Oscillating universe in Hořava-Lifshitz gravity

We study the dynamics of isotropic and homogeneous universes in the generalized Hořava-Lifshitz gravity, and classify all possible evolutions of vacuum spacetime. In the case without the detailed balance condition, we find a variety of phase structures of vacuum spacetimes depending on the coupling constants as well as the spatial curvature $K$ and a cosmological constant $\Lambda$. A bounce universe solution is obtained for $\Lambda >0$, $K=\pm 1$ or $\Lambda =0$, $K=-1$, while an oscillation spacetime is found for $\Lambda \ge 0$, $K=1$, or $\Lambda <0$, $K=\pm 1$. We also propose a quantum tunneling scenario from an oscillating spacetime to an inflationary universe, resulting in a macroscopic cyclic universe.

Figure 1

Phase diagram of spacetimes for $\Lambda =0$. The oscillating universe is found only for the case of $K=1$. The stable and unstable static universes (${\mathcal{S}}_{\mathrm{s}}$ and ${\mathcal{S}}_{\mathrm{u}}$) exist on the boundary ${\Gamma }_{0,1}$ and ${\Gamma }_{0,-1}$, respectively. On ${\Gamma }_{0,-1}$, we also find dynamical universes with an asymptotically static spacetime: ${\mathcal{S}}_{\mathrm{u}}\Rightarrow \mathcal{B}\mathcal{C}$, ${\mathcal{S}}_{\mathrm{u}}\Rightarrow \mathcal{M}$, $\mathcal{B}\mathcal{B}\Rightarrow {\mathcal{S}}_{\mathrm{u}}$, or $\mathcal{M}\Rightarrow {\mathcal{S}}_{\mathrm{u}}$.

Figure 2

The potential $\mathcal{U}(\tilde{a})$ for a stable static universe and an oscillating universe near the $g_{\mathrm{r}}$ axis. The “coupling” constants are chosen as $g_{\mathrm{r}}=1$ and $g_{\mathrm{s}}=-1/12$ on ${\Gamma }_{0,1}$ for a static universe, for which the radius is shown by $a_S=1/\sqrt{6}$. We also show the case with $g_{\mathrm{r}}=1$ and $g_{\mathrm{s}}=-0.001$ for an oscillating universe, for which maximum and minimum radii are given by $a_{\max }=0.576481$ and $a_{\min }=0.0316705$, respectively.

Figure 3

Phase diagram of spacetimes for $\Lambda >0$. The oscillating universe is found only for the case of $K=1$. The static universes (${\mathcal{S}}_{\mathrm{s}}$ and ${\mathcal{S}}_{\mathrm{u}}$) exist on the boundaries ${\Gamma }_{1,K(\pm )}$. We also find dynamical universes with an asymptotically static spacetime: ${\mathcal{S}}_{\mathrm{u}}\Rightarrow 𝒹\mathcal{S}$ or ${\mathcal{S}}_{\mathrm{u}}\Rightarrow \mathcal{B}\mathcal{C}$ on ${\Gamma }_{1,1(+)}({\tilde{g}}_{\mathrm{s}}\ge 0)$; ${\mathcal{S}}_{\mathrm{u}}\Rightarrow 𝒹\mathcal{S}$ or ${\mathcal{S}}_{\mathrm{u}}\Rightarrow Bounce\Rightarrow {\mathcal{S}}_{\mathrm{u}}$ on ${\Gamma }_{1,1(+)}({\tilde{g}}_{\mathrm{s}}<0)$; ${\mathcal{S}}_{\mathrm{u}}\Rightarrow 𝒹\mathcal{S}$ or ${\mathcal{S}}_{\mathrm{u}}\Rightarrow \mathcal{B}\mathcal{C}$ on ${\Gamma }_{1,-1}$.

Figure 4

The potential $\mathcal{U}(\tilde{a})$ for stable and unstable static universes [the solid dark (blue) and the dashed dark (blue) curves], and that for an oscillating universe near the ${\tilde{g}}_{\mathrm{r}}$ axis [the light (orange) curve]. The constants are chosen as ${\tilde{g}}_{\mathrm{r}}=0.8$ and ${\tilde{g}}_{\mathrm{s}}=-0.0643206$ on ${\Gamma }_{1,1(-)}$, and $-0.0245683$ on ${\Gamma }_{1,1(+)}$, for static universes, for which radii are given by ${\tilde{a}}_S$, and ${\tilde{g}}_{\mathrm{r}}=0.7$ and ${\tilde{g}}_{\mathrm{s}}=-0.001$ for an oscillating universe, for which maximum and minimum radii are given by ${\tilde{a}}_{\max }$ and ${\tilde{a}}_{\min }$, respectively. We also find ${\mathcal{S}}_{\mathrm{u}}\Rightarrow Bounce\Rightarrow {\mathcal{S}}_{\mathrm{u}}$, for which the bounce radius is given by ${\tilde{a}}_{\min }$.

