Cosmological perturbation theory, part 2

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.


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