Mixmaster universe in fourth-order gravity theories

Abstract

We analyze the behavior of the Bianchi type-IX cosmological model in the full fourth-order theory of gravity in four space-time dimensions on approach to the singularity both analytically and numerically. By directly analyzing the full fourth-order field equations we prove explicitly that the Belinski-Khalatnikov-Lifshitz (BKL) analytic solution (and the accompanying bounce law) is also a solution to the more general case that we consider. But it appears to be nongeneric, because a perturbation analysis shows that power-law Kasner asymptotes are unstable on approach to the singularity. On the other hand, we find that the model possesses a stable isotropic monotonic power-law asymptotic solution and accordingly we show that there is a general solution which is nonchaotic near the space-time singularity. Numerical experiments confirm the existence of such a behavior. The inclusion of matter fields does not alter the evolution in the neighborhood of the singularity. The role of space-time dimensionality on the chaotic evolution of the mixmaster universe in these theories is also spelled out. In particular, we show that it is impossible to build a mixmaster universe in space-time dimensions higher than four in the full fourth-order gravity theory based on the BKL approximation scheme.