The BKL scattering conjecture for the Einstein-scalar field system

Let $\mathcal{E}_\Phi$ be the graded Lie algebra encoding the Einstein-scalar field equations, and let $\mathcal{A}$ be the associated graded Lie algebra of its filtration. For a graded Lie algebra, write $\operatorname{MC}(\mathcal{A})$ and $\operatorname{MC}(\mathcal{E}_\Phi)$ for the Maurer–Cartan elements, modulo gauge equivalence denoted by $\sim$. Let $\mathcal{U}$ be a subset of $\frac{\operatorname{MC}(\mathcal{A})}{\sim}$.

BKL scattering conjecture. Scattering at $\tau to \infty$ from the dynamical system defined by $\mathcal{A}$ to that defined by $\mathcal{E}_\Phi$ defines a smooth map

$$U \subseteq \frac{\operatorname{MC}(\mathcal{A})}{\sim} to \frac{\operatorname{MC}(\mathcal{E}_\Phi)}{\sim}$$

that is a diffeomorphism onto its image in suitable function spaces. The domain $U$ is open and contains the anisotropic spatially homogeneous elements with positive Kasner parameters. The nondegenerate elements in the image are semiglobal solutions with a curvature singularity reached in finite proper time and with particle horizons.

This is an informal formulation of a strong implementation of the BKL heuristics for the Einstein-scalar field system. The source presents it as a conjectural framework; no resolution is supplied.