Periodic orbits in Ho\v{r}ava-Lifshitz cosmologies

TL;DR

This paper investigates the dynamics of Hořava-Lifshitz cosmological models, revealing new periodic orbits in certain cases that differ from traditional BKL behavior, using both analytical and computer-assisted methods.

Contribution

It introduces a computer-assisted approach to prove the existence of periodic orbits in Hořava-Lifshitz cosmologies, expanding understanding beyond classical BKL dynamics.

Findings

Existence of periodic orbits in type VIII and IX models.

New behavior not explained by BKL Kasner bouncing.

Periodic orbits are far from the Mixmaster attractor.

Abstract

We consider spatially homogeneous Ho\v{r}ava-Lifshitz (HL) models that perturb General Relativity (GR) by a parameter $v\in (0,1)$ such that GR occurs at $v=1/2$. We describe the dynamics for the extremal case $v=0$, which possess the usual Bianchi hierarchy: type $\mathrm{I}$ (Kasner circle of equilibria), type $\mathrm{II}$ (heteroclinics that induce the Kasner map) and type $\mathrm{VI_0},\mathrm{VII_0}$ (further heteroclinics). For type $\mathrm{VIII}$ and $\mathrm{IX}$, we use a computer-assisted approach to prove the existence of periodic orbits which are far from the Mixmaster attractor and thereby we obtain a new behaviour which is not described by the BKL picture of bouncing Kasner-like states.