### 6.3 Asymptotics for a phase of accelerated expansion

Fuchsian techniques cannot only be used to construct singular spacetimes; they can also be used to
construct spacetimes which are future geodesically complete and which exhibit accelerated expansion at late
times. A solution of the Einstein equations with a foliation of spacelike hypersurfaces whose mean curvature
is negative can be thought of as an expanding cosmological model. Supposing, for simplicity, that the
hypersurfaces are compact their volume satisfies . Associated to the volume
is a length scale , (This formula applies to the case of three space dimensions. In dimensions
it should be .) Expansion corresponds to which is equivalent to . The defining
condition for accelerated expansion is . This is equivalent to . If
the leaves of the foliation are not compact this can be taken as the definition of accelerated
expansion.
In [304] Fuchsian techniques were used to construct solutions of the Einstein vacuum equations with
positive cosmological constant in any dimension which have accelerated expansion at late times and are not
assumed to have any symmetry. Detailed asymptotic expansions are obtained for the late-time behaviour of
these solutions. In the case of three spacetime dimensions these expansions were first written down by
Starobinsky [326]. These spacetimes are closely related to those discussed in Section 5.1. In even spacetime
dimensions they have asymptotic expansions in powers of where , but in odd
dimensions there are in general terms containing a positive power of multiplied by a power of
.