Semiglobal non-oscillatory big bang singular spacetimes for the Einstein-scalar field system

Abstract

We construct semiglobal singular spacetimes for the Einstein equations coupled to a massless scalar field. Consistent with the heuristic analysis of Belinskii, Khalatnikov, Lifshitz or BKL for this system, there are no oscillations due to the scalar field. (This is much simpler than the oscillatory BKL heuristics for the Einstein vacuum equations.) Prior results are due to Andersson and Rendall in the real analytic case, and Rodnianski and Speck in the smooth near-spatially-flat-FLRW case. Similar to Andersson and Rendall we give asymptotic data at the singularity, which we refer to as final data, but our construction is not limited to real analytic solutions. This paper is a test application of tools (a graded Lie algebra formulation of the Einstein equations and a filtration) intended for the more subtle vacuum case. We use homological algebra tools to construct a formal series solution, then symmetric hyperbolic energy estimates to construct a true solution well-approximated by truncations of the formal one. We conjecture that the image of the map from final data to initial data is an open set of anisotropic initial data.

arXiv comment41 pages

References in corpus (10)

• Quiescent cosmological singularities
• Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity
• Curvature blow up in Bianchi VIII and IX vacuum spacetimes
• A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
• Aperiodic oscillatory asymptotic behavior for some Bianchi spacetimes
• Strongly Focused Gravitational Waves
• The BKL Conjectures for Spatially Homogeneous Spacetimes
• Bianchi VIII and IX vacuum cosmologies: Almost every solution forms particle horizons and converges to the Mixmaster attractor
• The graded Lie algebra of general relativity
• Filtered expansions in general relativity II