On the structure of big bang singularities in spatially homogenous solutions to the Einstein non-linear scalar field equations

The subject of this article is the structure of big bang singularities in spatially homogeneous solutions to the Einstein non-linear scalar field equations. In particular, we focus on Bianchi class A; i.e., developments arising from left invariant initial data on unimodular $3$-dimensional Lie groups. We prove that solutions are either vacuum or matter dominated, depending on whether the limit of an expansion normalised normal derivative of the scalar field is zero or not, respectively. The main result concerning the asymptotics in the direction of the singularity is, essentially, that solutions induce data on the singularity, with two exceptions: vacuum dominated Bianchi type VIII and IX without additional symmetries (they are neither isotropic nor locally rotationally symmetric) exhibit BKL-type oscillations. Disregarding the exceptions, there is in fact a bijection between initial data on the singularity and developments. Initial data on the singularity thus play a central role in the analysis; they both parameterise developments and give optimal asymptotic information. However, the main point of the article is to prove that the set of isometry classes of initial data on the singularity (of a fixed Bianchi type and symmetry (such as isotropy, local rotational symmetry etc.)) has a smooth structure; that the set of isometry classes of developments (similarly restricted) has a smooth structure which fits together with the natural smooth structure of isometry classes of regular initial data with fixed mean curvature; and that the Einstein flow generates a diffeomorphism between the two sets. However, the article contains substantial additional information, such as, e.g., the construction of a large class of spatially locally homogeneous solutions that can be demonstrated to be globally non-linearly stable (in the absence of symmetries) both to the future and to the past.