit

TL;DR

The paper develops a gauge-invariant framework for the collisionless Boltzmann equation to second order in cosmological perturbations, defining a gauge-invariant distribution function and brightness for radiation and deriving the corresponding second-order Boltzmann equation and fluid limit. Using a knight-diffeomorphism approach and tetrad formalism, it constructs gauge-invariant variables consistent with general covariance and analyzes their transformation properties. It shows how the fluid description emerges from the kinetic theory when anisotropic stress is neglected, and it clarifies discrepancies in prior literature that arose from tetrad versus canonical-basis identifications. The framework provides a robust foundation for studying second-order effects and non-Gaussianities in the CMB, with potential applications to high-precision cosmology.

Abstract

This article investigates the collisionless Boltzmann equation up to second order in the cosmological perturbations. It describes the gauge dependence of the distribution function and the construction of a gauge invariant distribution function and brightness, and then derives the gauge invariant fluid limit.