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

Abstract

We obtain finite-time existence for the massless Boltzmann equation, with a range of soft cross-sections, in an FLRW background with data given at the initial singularity. In the case of positive cosmological constant we obtain long-time existence in proper-time for small data as a corollary. * holee@khu.ac.kr † ernesto.nungesser@icmat.es ‡ tod@maths.ox.ac.uk and then data is given actually at what was the singularity. The question naturally arises of extending the work of [1] to the Einstein-Boltzmann equations, that is to say the Einstein equations with collisional matter as source.Mathematically, with any of these matter models, the problem is to find an extended set of conformal Einstein equations in the conformally extended manifold for which finite-time existence can be proved with data at the bang surface. This was achieved for a range of polytropic perfect fluids in [2], for massless Einstein-Vlasov with a spatially homogeneous metric in [3], and for massless Einstein-Vlasov without symmetry in [1]. For both kinds of source, the conformal Einstein equations can be formulated as a Fuchsian system with the pole located at the bang surface but the appropriate Cauchy data are strikingly different in the two cases: for the perfect fluid case the data are simply a Riemannian 3-metric, in fact the metric of the bang surface, with no separate degrees of freedom for the fluid, while for the Einstein-Vlasov case there is a single datum, the initial distribution function subject only to non-negativity and a vanishing dipole condition. The initial metric is extracted from the initial distribution function.The question was raised 1 as to whether the Einstein-Boltzmann case, which is to say the Einstein-Vlasov case with the inclusion of a collision term in the Vlasov equation, might make a bridge between these two cases, with perfect fluids as one limit and Vlasov as the other, depending on the scattering crosssection. This possibility was explored informally in [20] and received some support. The intention now is to proceed more rigorously. The problem is difficult because the collision term inevitably has singularities which have to be dealt with (these are visible in equations (12) and (13) below).One considers massless particles because, near the bang, one expects the particles to be so energetic that their rest-mass, even if nonzero, would have negligible effect. Likewise, near the bang the cosmological constant Λ would be expected to be physically irrelevant, and there have been studies [21] to indicate that indeed, in the cases so far studied, the inclusion of nonzero rest-mass or Λ have negligible effect, but we include the case of a positive cosmological constant to make contact with Penrose's conformal cyclic cosmology (or CCC,[16]). In that theory the rest-mass of all elementary particles is zero near the bang and in the remote future.There are rather few mathematical results on the Einstein-Boltzmann system since the original existence results of [5], and fewer still on massless Einstein-Boltzmann, where th...

Relativistic BGK model for massless particles in the FLRW spacetime

“…We refer to [21] for the study of exact solutions and to [3,4] in the Boltzmann case. For an introduction to the Boltzmann or Vlasov equation for massless particles we refer to [2,19,20,27]. This paper is organized as follows.…”

Relativistic BGK model for massless particles in the FLRW spacetime

“…for Λ = 0, C 2 sinh(2Ht) for Λ > 0, (1.6) where C 1 and C 2 are some positive constants depending on initial data, and Λ = 3H 2 (see [20] for more details). Hence, we may assume that an FLRW background is given and only need to study the BGK model.…”

“…We refer to [22] for the study of exact solutions and to [3,4] in the Boltzmann case. For an introduction to the Boltzmann or Vlasov equation for massless particles, we refer to [2,19,20,21,28].…”

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

“…In this paper, the spatially homogeneous relativistic Boltzmann equation for massless particles will be referred to as the massless Boltzmann equation, for simplicity. The massless Boltzmann equation has recently been studied in [8,9], where an analytic solution has been found, and in [26], where a local existence was obtained. The main interest in [26] was to study the isotropic singularity problem, for which one needs to establish a well-posed Cauchy problem with data at t = 0, where the initial singularity is located (see [3,4,5,35,36] for more details).…”

“…The massless Boltzmann equation has recently been studied in [8,9], where an analytic solution has been found, and in [26], where a local existence was obtained. The main interest in [26] was to study the isotropic singularity problem, for which one needs to establish a well-posed Cauchy problem with data at t = 0, where the initial singularity is located (see [3,4,5,35,36] for more details). In this paper, we also study the Cauchy problem for the massless Boltzmann equation, but (a) data will be given at a finite time after the initial singularity, say t = t 0 > 0, (b) we will obtain the global existence, and (c) the scattering kernel in this paper will differ from the one in [26], which was the type of soft potentials:…”