Archive for June, 2009

The Mixmaster model

June 24, 2009

The Mixmaster model arises in the study of solutions of the vacuum Einstein equations which are spatially homogeneous. To say that a solution is spatially homogeneous means that it is invariant under a group of symmetries whose orbits are three-dimensional and spacelike. This implies that it can be described by functions depending only on time and that the Einstein equations reduce to a system of ordinary differential equations. The details of these equations depend on the group defining the symmetry. In all but one case this group can be taken to be three-dimensional. Then classifying the symmetry type essentially comes down to classifying all three-dimensional Lie algebras.This was done by Bianchi in the 1890’s. He introduced types I to IX and this terminology is used in the study of the Einstein equations to this day. Types VI and VII are actually one-parameter families of non-isomorphic Lie algebras and are therefore denoted by VI{}_h and VII{}_h, where h is the parameter. The most general Bianchi types (in the sense that the solutions depend on the largest number of parameters) are types VIII, IX and VI{}_{-\frac19}. All Bianchi types contain solutions with initial singularities and the three types just listed exhibit complicated oscillatory behaviour in the approach to the singularity. The case which has been studied most is Bianchi IX and this was christened the Mixmaster model by Charles Misner in 1969. He named it after a kind of food mixer which was popular at the time. There has been a great deal of heuristic and numerical work on the dynamics of this system of ODE. Rigorous mathematical results have been relatively rare. The most notable results in this direction are those arising from the PhD thesis of Hans Ringström, some of which will now be described.

It is useful in the study of Bianchi models to distinguish between the Lie algebras which are unimodular (trace of the structure constants zero) and the rest. These are called Class A and Class B respectively. All models of Class A can be incorporated into a single dynamical system. The dynamical systems for types VIII and IX occupy open subsets of the state space and the other types are represented on lower dimensional submanifolds on the boundaries of these open sets. Class A consists of the types I, II, VI{}_0, VII{}_0, VIII and IX. It was shown by Ringström that the limit points of solutions of Bianchi type IX in the approach to the singularity (technically the \alpha-limit points) are of type I and II. For almost all initial data it was proved that the approach to the singularity is oscillatory in the sense that there are at least three distinct \alpha-limit points. More details can be found in this paper. The wider context of these results and aspects of the history of the problem are discussed in the paper ‘Mixmaster: Fact and Belief‘ by Mark Heinzle and Claes Uggla.

Recently there has been renewed interest in these problems. A project which I have started in cooperation with the research group of Bernold Fiedler has generated a lot of activity. The central idea is to bring together knowledge about the concrete application with sophisticated techniques from the theory of dynamical systems. What are the problems to be solved? A first question is whether the direct analogues of the Bianchi IX results hold for type VIII. This has not been proved and there are serious reasons for doubting if it is even true. The key issue, at least at our present state of understanding is whether Bianchi VIII solutions may have \alpha-limit points of type VI{}_0. For type VI{}_{-\frac19} almost nothing is known about the corresponding questions on a rigorous level.

Misner’s original motivation for introducing the Mixmaster model was to find an explanation for the observed isotropy of the universe. Whether this explanation can work depends on the nature of the causal structure of the geometry near the singularity. These days it is usual to explain the isotropy of the universe on the basis of inflation and so Misner’s proposal no longer has the same direct physical significance. The mathematical problem remains and has lost none of its intrinsic fascination. In this context it is desirable to understand the detailed structure of the \alpha-limit sets of generic and non-generic solutions. In particular there are three special points in the phase space, the flat Kasner solutions. Whether Misner’s mechanism is effective depends on how much time solutions spend close to these special points. Quantifying this requires a very detailed understanding of the asymptotics of solutions near the singularity. This issue is under intensive investigation. I hope to write about the results here when they are ready to come into the public domain.

These questions are of interest for the understanding of solutions far beyond the homogeneous case. This has to do with the picture of general singularities in solutions of the Einstein equations developed by Belinkii, Khalatnikov and Lifshitz, which is one of the most outstanding challenges in mathematical relativity.


Cosmological perturbation theory, part 2

June 16, 2009

In a previous post which I wrote several months ago I promised some more information about the work which Paul Allen and I have been doing on cosmological perturbation theory. Next Thursday I will give a talk on the subject at a conference in Lisbon. This work has been delayed by other commitments but now we posted a paper as arXiv:0906.2517. Here I will explain some of the results. In this paper we study the equation which describes so-called scalar perturbations. I will not attempt to explain the name but instead just say that these are the type of perturbations which cosmologists use to describe processes like the formation of the distribution of galaxies. In the simplest case these are described by the equation \Phi''+\frac{6(1+w)}{1+3w}\frac1{\eta}\Phi'=w\Delta\Phi. Here \Phi is a real valued function on {\rm R}\times T^3 where T^3 is the three-dimensional torus. The prime denotes a derivative with respect to a time coordinate \eta and \Delta is the Laplacian. The constant parameter w comes from the equation of state of the fluid, assumed to be of the form p=w\rho.

