Wald's result is only dependent on energy conditions and uses no details of the matter field equations. The question remains whether solutions corresponding to initial data for the Einstein equations with positive cosmological constant, coupled to reasonable matter, exist globally in time under the sole condition that the model is originally expanding. It can be shown that this is true for various matter models using the techniques of . Suppose we have a solution on an interval . It follows from  that the mean curvature is increasing and no greater than . Hence, in particular, is bounded as t approaches . Now we wish to verify condition (7) of . This says that if the mean curvature is bounded as an endpoint of the interval of definition of a solution is approached then the solution can be extended to a longer interval. As in  it can be shown that if is bounded, then , , and are bounded. Thus, in the terminology of , it is enough to check (7)' for a given matter model in order to get the desired global existence theorem. This condition involves the behaviour of a fluid in a given spacetime. Since the Euler equation does not contain , the result of  applies directly. It follows that global existence holds for perfect fluids and mixtures of non-interacting perfect fluids. A similar result holds when the matter is described by collisionless matter satisfying the Vlasov equation. Here it suffices to note that the proof of Lemma 2.2 of  generalizes without difficulty to the case where a cosmological constant is present.
The effect of a cosmological constant can be mimicked by a suitable exotic matter field that violates the strong energy condition: for example, a nonlinear scalar field with exponential potential. In the latter case, an analogue of Wald's theorem has been proved by Kitada and Maeda . For a potential of the form with smaller than a certain limiting value, the qualitative picture is similar to that in the case of a positive cosmological constant. The difference is that the asymptotic rate of decay of certain quantities is not the same as in the case with positive . In  it is discussed how the limiting value of can be increased. The behaviour of homogeneous and isotropic models with general has been investigated in .
Both models with a positive cosmological constant and models with a scalar field with exponential potential are called inflationary because the rate of (volume) expansion is increasing with time. There is also another kind of inflationary behaviour that arises in the presence of a scalar field with power law potential like or . In that case the inflationary property concerns the behaviour of the model at intermediate times rather than at late times. The picture is that at late times the universe resembles a dust model without cosmological constant. This is known as reheating. The dynamics have been analysed heuristically by Belinskii et al. . Part of their conclusions have been proved rigorously in . Calculations analogous to those leading to a proof of isotropization in the case of a positive cosmological constant or an exponential potential have been done for a power law potential in . In that case, the conclusion cannot apply to late time behaviour. Instead, some estimates are obtained for the expansion rate at intermediate times.
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 , 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 local homogenization 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 . This choice, which is different from that discussed in , simplifies the algebra as much as possible. Letting shows that the above metric can be written in the form . 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 . Thus, the trace-free part of the second fundamental forms decays exponentially, as desired.
There have been several numerical studies of inflation in inhomogeneous spacetimes. These are surveyed in Section 3 of .
|Theorems on Existence and Global Dynamics for the
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
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