The Bianchi IX Attractor in Modified Gravity

TL;DR

This paper extends the understanding of vacuum anisotropic models in modified gravity theories, showing that Bianchi IX solutions exhibit Mixmaster-like chaotic behavior similar to general relativity, with unique convergence properties.

Contribution

It proves an analogue of the Ringström attractor theorem for modified gravity models, demonstrating convergence to Mixmaster attractors in these theories.

Findings

Solutions converge to Mixmaster attractors in modified gravity.

No solutions converge to non-Mixmaster sets in these theories.

The behavior generalizes known GR results to a broader class of gravity models.

Abstract

We consider vacuum anisotropic spatially homogeneous models in certain modified gravity theories (such as Ho\v{r}ava-Lifshitz, $\lambda$-$R$ or $f(R)$ gravity), which are expected to describe generic spacelike singularities for these theories. These models perturb the well-known Bianchi models in general relativity (GR) by a parameter $v\in (0,1)$ with GR recovered at $v=1/2$. We prove an analogue of the well-known Ringstr\"om attractor theorem in GR to the supercritical theories: for any $v\in (1/2,1)$, all solutions of Bianchi type $\mathrm{IX}$ converge to an analogue of the Mixmaster attractor, consisting of Bianchi type I solutions (Kasner states) and heteroclinic chains of Bianchi type II solutions. In contrast to GR, there are no solutions that converge to a different set other than the Mixmaster (such as the locally rotationally symmetric solutions in GR).