The Bianchi IX Attractor in Modified Gravity

Abstract

We consider vacuum anisotropic spatially homogeneous models in certain modified gravity theories (such as Hořava-Lifshitz, $λ$-$R$ 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 Ringström 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).

Figures (5)

• Figure 2.1: Left: Bianchi type II solutions are heteroclinic orbits in the hemisphere $\mathcal{B}_{N_1}$ with $\alpha$-limit sets within int$(A_1)$ and $\omega$-limit sets in $A_1^c$. Right: Projection of the Bianchi type II solutions into the $\Sigma$-plane.
• Figure 2.2: The common stable set $S_{\mathrm{VI}_0,\mathrm{VII}_0}$ for Bianchi types $\mathrm{VI}_0$ and $\mathrm{VII}_0$. In addition, projected onto $(\Sigma_+,\Sigma_-)$-space, there are illustrative heteroclinic chains located on the $\mathrm{II}_2\cup\mathrm{II}_3\cup\mathrm{K}^\ocircle$ boundary. In particular, $v\in(1/2,1)$ admits a heteroclinic cycle/chain with period 2, which resides on the projected line between $\mathrm{Q}_2/v$ and $\mathrm{Q}_3/v$ characterized by $\Sigma_+=-1/(2v)$.
• Figure 2.3: The constrained two-dimensional dynamics of $\mathcal{LRS}_1$ (gray) described by the equation \ref{['LRSIXeq']} and its corresponding boundary sets (bold): the lines $\mathrm{LRS}^\pm$ in \ref{['LRS']} and the heteroclinic from $\mathrm{Q}_1$ to $\mathrm{T}_1$ within $\mathcal{B}_{N_1}$ in \ref{['BII_N_1']}. The dynamics of each case (subcritical, critical or supercritical) is described in Lemma \ref{['lem:LRS3d']}
• Figure 3.1: Numerical stable manifold $W^{\mathrm{ss}}(p_{\mathrm{VI}_0})$ for $v = 0.75$ from Lemma \ref{['lem:strong-stable-pVI']}, which consists of two symmetric solutions with initial data satisfying \ref{['Wsspviv3']} with $\Sigma_-=10^{-10}$ and $\Sigma_-= - 10^{-10}$. Left: Projection of the solutions into the $\Sigma$-plane containing the Kasner circle, the fixed point $p_{\mathrm{VI}_0}$ (gray dot), and the numerical $\alpha$-limit sets (blue dots) of the set $W^{\mathrm{ss}}(p_{\mathrm{VI}_0})$. Here, the light gray disks represent regions where the cross-terms $N_\alpha N_\beta$ grow, see HellLappicyUggla; in particular, $p_{\mathrm{VI}_0}$ lies in the extremum of a disk, whereas the $\alpha$-limit sets lie in the intersection of two disks. Middle and right: The backwards evolution of the solution with initial data $\Sigma_-=10^{-10}$ with variables $\Sigma_-,\Sigma_+$ (which are bounded) and $N_1,N_2,N_3$ ($N_3$ goes to 0 and $N_1\approx N_2$ grow).
• Figure 3.2: Numerical stable manifold $W^{\mathrm{ss}}(p_{\mathrm{VI}_0})$ as $v \to 1/2$. Left: Projection of the solutions into the $\Sigma$-plane. Middle and right: For $v=0.501$, the backwards evolution of the solution with initial data $\Sigma_-=10^{-10}$ and variables $\Sigma_-,\Sigma_+$ (which are bounded), $N_3$ (which tracks a type $\mathrm{II}$ or $\mathrm{VI}_0$ trajectory and then goes to 0), and $N_1,N_2$ (which grow following a LRS solution, $N_1\approx N_2$).

Theorems & Definitions (7)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6