The Cosmic No-Hair conjecture for scalar fields with positive constant potential

Let $\mathcal A$ denote the class of initial data such that the corresponding maximal globally hyperbolic developments are future causally geodesically complete solutions to the Einstein non-linear scalar field equations with a positive constant potential $V_{\operatorname{const}}$. A solution is future asymptotically de Sitter-like when its late-time geometry approaches de Sitter geometry in the sense specified by the preceding definition. Cosmic No-Hair conjecture. Every generic element of $\mathcal A$ has a maximal globally hyperbolic development which is future asymptotically de Sitter-like.

This is a precise formulation of the expectation that expanding cosmological solutions isotropise and become de Sitter-like as perceived by observers. The paper proves the conjecture for certain ${{\mathbb T}}^2$-symmetric solutions, but no general resolution is supplied here.