The cosmic no-hair conjecture for generic Einstein developments

Let $(M,g)$ be a cosmological spacetime solving Einstein's equations with a positive cosmological constant $\alpha0\Lambda>0$. A spacetime is future asymptotically de Sitter like if, along every future-oriented inextendible causal curve, the relevant causal past is modelled on a de Sitter region and its induced metric and second fundamental form converge to those of de Sitter space in the $C^N$ norms defined using the de Sitter metric. Let $\mathcal{A}$ denote the class of initial data whose corresponding maximal globally hyperbolic developments are future causally geodesically complete solutions for a fixed matter model.

Cosmic no-hair conjecture. Generic elements of $\mathcal{A}$ yield maximal globally hyperbolic developments that are future asymptotically de Sitter like.

This is the formal version of the expectation that positive cosmological constant causes generic expanding solutions to lose their matter-dependent anisotropies and inhomogeneities at late times. The paper establishes this behaviour for the $T^3$-Gowdy symmetric Einstein–Vlasov solutions considered, but the generic statement over the full class $\mathcal{A}$ remains broader than that result.