Late-time behaviour of the Einstein–Boltzmann system with a positive cosmological constant

Abstract

In this paper we study the Einstein-Boltzmann system for Israel particles with a positive cosmological constant. We consider spatially homogeneous solutions of Bianchi types except IX and obtain future global existence and asymptotic behaviour of solutions to the Einstein-Boltzmann system. The result shows that the solutions converge to the de Sitter solution at late times. * holee@khu.ac.kr † ernesto.nungesser@icmat.es 1 collision operator we consider the one for Israel particles. The main result of this paper is given in Theorem 1, and the proof is to use the energy method as in [4].This paper is organised as follows. In Section 2 we introduce the notations and terminology which will be assumed in this paper. In the second part of the section we introduce the Einstein-Boltzmann system for Israel particles with a positive cosmological constant having Bianchi symmetry. The main theorem will also be stated in this part. In Section 3 and 4 we study the Einstein equations and the Boltzmann equation, respectively, and show that solutions have certain desired properties. At the end of Section 4 we combine the results to prove the main theorem. In Section 5 we discuss the result of the present paper and related problems.

The massless Einstein–Boltzmann system with a conformal gauge singularity in an FLRW background

“…Furthermore, as is familiar, these are the only solutions giving zero on the right. They are the equilibrium solutions and α, β can be related to the constants of integration found in (11):…”

The spatially homogeneous Boltzmann equation for massless particles in an FLRW background

“…The restriction was removed in [23], but the argument applies only to the case of the scattering kernel for Israel particles. These results have been extended to the Bianchi cases [22,24,25], and we also refer to [29,30,31] for a different approach. The purpose of this paper is to study the global existence of small solutions to the spatially homogeneous relativistic Boltzmann equation in an FLRW background, and the results of the paper will be an improvement of [21,23], in the sense that the unphysical restriction of [21] will be removed, and the global existence will be proved for a wider class of scattering kernels.…”

Small solutions of the Einstein–Boltzmann-scalar field system in a spatially flat FLRW spacetime

“…We extend the work in Refs. [18,19], where a cosmological constant was considered, to the scalar field case, and the work in Ref. [16], where the FLRW case was treated, to the Bianchi case.…”

“…The argument is basically the same as in Refs. [18,19], but in this paper we present a more detailed proof, for instance the higher order derivatives of post-collision momenta are estimated in Lemma 5 in a mathematically rigorous way. In Section 2.2, we study the coupled Einstein-Boltzmann-scalar field system.…”

“…More specifically, we assume that the spacetime is of Bianchi type I-VIII, that the matter is described by Israel particles and that there exists a scalar field with a potential which has a positive lower bound. This represents a generalization of the work [19], where a cosmological constant was considered, and a generalization of [16], where a spatially flat FLRW spacetime was considered. We obtain the global existence and asymptotic behavior of classical solutions to the Einstein-Boltzmann-scalar field system for small initial data.…”