Well-posedness of anisotropic and homogeneous solutions to the Einstein-Boltzmann system with a conformal-gauge singularity

TL;DR

This paper proves the well-posedness of the Einstein-Boltzmann system with a conformal-gauge singularity in Bianchi I space-time, using conformal rescaling and singular moment estimates to handle singularities.

Contribution

It establishes the existence and uniqueness of solutions to the Einstein-Boltzmann system at the initial singularity with a novel approach involving conformal rescaling and singular weights.

Findings

Initial value problem is well-posed at the conformal singularity.

Existence of solutions is proven under differentiability and eigenvalue conditions.

Singularities in momentum variables are managed with singular weights.

Abstract

We consider the Einstein-Boltzmann system for massless particles in the Bianchi I space-time with scattering cross-sections in a certain range of soft potentials. We assume that the space-time has an initial conformal gauge singularity and show that the initial value problem is well posed with data given at the singularity. This is understood by considering conformally rescaled equations. The Einstein equations become a system of singular ordinary differential equations, for which we establish an existence theorem which requires several differentiability and eigenvalue conditions on the coefficient functions together with the Fuchsian conditions. The Boltzmann equation is regularized by a suitable choice of time coordinate, but still has singularities in momentum variables. This is resolved by considering singular weights, and the existence is obtained by exploiting singular moment estimates.