Cosmologies with positive λ: hierarchies of future behaviour

Abstract

Smooth Cauchy data on S 3 for the Einstein- λ -vacuum field equations with cosmological constant λ > 0 that are sufficiently close to de Sitter data develop into a solution that admits a smooth conformal boundary J + in its future. The conformal Einstein equations determine a smooth conformal extension across J + that defines on ‘the other side’ again a λ -vacuum solution. In this article, we discuss to what extent these properties generalize to the future asymptotic behaviour of solutions to the Einstein- λ equations with matter . We study Friedmann–Lemaitre–Robertson–Walker (FLRW) solutions and the Einstein- λ equations coupled to conformally covariant matter transport equations, to conformally privileged matter equations, and to conformally non-covariant matter equations. We present recent results on the Einstein- λ -perfect-fluid equations with a nonlinear asymptotic dust or asymptotic radiation equation of state. This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.