Figure 5

Phase diagram of spacetimes for $\Lambda <0$. The oscillating universe is found for both $K=\pm 1$. The static universe exists on the boundary ${\Gamma }_{-1,1(-)}$ ($K=1$) and on ${\Gamma }_{-1,-1(\pm )}$ ($K=-1$). In the branch of unstable static universe on ${\Gamma }_{-1,-1(+)}$, we also find dynamical universes with an asymptotically static spacetime; $\mathcal{B}\mathcal{B}\Rightarrow {\mathcal{S}}_{\mathrm{u}}$, ${\mathcal{S}}_{\mathrm{u}}\Rightarrow \mathcal{B}\mathcal{C}$, or ${\mathcal{S}}_{\mathrm{u}}\Rightarrow Bounce\Rightarrow {\mathcal{S}}_{\mathrm{u}}$.

Figure 6

The potential $\mathcal{U}(\tilde{a})$ for a stable static universe [dark (blue) curve] and an oscillating universe near the ${\tilde{g}}_{\mathrm{r}}$ axis [light (orange) curve] in the case of $K=1$. We set ${\tilde{g}}_{\mathrm{r}}=0.8$ and ${\tilde{g}}_{\mathrm{s}}=-0.0477674$ on ${\Gamma }_{-1,1}$ for a static universe with the radius ${\tilde{a}}_S=0.337461$, and ${\tilde{g}}_{\mathrm{r}}=0.8$ and ${\tilde{g}}_{\mathrm{s}}=-0.001$ for an oscillating universe, for which maximum and minimum radii are given by ${\tilde{a}}_{\max }=0.466615$ and ${\tilde{a}}_{\min }=0.035439$, respectively.

Figure 7

The potential $\mathcal{U}(\tilde{a})$ for a stable static universe [dark (blue) curve] and an oscillating universe near the ${\tilde{g}}_{\mathrm{r}}$ axis [light (orange) curve] for $K=-1$. We set ${\tilde{g}}_{\mathrm{r}}=0.2$ and ${\tilde{g}}_{\mathrm{s}}=-0.581008$ on ${\Gamma }_{-1,-1}$ for a static universe, for which the radius is given by ${\tilde{a}}_S=0.835752$, and ${\tilde{g}}_{\mathrm{r}}=0.2$ and ${\tilde{g}}_{\mathrm{s}}=-0.001$ for an oscillating universe, for which maximum and minimum radii are given by ${\tilde{a}}_{\max }=1.03075$ and ${\tilde{a}}_{\min }=0.0683656$, respectively.

Figure 8

The potential $\mathcal{U}(\tilde{a})$ for stable and unstable static universes [solid dark (blue) and dashed dark (blue) curve, respectively] and an oscillating universe on the ${\tilde{g}}_{\mathrm{r}}$ axis [light (orange) curve] for $K=-1$. We set ${\tilde{g}}_{\mathrm{r}}=-0.5$ and ${\tilde{g}}_{\mathrm{s}}=-0.134123$ on ${\Gamma }_{-1,-1(-)}$, and ${\tilde{g}}_{\mathrm{s}}=0.0230119$ on ${\Gamma }_{-1,-1(+)}$ for static universes, for which the radius is given by ${\tilde{a}}_S=0.754344$, and ${\tilde{g}}_{\mathrm{r}}=-0.5$ and ${\tilde{g}}_{\mathrm{s}}=0$ for an oscillating universe, for which maximum and minimum radii are given by ${\tilde{a}}_{\max }=0.888074$ and ${\tilde{a}}_{\min }=0.459701$, respectively. We also find ${\mathcal{S}}_{\mathrm{u}}\Rightarrow Bounce\Rightarrow {\mathcal{S}}_{\mathrm{u}}$, for which the bounce radius is given by ${\tilde{a}}_T=0.897072$.

Figure 9

The potential $\mathcal{U}(a)$ for the coupling constants which provide both oscillating and bounce universes ($\Lambda >0$ and $K=1$). After a number of oscillations between ${\tilde{a}}_{min}$ and ${\tilde{a}}_{max}$, the universe may evolve into the de Sitter phase via a quantum tunneling process to the bounce point $a_T$.

Figure 10

The value of the Euclidian action $S_E$ in terms of the coupling constant ${\tilde{g}}_s$. $e^{-S_E}$ provides a quantum tunneling probability.