Bianchi I solutions of the Einstein-Boltzmann system with a positive cosmological constant

Abstract

In this paper we study the future global existence and late-time behaviour of the EinsteinBoltzmann system with Bianchi I symmetry and a positive cosmological constant Λ > 0. For the Boltzmann equation we consider the scattering kernel of Israel particles which are the relativistic counterpart of Maxwellian particles. Under a smallness assumption on initial data in a suitable norm we show that solutions exist globally in time and isotropize at late times.

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

“…In the Boltzmann case we obtained similar results in [4,5] where the FLRW and the Bianchi I cases were studied. In this paper we extend [4,5] to all Bianchi cases except type IX. For the scattering kernel of the collision operator we consider the one for Israel particles.…”

“…Let a distribution function f be given and satisfy (10). Suppose that χ ab (t 0 ) and k ab (t 0 ) are initial data of the evolution equations (2)-( 3), which satisfy the constraint equations (4)- (5). Then, the evolution equations (2)-( 3) admit global-in-time solutions with the following asymptotic behaviour:…”

“…For the proofs of Lemma 2, 3, and 4 below we refer to [5] where the Bianchi I case is studied. Since the asymptotic behaviour of the spatial metric assumed in Section 4.1 is the same with the one considered in the Bianchi I case, the proof given in [5] still applies to Lemma 2.…”

“…For the proofs of Lemma 2, 3, and 4 below we refer to [5] where the Bianchi I case is studied. Since the asymptotic behaviour of the spatial metric assumed in Section 4.1 is the same with the one considered in the Bianchi I case, the proof given in [5] still applies to Lemma 2. To prove the results of Lemma 3 and 4 one can use the orthonormal frame approach, hence they are exactly the same with the ones given in [5].…”

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.…”