The last post was sparked off by a talk I heard at a conference in Oberwolfach. Here I will write about a topic where another talk at that conference looks like a big step forward. This was by Jared Speck. He was describing work of his with Igor Rodnianski which is not yet fully written up.
These days there is a wide consensus among astrophysicists that there is strong observational evidence to indicate that the expansion of the universe is accelerated. In other words it is not only the case that all distant galaxies are moving away from us (and from each other) but the velocity of recession is actually increasing. In the standard view this is only consistent with general relativity if there is a positive cosmological constant or some exotic matter called dark energy. For convenience I will not distinguish between these two in what follows. Dark energy leads to accelerated expansion and accelerated expansion causes spatial irregularities to be damped. The
geometry of spacetime and the matter distribution are smoothed. This kind of idea can be turned into a precise mathematical statement (maybe not uniquely) called the cosmic no hair theorem. From a mathematical point of view this is rather a conjecture than a theorem – at least it has been that way for most of the time it has existed. The name originates from a phrase of John Wheeler, ‘a black hole has no hair’. The idea of this was that a particular solution of the Einstein equations describing a black hole, the Kerr solution, should be attractor for the evolution of more general solutions containing a black hole. In other words a general class of solutions should evolve so as to look more and more like the Kerr solution. The Kerr solution depends only on two parameters. Thus in this scenario all the details get lost dynamically, leaving a very simple object with no complicated features, no hair. In models for an expanding universe with positive cosmological constant the smoothing process mentioned above also seems to drive all solutions towards an attractor, the de Sitter solution. It is this analogy which gave rise to the name ‘cosmic no hair theorem’.
The mathematical formalization of the cosmic no hair theorem says that a solution of the Einstein-matter equations with positive cosmological constant converges to the de Sitter solution at late times in a suitable sense. A weaker statement is that this should be true for solutions which start close to the de Sitter solution. The latter version can also be thought of as a kind of stability statement for de Sitter space. In the case of the vacuum Einstein equations the stability of de Sitter space was proved by Helmut Friedrich in 1986. Since our universe is certainly not empty the relevance of this result to cosmology is not immediately obvious. It turns out, however, that there are reasons to believe that the cosmological constant can often have a dominant effect on the late-time cosmological expansion which tends to make the effect of the matter into a higher-order correction. It is important to confirm these ideas by a theorem which includes the effect of matter. The most commonly used matter model in cosmology is a perfect fluid with linear equation of state. It contains a parameter which is often restricted by an inequality corresponding to perfect fluids which are less stiff than radiation. The result of Rodnianski and Speck is a form of the cosmic no hair theorem for precisely this class of matter models. The proofs build on previous work of Hans Ringström. Friedrich’s proof uses a technique (the conformal method) which is very powerful but rather rigid. It is difficult to see how to modify the proof to include matter such as a fluid, or indeed to replace the cosmological constant by some other kind of dark energy, such as a nonlinear scalar field. Ringström introduced more flexible methods which allowed him to obtain a version of the cosmic no hair theorem for dark energy modelled by certain types of nonlinear scalar field. His methods open up the perspective of including matter and this is what Rodnianski and Speck have now done. These methods use energy estimates, the workhorse of the theory of nonlinear hyperbolic equations, in a clever way. (I might say more clever than I am, since I once tried very hard to do this, without success.)
The result of Rodnianski and Speck is restricted to the case of irrotational fluids. I see no fundamental reason why this should be necessary. Nevertheless there is a clear technical reason – in the irrotational case the Euler equation of the fluid is equivalent to a nonlinear wave equation. On the level of formal power series the case with rotation works out, as shown in a paper of mine (Ann. H. Poincare 5, 1041). Another question is what happens for large data. In that case there are various restrictions.For sufficiently large data it is to be expected that black holes would be formed (even in the vacuum case). Moreover, the fluid can be expected to form shocks which means that the solution cannot be continued, at least in the realm of smooth solutions. I find it remarkable that the expansion caused by a positive cosmological constant is strong enough to suppress formation of shocks in a small data regime. There is just one result available on this subject for large initial data and inhomogeneous solutions. In this work, due to Blaise Tchapnda and myself (arising from Blaise’s PhD thesis, Class. Quantum Grav. 20, 3037) we treated plane-symmetric solutions of the Einstein-Vlasov system with positive cosmological constant. In this case the symmetry prevents formation of black holes and the choice of matter model allows any analogue of shocks to be avoided.