Deriving Boltzmann Equations from Kadanoff-Baym Equations in Curved Space-Time

TL;DR

This paper derives covariant Boltzmann equations from Kadanoff-Baym equations in curved space-time, clarifying the impact of the universe's expansion on collision terms relevant for baryogenesis models.

Contribution

It provides a first-principles derivation of covariant Boltzmann equations in curved space-time, showing the metric dependence only on the left-hand side and justifying previous vacuum-based calculations.

Findings

Collision terms are metric-independent at tree level.

Derived equations are covariant generalizations of Minkowski space equations.

Loop contributions involve integrals over distribution functions.

Abstract

To calculate the baryon asymmetry in the baryogenesis via leptogenesis scenario one usually uses Boltzmann equations with transition amplitudes computed in vacuum. However, the hot and dense medium and, potentially, the expansion of the universe can affect the collision terms and hence the generated asymmetry. In this paper we derive the Boltzmann equation in the curved space-time from (first-principle) Kadanoff-Baym equations. As one expects from general considerations, the derived equations are covariant generalizations of the corresponding equations in Minkowski space-time. We find that, after the necessary approximations have been performed, only the left-hand side of the Boltzmann equation depends on the space-time metric. The amplitudes in the collision term on the right--hand side are independent of the metric, which justifies earlier calculations where this has been assumed implicitly. At tree level, the matrix elements coincide with those computed in vacuum. However, the loop contributions involve additional integrals over the the distribution function.