Homoclinic chaos in axisymmetric Bianchi-IX cosmological models with an ad hoc quantum potential

In this work we study the dynamics of the axisymmetric Bianchi-IX cosmological model with a term of quantum potential added. As it is well known, this class of Bianchi-IX models is homogeneous and anisotropic with two scale factors, $A(t)$ and $B(t)$, derived from the solution of Einstein’s equation for general relativity. The model we use in this work has a cosmological constant and the matter content is dust. To this model we add a quantum-inspired potential that is intended to represent short-range effects due to the general relativistic behavior of matter in small scales and play the role of a repulsive force near the singularity. We find that this potential restricts the dynamics of the model to positive values of $A(t)$ and $B(t)$ and alters some qualitative and quantitative characteristics of the dynamics studied previously by several authors. We make a complete analysis of the phase space of the model finding critical points, periodic orbits, stable/unstable manifolds using numerical techniques such as Poincaré section, numerical continuation of orbits, and numerical globalization of invariant manifolds. We compare the classical and the quantum models. Our main result is the existence of homoclinic crossings of the stable and unstable manifolds in the physically meaningful region of the phase space [where both $A(t)$ and $B(t)$ are positive], indicating chaotic escape to inflation and bouncing near the singularity.

Figure 1

Phase space portrait of the invariant plane $A=B$, $p_A=p_B/2$. (a) The separatrices $S$ are characterized by the energy $E_0=E_{\mathrm{crit}}=1/\sqrt{4\Lambda }$. The point $A=B=0$ is a degenerate singularity. (b) The invariant plane for $\sigma \ne 0$: note that the quantum term removes the singularity $A=B=0$ and the trajectories bounce back in the vicinity of this point.

Figure 2

Projection of the center manifold with $\sigma =0.01$: (top) $(A,p_A)$ plane; (bottom) $(B,p_B)$ plane. Note that the periodic orbits are qualitatively similar to classical ones.

Figure 3

$(A,p_A)$ projections of unstable cylinder manifolds: branch of the unstable manifold which goes towards $A=0$: for (top) $\sigma =0$ and (bottom) $\sigma =0.01$. Note that the addition of the quantum potential makes the unstable manifold avoid the collapse by bouncing back towards the singularity.

Figure 4

From top to bottom, (top) $\sigma =0$ (classical case) four cuts $(A,p_A)$ of the unstable and unstable cylinder in the surface of section ($p_B=0$, ${\dot{p}}_B<0$), for $E_0=0.9706090$, 0.906 208 28, 0.707 897 78, and 0.566 396 00; (bottom) four cuts as above for $\sigma =0.01$ for the $E_0=1.0095581$, 1.007 060 88, 1.002 744 54, and 0.996 854 69. Note that the quantum term brings the Poincaré section of the tubes to the $A>0$ region while they are in the $A<0$ region in the classical case (see top figure).

Figure 5

$(A,p_A)$ first iteration of the Poincaré map in the surface of section ($p_B=0$, ${\dot{p}}_B<0$) of orbits on the unstable and stable cylinder manifolds to the periodic orbit (p.o.) with $E_0=1.00706088$. Note that they are mirror images of each other with respect to $p_A=0$ due to the symmetry $(A,p_A,t)\leftarrow (A,-p_A,-t)$ in the Poincaré surface of section. The two intersections represent the one-turn homoclinic orbits.

Figure 6

$(A,p_A)$ (top) second and (bottom) third iterations of the Poincaré map in the surface of section ($p_B=0$, ${\dot{p}}_B<0$) of orbits on the unstable manifolds to the p.o. with $E_0=1.00706088$. Note that the external shape of the second cut is practically the same as the first cut shown in Fig. 5. The external part of the third cut mimics the second one.

Figure 7

$(A,p_A)$ superposition of the first and second iterations of the Poincaré map in the surface of section ($p_B=0$, ${\dot{p}}_B<0$) of orbits on the unstable and stable manifolds to the p.o. with $E_0=1.00706088$.

Figure 8

The unstable manifold branch (top) at the onset of very large values of $p_A$ at $E_0=1.00274454$; (bottom) for $E_0=0.99585469$, that is, for a larger value of energy the density of trajectories that approach $A=0$ increases, but part of them approach the periodic orbit and escape to the de Sitter region.