Abstract

In cosmology an important role is played by homogeneous and isotropic solutions of the Einstein-Euler equations and linearized perturbations of these. In this talk I present results of Paul Allen and myself on the asymptotic behaviour of certain linearized solutions (so-called ‘scalar perturbations’) both in the approach to the initial singularity and at late times. The main equation of interest is a linear hyperbolic equation whose coefficients depend only on time. Expansions for the solutions are obtained in both asymptotic regimes. In both cases it is shown how general solutions can be parametrized by certain functions which are coefficients in the asymptotic expansion. The central technique in the proofs is that of energy estimates. An interesting feature of the results is a bifurcation observed in the late-time asymptotics for an asymptotically polytropic equation of state, occurring for the polytropic index corresponding to a radiation fluid.