Onset of synchronization in coupled Mixmaster oscillators

We consider the problem of asymptotic synchronization of different spatial points coupled to each other in inhomogeneous space-time and undergoing chaotic Mixmaster oscillations towards the singularity. We demonstrate that for couplings larger than some threshold value, two Mixmaster spatial points [Formula: see text], with [Formula: see text] in the past of [Formula: see text], synchronize and thereby proceed in perfect unison towards the initial singularity. We further show that there is a Lyapunov function for the synchronization dynamics that makes different spatial points able to synchronize exponentially fast in the past direction. We provide an elementary proof of how an arbitrary spatial point responds to the mean field created by the oscillators, leading to their direct interaction through spontaneous synchronization. These results ascribe a clear physical meaning of early-time synchronization leading to a resetting effect for the two BKL maps corresponding to two distinct oscillating spatial points, as the two maps converge to each other to become indistinguishable at the end of synchronization. Our results imply that the universe generically organizes itself through simpler, synchronized, states as it approaches the initial singularity. A discussion of further implications of early-time inhomogeneous Mixmaster synchronization is also provided. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.