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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 trk is negative can be thought of as an expanding cosmological model. Supposing, for simplicity, that the hypersurfaces are compact their volume V (t) satisfies dV /dt = - (trk)V. Associated to the volume V is a length scale l = V 1/3, (This formula applies to the case of three space dimensions. In n dimensions it should be l = V 1/n.) Expansion corresponds to l > 0 which is equivalent to trk < 0. The defining condition for accelerated expansion is ¨l > 0. This is equivalent to - d/dt(trk) + 1(trk)2 > 0 3. If the leaves of the foliation are not compact this can be taken as the definition of accelerated expansion.

In [304Jump To The Next Citation Point] 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 [326Jump To The Next Citation Point]. These spacetimes are closely related to those discussed in Section 5.1. In even spacetime dimensions they have asymptotic expansions in powers of e- Ht where V~ ---- H = /\/3, but in odd dimensions there are in general terms containing a positive power of t multiplied by a power of - Ht e.


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