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The subject of the article is linear systems of wave equations oncosmological backgrounds with convergent asymptotics. The condition ofconvergence corresponds to the requirement that the second fundamental form,when suitably normalised, converges. The model examples are the Kasnersolutions. The main result of the article is optimal energy estimates. However,we also derive asymptotics and demonstrate that the leading order asymptoticscan be specified.  It is sometimes argued that if the factors multiplying the spatialderivatives decay exponentially (for a system of wave equations), then thespatial derivatives can be ignored. This line of reasoning is incorrect: wegive examples of equations such that  1) the factors multiplying the spatial derivatives decay exponentially,  2) the factors multiplying the time derivatives are constants,  3) the energies of individual modes of solutions asymptotically decayexponentially, and  4) the energies of generic solutions grow as $e^{e^{t}}$ as $t\rightarrow\infty$.  When the factors multiplying the spatial derivatives grow exponentially, theFourier modes of solutions oscillate with a frequency that grows exponentially.To obtain asymptotics, we fix a mode and consider the net evolution over oneperiod. Moreover, we replace the evolution (over one period) with a matrixmultiplication. We cannot calculate the matrices, but we approximate them. Toobtain the asymptotics we need to calculate a matrix product where there is nobound on the number of factors, and where each factor can only be approximated.Nevertheless, we obtain detailed asymptotics. In fact, it is possible toisolate an overall behaviour (growth/decay) from the (increasingly violent)oscillatory behaviour. Moreover, we are also in a position to specify theleading order asymptotics.

Linear systems of wave equations on cosmological backgrounds with convergent asymptotics