Cosmologies with positive λ: hierarchies of future behaviour

Smooth Cauchy data on [Formula: see text] for the Einstein-[Formula: see text]-vacuum field equations with cosmological constant [Formula: see text] that are sufficiently close to de Sitter data develop into a solution that admits a smooth conformal boundary [Formula: see text] in its future. The conformal Einstein equations determine a smooth conformal extension across [Formula: see text] that defines on 'the other side' again a [Formula: see text]-vacuum solution. In this article, we discuss to what extent these properties generalize to the future asymptotic behaviour of solutions to the Einstein-[Formula: see text] equations with matter. We study Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions and the Einstein-[Formula: see text] 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-[Formula: see text]-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'.