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5.1 Stability of de Sitter space

In [141Jump To The Next Citation Point], Friedrich proved a result on the stability of de Sitter space. He gives data at infinity but the same type of argument can be applied starting from a Cauchy surface in spacetime to give an analogous result. This concerns the Einstein vacuum equations with positive cosmological constant and is as follows. Consider initial data induced by de Sitter space on a regular Cauchy hypersurface. Then all initial data (vacuum with positive cosmological constant) near enough to these data in a suitable (Sobolev) topology have maximal Cauchy developments that are geodesically complete. The result gives much more detail on the asymptotic behaviour than just this and may be thought of as proving a form of the cosmic no hair conjecture in the vacuum case. (This conjecture says roughly that the de Sitter solution is an attractor for expanding cosmological models with positive cosmological constant.) This result is proved using conformal techniques and, in particular, the regular conformal field equations developed by Friedrich. An alternative proof of this result which extends to all higher even dimensions was given in [6]. For some comments on the case of odd dimensions see [304Jump To The Next Citation Point].

There are results obtained using the regular conformal field equations for negative or vanishing cosmological constant [143146Jump To The Next Citation Point], but a detailed discussion of their nature would be out of place here (cf. however Section 9.1).


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