Future global existence and asymptotic behaviour of solutions to the Einstein–Boltzmann system with Bianchi I symmetry

Abstract

In this paper we study the Einstein-Boltzmann system with Bianchi I symmetry. We show that for small initial data the corresponding solutions of the Einstein-Boltzmann system are future geodesically complete and that they isotropize and have a dust-like behaviour at late times. Detailed information about the metric and the matter terms is obtained, and the results show that the solutions tend asymptotically to the Einstein-de Sitter

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

“…This means that we can go from our frame to an orthonormal frame and back with transformation matrices which satisfy |e a b | ≤ Ce γt and |(e −1 ) a b | ≤ Ce −γt . For details on this we refer to [4].…”

“…Proof. We refer to [2,4] for the proofs. Note thatp i is the i-th component of the momentum in an orthonormal frame.…”

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

“…Another natural generalisation would be to consider the case of a vanishing cosmological constant. There has been recent progress on that matter [1,10] which in combination with our previous work [3] could lead to results in that direction. Finally the well-posedness of solutions to the Einstein-Boltzmann system with an isotropic singularity (see [11] for the Vlasov case) is still open and seems an interesting and challenging problem.…”

Kantowski–Sachs cosmology with Vlasov matter

“…For massless particles, ie for solutions in X 0 , Lemma 1 is trivial; cf Subsection 4.2. From (16) we see that for solutions in X \(X 0 ∪X 1 ) the sign of l ′ is dictated by the sign of H D . From (34) we thus know that Kantowski-Sachs solutions with massive Vlasov matter satisfy…”

“…The flow at the 'equator' is then characterised by dZ = 0, and given by the with P , Q and S given by the right hand sides of (13), (14) and (16).…”