Hořava-Lifshitz bouncing Bianchi IX universes: A dynamical system analysis

We examine the Hamiltonian dynamics of bouncing Bianchi IX cosmologies with three scale factors in Hořava-Lifshitz (HL) gravity. We assume a positive cosmological constant plus noninteracting dust and radiation as the matter content of the models. In this framework the modified field equations contain additional terms which turn the dynamics nonsingular. The six-dimensional phase space presents (i) two critical points in a finite region of the phase space, (ii) one asymptotic de Sitter attractor at infinity and (iii) a two-dimensional invariant plane containing the critical points; together they organize the dynamics of the phase space. We identify four distinct parameter domains $A$, $B$, $C$ and $D$ for which the pair of critical points engenders distinct features in the dynamics, connected to the presence of centers of multiplicity 2 and saddles of multiplicity 2. In the domain $A$ the dynamics consists basically of periodic bouncing orbits, or oscillatory orbits with a finite number of bounces before escaping to the de Sitter attractor. The center with multiplicity 2 engenders in its neighborhood the topology of stable and unstable cylinders $R\times S^3$ of orbits, where $R$ is a saddle direction and $S^3$ is the center manifold of unstable periodic orbits. We show that the stable and unstable cylinders coalesce, realizing a smooth homoclinic connection to the center manifold, a rare event of regular/nonchaotic dynamics in bouncing Bianchi IX cosmologies. The presence of a saddle of multiplicity 2 in the domain $B$ engenders a high instability in the dynamics so that the cylinders emerging from the center manifold about $P_2$ towards the bounce have four distinct attractors: the center manifold itself, the de Sitter attractor at infinity and two further momentum-dominated attractors with infinite anisotropy. In the domain $C$ we examine the features of invariant manifolds of orbits about a saddle of multiplicity 2 $P_2$. The presence of the saddle of multiplicity 2 engenders bifurcations of the invariant manifold as the energy $E_0$ of the system increases relative to the energy $E_{c{r}_2}$ of $P_2$: (i) for $E_0<E_{c{r}_2}$ the invariant manifold has the topology $S^3$; (ii) for $E_0=E_{c{r}_2}$ two points of $S^3$ pinch into the point $P_2$, so that the invariant manifold contains infinitely many orbits homoclinic to $P_2$; (iii) for $E_0>E_{c{r}_2}$ the center manifold bifurcates into a 3-torus; (iv) for $E_0$ sufficiently large the 3-torus bifurcates into three $S^3$, an invariant manifold multiply connected. Such structures were not yet observed in the literature. The domain $D$ is not examined as most of its features are present already in the previous domains.

Figure 1

The invariant plane. Here we fixed the parameters $\mathrm{\Lambda }=1$, $A_2=0.05$ and $A_3=0.005$. Dashed, solid and dotted lines correspond to $E_0=0.250$, $E_0=0.2201517192605279$ and $E_0=0.185$, respectively. The second value of the energy is the energy of the critical point $P_2$, and the solid line constitutes a homoclinic connection of $P_2$ to itself. The critical points are given by $P_1=(0.2348551826828089,0)$ and $P_2=(0.51007113736321,0)$. The graph was made in the canonical variables $(x,p_x)$ introduced in Sec. 4.

Figure 2

The parameter space $({\alpha }_{31},{\alpha }_{32},{\alpha }_{21})$ and the four three-dimensional domains corresponding to the configurations ($A$), ($B$), ($C$) and ($D$) of the critical points.

Figure 3

Section $p_z=0,z=1$ (left) and section $p_y=0,y=1$ (right) of the center manifold $S^3$ (60), for $E_0=9.89$, 9.0, 6.0, 5.0, about the critical point $P_2$, with parameters (61) in ($A$). The energy of $P_2$ is $E_{c{r}_2}=9.89949505925050$.

Figure 4

Left: The evolution of the scale factor $x(t)$ for a periodic orbit having infinite bounces, with parameters (61) in ($A$) and initial conditions on a $S^1$ section of the center manifold. Right: The evolution of the oscillatory mode $p_z(t)$. We note that the frequency of the mode increases substantially as the orbit bounces.

Figure 5

Plot of the evolution of the anisotropy parameter ${\sigma }^2$ related to the periodic orbit with infinite bounces displayed in Fig. 4, showing a relatively large amplification in the oscillations when the orbit visits a neighborhood of the bounces.

Figure 6

The evolution of the scale factor $x(t)$ for an orbit with parameters (63) in ($A$) exhibiting six bounces before the orbit escapes to the de Sitter attractor at infinity.

Figure 7

The evolution of the oscillatory modes $p_z(t)$ (left) and $z(t)$ (right) of the orbit of Fig. 6. The frequency of the modes increases substantially in the neighborhood of the bounces. As the orbit approaches the de Sitter attractor the variables approach the constant values $(z\sim 1,p_z\sim 0)$ as expected.

Figure 8

Left: Plot of the evolution of the shear parameter ${\sigma }^2$ for the orbit of Fig. 6. Right: Amplification of the final part of the signal showing that the shear becomes zero as the orbit reaches the de Sitter attractor.

Figure 9

A numerical illustration of the unstable cylinder (gray) and the stable cylinder (black), both spanned by 10 orbits with initial conditions taken on a circle in the domain $(z,p_z)$ of the center manifold for $E_0=9.8994$, emerging towards the bounce. The parameter configuration is given in (61). This numerical simulation was implemented for a time domain corresponding to just one bounce, with initial conditions ($x_0=14.142135621521241$, $p_{x_0}=0$, $y_0=1$, $p_{y_0}=0$), the $(z,p_z)$ coordinates being a solution of the constraint (60).

