Approximate monotonicity in stable Big Bang formation
Stable Big Bang formation has been established in various settings where BKL type oscillations are silenced by the presence of matter or symmetry. It was first achieved in the seminal work of Rodnianski and Speck '14 which treated nearly spatially isotropic solutions. Their method of proof relied on approximate monotonicity, that is to say cancellations in the energy estimates, specifically, in the borderline non-integrable terms. The full range of anisotropic Kasner exponents was later treated using a more robust method that allows for highly degenerate top order estimates. Although sufficient for singularity formation, it is not clear whether the latter degeneracy is an actual phenomenon of dynamical Big Bang singularities, e.g. due to loss of regularity in the asymptotic data, or merely a crude handling of the borderline terms in the estimates. We will present ongoing work with Jared Speck which shows that approximate monotonicity is in fact manifest in the full sub-critical regime. As a consequence, we are able to close estimates at a fixed regularity and show that distinct Kasner exponents at the singularity lose only one derivative relative to initial data.

