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7.6 Inhomogeneous spacetimes with accelerated expansion

Consider what happens to Wald’s proof in an inhomogeneous spacetime with positive cosmological constant. His arguments only use the Hamiltonian constraint and the evolution equation for the mean curvature. In Gauss coordinates spatial derivatives of the metric only enter these equations via the spatial scalar curvature in the Hamiltonian constraint. Hence, as noticed in [196], Wald’s argument applies to the inhomogeneous case, provided we have a spacetime that exists globally in the future in Gauss coordinates and which has everywhere non-positive spatial scalar curvature. Unfortunately, it is hard to see how the latter condition can be verified starting from initial data. It is not clear whether there is a non-empty set of inhomogeneous initial data to which this argument can be applied.

In the vacuum case with positive cosmological constant, the result of Friedrich discussed in Section 5.1 proves isotropization of inhomogeneous spacetimes, i.e. that all generalized Kasner exponents corresponding to a suitable spacelike foliation tend to 1/3 in the limit. To see this, consider (part of) the de Sitter metric in the form - dt2 + e2t(dx2 + dy2 + dz2). (Here, to simplify the algebra, we have chosen /\ = 3.) This choice of the metric form, which is different from that discussed in [141], simplifies the algebra as much as possible. Letting t = e-t shows that the above metric can be written in the form -2 2 2 2 2 t (- dt + dx + dy + dz ). This exhibits the de Sitter metric as being conformal to a flat metric. In the construction of Friedrich the conformal class and conformal factor are perturbed. The corrections to the metric in terms of coordinate components are of relative order t = e -t. Thus, the trace-free part of the second fundamental form decays exponentially, as desired.

Inflationary asymptotics has been proved in the case of inhomogeneous solutions of the Einstein-Vlasov system with positive cosmological constant and three Killing vectors. This was done under the assumption of plane symmetry in [338Jump To The Next Citation Point] and for a restricted class of spherically symmetric solutions in [337]. The spacetimes were shown to be future geodesically complete and to have an asymptotic behaviour which resembles that of the de Sitter solution in leading order. Detailed information was obtained on the asymptotics of the matter fields. The results of [340] on local existence and continuation criteria for solutions of the Einstein-Vlasov-scalar field system can be thought of as a first step towards generalizing the results of [338] by replacing the cosmological constant by a scalar field.

There have been several numerical studies of inflation in inhomogeneous spacetimes. These are surveyed in Section 3 of [26]. An interesting effect which can occur in the inhomogeneous case is the formation of domain walls. Consider a potential which has two minima and suppose that the evolution at different spatial points decouples at late times. Then it may happen that in one spatial region the scalar field falls into one minimum of the potential while in another region it falls into the other minimum. In between the spatial derivatives must be relatively large in a small region forming the boundary of the two regions. This boundary is a domain wall. It would be very interesting to prove the formation of domain walls in some case.

There are heuristic results on the asymptotics of inhomogeneous solutions which are general in the sense that they have no symmetry and depend on the same number of free functions as the general solution. In [326Jump To The Next Citation Point] this kind of analysis was done for solutions of the Einstein equations in vacuum or coupled to a fluid with non-stiff linear equation of state. The mathematical interpretation of the formulae of [326Jump To The Next Citation Point] was elucidated in [304Jump To The Next Citation Point] where an analysis was done in the framework of formal power series. Compare also the results of Appendix B in [227Jump To The Next Citation Point]. In the vacuum case it was shown that there are large classes of solutions for which these series converge. A formal series analysis analogous to that of [304] was done for a scalar field with a positive minimum in [57]. It was also extended to the case of curvature-coupled scalar fields. An analysis on the level of [326] has been done for a scalar field with an exponential potential in [251].

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