Chaotic dynamics of the Bianchi IX universe in Gauss-Bonnet gravity

We investigate the dynamics of a closed Friedmann-Robertson-Walker universe and anisotropic Bianchi type-IX universe characterized by two scale factors in a gravity theory including a higher curvature (Gauss-Bonnet) term. The presence of the cosmological constant creates a critical point of saddle type in the phase space of the system. An orbit starting from a neighborhood of the separatrix will evolve toward the critical point, and it eventually either expands to the de Sitter space or collapses to the big crunch. In the closed Friedmann-Robertson-Walker model, the dynamics is reduced to hyperbolic motions in the two-dimensional center manifold, and the system is not chaotic. In the anisotropic model, anisotropy introduces the rotational mode, which interacts with the hyperbolic mode to present a cylindrical structure of unstable periodic orbits in the neighborhood of the critical point. Due to the nonintegrability of the system, the interaction of rotational and hyperbolic modes makes the system chaotic, making it impossible for us to predict the final fate of the universe. We find that the chaotic dynamics arises from the fact that orbits with even small perturbations around the separatrix oscillate in the neighborhood of the critical point before finally expanding or collapsing. The chaotic character is also evidenced by the fractal structures in the basins of attraction.

Figure 1

Phase space portrait of the invariant manifold $\mathcal{M}$ in conformal time $\mathrm{d}\eta =\mathrm{d}t/a$. The orbit $S$ is the separatrix and the point $P$ is the critical point.

Figure 2

Escape to inflation of 100 orbits around a point on the separatrix with coordinates $a=b=0.9$, $\dot{a}=\dot{b}=0.0518187725$, and $\sigma =\dot{\sigma }=0$. The coupling constant is $\lambda =16$. The energy surface is given by $E_0=0.4999999999$ and the radius of the sphere of initial conditions is $\delta ={10}^{-4}$.

Figure 3

Collapse of 100 orbits around the same point in Fig. 2 with $\lambda =16$, $E_0=0.4999999802$, and $\delta ={10}^{-4}$.

Figure 4

A magnified view of 100 orbits around the critical point in the $(a,\dot{a})$ plane for $\delta ={10}^{-4}$ and $E_0=0.4999999975$.

Figure 5

The basins of attraction in the $(\sigma ,\dot{\sigma })$ plane for the closed FRW model. The slice is through $a=0.9$ and $\dot{a}$ is fixed by the Hamiltonian constraint. We chose the parameters $\lambda =16$ and $E_0=0.4999999975$. The black regions correspond to collapse and the white regions inflation.

Figure 6

The basins of attraction in the $(\sigma ,\dot{\sigma })$ plane for the axisymmetric Bianchi IX model. The slice is through $a=b=0.9$, $\dot{a}=0.0518187725$, and $\dot{b}$ is fixed by the Hamiltonian constraint. The other parameters are the same as in Fig. 5.

Figure 7

A magnified view of Fig. 6.

Figure 8

The basins of attraction in the $(a,\dot{a})$ plane.