## 7 Asymptotics of Expanding Cosmological Models

The aim of this section is to present a picture of the dynamics of forever-expanding cosmological models,
by which we mean spacetimes that are maximal globally hyperbolic developments and which can be covered
by a foliation by Cauchy surfaces whose mean curvature is strictly negative. In contrast to the
approach to the big bang considered in Section 8, the spatial topology can be expected to play an
important role in the present considerations. Intuitively, it may well happen that gravitational waves have
time to propagate all the way around the universe. It will be assumed, as the simplest case, that the
spacetimes considered admit a compact Cauchy surface. Then the hypersurfaces of negative mean
curvature introduced above have finite volume and this volume is a strictly increasing function of
time.
Models with accelerated expansion are now an important subject in cosmology. They have been
important for a long time in connection with the early universe, where inflation plays a key role. More
recently they have acquired a new significance in view of accumulating observational evidence that the
expansion of the universe is accelerating at the present epoch. For these reasons it is particularly
interesting to consider solutions of the Einstein equations which are appropriate for modelling
cosmic acceleration. The last four Sections 7.3 to 7.6 are devoted to various aspects of this
topic. An introductory account which describes some of the physical background can be found
in [305].