The aim of the paper is to obtain information about the asymptotics of general solutions of this equation in the limits t\to 0 and t\to\infty. It may be noted that this equation bears a certain resemblance to the polarized Gowdy equation. In fact we are able to import a number of techniques which have been used in the study of the Gowdy equations to understand the solutions of the equation above. The solutions can be parametrized by certain asymptotic data in each asymptotic regime. For the limit t\to 0 this data consists of two free functions which are coefficients in an asymptotic expansion of the form \sum_i\Phi_i(\eta) \zeta_i(t). For the limit t\to\infty it turns out to be useful to distinguish between solutions which are constant on the hypersurfaces of constant \eta and those whose integral over the torus is zero for each fixed \eta. Here I restrict consideration to the latter. It then turns out that after rescaling by a suitable power of \eta the solution looks like a solution of the flat space wave equation plus a remainder term. Thus in fact the asymptotics in each direction bears a strong qualitative resemblance to the asymptotics in the corresponding direction for solutions of the polarized Gowdy equations.

The paper also proves results about more general equations of state. Of course in that case the equation above is replaced by a more complicated one. It is assumed that in the appropriate limit the equation of state is the sum of a linear term and an expression which has an asymptotic expansion in powers of the energy density \epsilon which are negligible with respect to linear terms in the given regime. In many cases the modification does not make much of a difference and the leading order asymptotics is the same as in the corresponding linear case. An exception is the late time behaviour when the coefficient of the linear term vanishes so that the leading term in f(\epsilon) in the limit \epsilon\to 0 is proportional to \epsilon^{1+\sigma} for some positive \sigma. There is a bifurcation at \sigma=\frac13. When \sigma is smaller than this value the asymptotics is similar to that in the linear case. It does however happen that in the leading term the time coordinate in the solution of the flat space wave equation has to be distorted. For \sigma>\frac13 there is a more radical change in the asymptotics. In that case the behaviour looks more like what happens in the limit t\to 0. Waves which continue to propagate for ever are replaced by waves which freeze.In a sense the natural time coordinate for describing the dynamics is one which brings infinity to a finite value. This is reminiscent of the behaviour of the gravitational field in perturbed de Sitter spacetimes.

The Vlasov-Poisson system

June 4, 2009

The Vlasov-Poisson system is a system of partial differential equations which comes up in mathematical physics. I have been involved quite a bit with these equations and related systems for many years now. In this post I want to reflect a little on what is and is not known about solutions of this system. One of the things which has stimulated me to think more about these questions just now is a lecture course on kinetic equations which I am giving at the Free University in Berlin. Because of the physics motivation the Vlasov-Poisson system is usually studied in three space dimensions. Here I will allow the space dimension n to be general. For convenience I also introduce a parameter \gamma which can take the values +1 and -1. The equations are \partial_t f+v\cdot \nabla_x f+\gamma\nabla_x U\cdot\nabla_v f=0 and \Delta U=\rho where \rho=\int f dv. Here t is time and x and v denote position and velocity variables, each belonging to {\bf R}^n. \Delta is the Laplacian. Because of their most frequent applications the cases \gamma=1 and \gamma=-1 are often called the plasma physics case and the stellar dynamics case respectively. A natural problem here is the initial value problem (Cauchy problem) with data prescribed for f.

Local existence in the Cauchy problem is known. For n\le 3 it is furthermore known that the local solution extends to a global in time solution, independently of the sign of \gamma. (The first proofs by two different methods were given by Pfaffelmoser and Lions/Perthame around 1991.) When n\ge 4 and \gamma=-1 there are large classes of smooth initial data for which global existence fails. More specifically, these equations have a conserved energy and when this energy is negative the corresponding smooth solution breaks down after finite time. The easiest way to realize that n=4 might be important is to look at scaling properties of the equations. For a discussion of the significance of scaling properties in general see Terry Tao’s post on the Navier-Stokes equations. In the case n=3 the potential and kinetic energies satisfy an inequality of the form |{\cal E}_{\rm pot}|\le C{\cal E}_{\rm kin}^{\frac12} and this plays an important role in the global existence proof. The essential feature is that the power on the right hand side is less than one. If similar arguments are carried out in the case n=4 then the power one half is replaced by the power one. Thus in a sense n=4 is the critical case. For n\ge 4 the global existence problem for the Vlasov-Poisson system with \gamma=1 is open. For n=4 it is a critical problem and might be solvable in a not too distant future. Similar remarks might be made about the relativistic Vlasov-Poisson system with massless particles in three space dimensions which is given by \partial_t f+\hat v\cdot \nabla_x f+\nabla_x U\cdot\nabla_v f=0 where \hat v=\frac{v}{|v|}. The analogue of this last system plays an important role in the recent work of Lemou, Méhats and Raphaël on the nature of singularities in solutions of the relativistic Vlasov-Poisson system with \gamma=-1.

Other open questions concern the behaviour of solutions of the Vlasov-Poisson system at late times. There are various results on this but they seem to be far from an exhaustive understanding of the asymptotics. Interesting questions include whether the density \rho is bounded any solution with \gamma=-1 and whether \|\rho\|_{L^\infty}=O(t^{-3}) in the case \gamma=1, as is known to be the case for small initial data.