Bianchi models with a free massless scalar field: invariant sets and higher symmetries

Abstract: We scrutinize the overall structure of the space of cosmological models of Bianchi type I-VII$h$ that contain a free massless scalar field, with a spatially homogeneous gradient $\nabla\mu \varphi$ that generally breaks isotropy, in addition to a standard perfect fluid. Specifically, state space is written as a union of disjoint invariant sets, each corresponding to a particular cosmological model that is classified with respect to the Bianchi type and the matter content. Subsets corresponding to higher-symmetry models, including all locally rotationally symmetric models and FLRW models, and models with a shear-free normal congruence, are also derived and classified. For each model the dimension ($d$) of the space of initial data is given, after fixing the orientation of the orthonormal frame uniquely relative to matter anisotropies and geometrical anisotropies.