Onset of synchronization in coupled Mixmaster oscillators

TL;DR

This paper demonstrates that in inhomogeneous spacetime, coupled Mixmaster oscillators synchronize near the singularity, leading to a unified, simpler state of the universe through exponential synchronization driven by a Lyapunov function.

Contribution

It introduces the concept of early-time synchronization of Mixmaster oscillators, providing an elementary proof and physical interpretation of how spatial points synchronize in inhomogeneous spacetime.

Findings

Synchronization occurs for coupling above a threshold.

Exponential synchronization driven by a Lyapunov function.

Synchronization leads to convergence of BKL maps.

Abstract

We consider the problem of asymptotic synchronization of different spatial points coupled to each other in inhomogeneous spacetime and undergoing chaotic Mixmaster oscillations towards the singularity. We demonstrate that for couplings larger than some threshold value, two Mixmaster spatial points $A,B$, with $A$ in the past of $B$, 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.