Figure 10

Poincaré maps of the first coalescence of the stable cylinder (black dots) and unstable cylinder (gray diamonds) in the surface of section ${\mathrm{\Sigma }}_b:(x=x_b,p_x=0)$ at the first bounce $x_b=4.565245$ (left), and in the surface of section ${\mathrm{\Sigma }}_2=(x=7,p_x=15.413)$ (right) shown in the plane $(z,p_z)$, for $E_0=9.8994$. Both cylinders were spanned by 346 orbits, with initial conditions taken on a circle in the domain $(z,p_z)$ of the center manifold. Initial conditions and parameters are the same as in Fig. 9.

Figure 11

Poincaré map of the first coalescence of the stable cylinder (black dots) and unstable cylinder (gray diamonds) in the surface of section ${\mathrm{\Sigma }}_1=(x=x_b,p_x=0)$ (at the first bounce $x_b=4.56523$) shown in the plane $(z,p_z)$ (top left); $(z,p_y)$ (top right); $(y,p_y)$ (bottom left); and $(y,p_z)$ (bottom right) for $E_0=9.8994$. Both cylinders were spanned by 173 orbits, with initial conditions taken on a circle in the domain $(z,p_z)$ of the center manifold. Here we fixed $\mathrm{\Lambda }=0.001$, $E_r=10$, ${\alpha }_{21}=-100$, $A_2=-80$, ${\alpha }_{31}={10}^{-4}$, ${\alpha }_{32}={10}^{-5}$, $A_3={10}^{-5}$ so that $E_{\mathrm{crit}}=9.899494937274579$. The $I{S}_1$ initial conditions are $x=14.142135621521241$, $p_{x_0}=0$, $y_0=z_0$, $p_{y_0}=3p_{z_0}$.

Figure 12

Poincaré map of the first coalescence of the stable cylinder (black dots) and unstable cylinder (gray diamonds) in the surface of section ${\mathrm{\Sigma }}_2=(x=7,p_x=15.4131)$ shown in the plane $(z,p_z)$ (top left); $(z,p_y)$ (top right); $(y,p_y)$ (bottom left); and $(y,p_z)$ (bottom right) for $E_0=9.8994$. Both cylinders were spanned by 173 orbits, with initial conditions taken on a circle in the domain $(z,p_z)$ of the center manifold. Here we fixed $\mathrm{\Lambda }=0.001$, $E_r=10$, ${\alpha }_{21}=-100$, $A_2=-80$, ${\alpha }_{31}={10}^{-4}$, ${\alpha }_{32}={10}^{-5}$, $A_3={10}^{-5}$ so that $E_{cr}=9.899494937274579$. The $I{S}_1$ initial conditions are $x=14.142135621521241$, $p_{x_0}=0$, $y_0=z_0$, $p_{y_0}=3p_{z_0}$.

Figure 13

Numerical illustration of the phase space in a neighborhood of the center-saddle-saddle $P_1$, corresponding to the parameter configuration (67). Here we display the section $(x=x_{c{r}_1}=7.071067829543,p_x=0,p_y=0)$ of the five-dimensional phase space for $E_0=E_{c{r}_1}$. The critical point $P_1$ is located at the common vertex of the cones into which the 4-hyperboloid degenerates for $E_0=E_{cr}$. The high instability of the dynamics outside the invariant plane is due to the presence of a saddle of multiplicity 2 at $P_1$.

Figure 14

Numerical illustration of the unstable cylinder (gray) and stable cylinder (black), that emerge from a neighborhood of $P_2$ towards the bounce, spanned each by 26 orbits with initial conditions taken on a circle in the domain $(z,p_z)$ of the center manifold about $P_2$ for $E_0=9.8994$, corresponding to the parameters (67). The projection of the figure in the plane $(x,p_x)$ “shadows” the separatrix of the invariant plane. Due to the instability in the dynamics connected to the saddle with multiplicity 2 in $P_1$, parts of the orbits escape to two additional $p_z$-momentum attractors with an infinitely large anisotropy parameter, at $x=\text{const}$, $z=\text{const}$ and $p_z\to \pm \infty$, as shown in the figure.

Figure 15

Section $(p_z=0,z=1)$ (left) and section $(p_y=0,y=1)$ (right) of the three-dimensional center manifold (74) for $E_0=0$ (solid), 0.00001 (dashed) and 0.000014 (dashed-dotted), corresponding to the parameter configuration (69) with $q<0$. For these energies $E_0<E_{c{r}_2}$ the manifold is topologically a 3-sphere enclosing the critical point $P_2$ which is a saddle-saddle-saddle. As $E_0\to E_{c{r}_2}$ the curves pinch at the critical point $P_2$, corresponding to $(y=1,p_y=0)$ (left panel) and $(z=1,p_z=0)$ (right panel).

Figure 16

The two-dimensional sections $p_y=0$ of the three-dimensional center manifold (74) for several significative energies: (i) $E_0=0$ (top left): (74) has the topology of an $S^3$ enclosing the saddle with multiplicity 2, a pattern which holds for the domain $0\le E_0<E_{c{r}_2}$; (ii) $E_0=E_{c{r}_2}$ (top right): the saddle with multiplicity 2 now belongs to (74), being the one point connection of two leaves of the manifold and breaking the $S^3$ topology, so that this manifold contains infinitely many homoclinic orbits; (iii) $E_0=0.0000153>E_{c{r}_2}$ (bottom left): showing the bifurcation of $S^3$ into a three-dimensional torus, with the critical point outside the 3-torus; (iv) for larger values, $E_0=0.000025$ (bottom right): the center manifold becomes multiply connected. The parameters of this configuration are given in (69).