Emergent universe from the Hořava-Lifshitz gravity

We study the stability of the Einstein static universe in the Hořava-Lifshitz (HL) gravity and a generalized version of it formulated by Sotiriou, Visser, and Weifurtner. We find that, for the HL cosmology, there exists a stable Einstein static state if the cosmological constant $\Lambda$ is negative. The universe can stay at this stable state eternally and thus the big bang singularity can be avoided. However, in this case, the Universe can not exit to an inflationary era. For the Sotiriou, Visser, and Weifurtner HL cosmology, if the cosmic scale factor satisfies certain conditions initially, the Universe can stay at the stable state past eternally and may undergo a series of infinite, nonsingular oscillations. Once the parameter of the equation of state $w$ approaches a critical value, the stable critical point coincides with the unstable one, and the Universe enters an inflationary era. Therefore, the big bang singularity can be avoided and a subsequent inflation can occur naturally.

Figure 1

The evolutionary curve of the scale factor with time (left) and the phase diagram in space $(a,\dot{a})$ (right) for the case $\Lambda <0$ in a Planck unit and with $w=-0.90$, $\Lambda =-0.6$, $\alpha =-1$.

Figure 2

Regions of existence in the $(w,{\overline{\beta }}_1)$ parameter space. The left panel shows the existence region for Point $D$, while the right panel for Point $C$.

Figure 3

Regions of stability in the $(w,{\overline{\beta }}_1)$ parameter space for Point $D$ (left panel), and the evolution of Points $C$ and $D$ with the decreasing of $w$ (right panel). In the right panel, $\Lambda =0.6$ and ${\overline{\beta }}_1=-1$.

Figure 4

The phase transition from a stable state to an inflation by assuming a slowly deceasing equation of state ($w(t)=0.280-0.001t$) for the HL cosmology with the initial conditions $a(0)=0.5$ and $\dot{a}(0)=0$. The parameters are set as $\Lambda =0.6$ and ${\overline{\beta }}_1=-1$.

Figure 5

Regions of existence for Point $E$ in the $(w,{\overline{\beta }}_2)$ parameter space. The left, middle, and right panels show the case with ${\overline{\beta }}_1=2$, 0, and $-2$, respectively.

Figure 6

Regions of existence for Point $F$ in the $(w,{\overline{\beta }}_2)$ parameter space. The left and right panels show the case with ${\overline{\beta }}_1=0$ and $-2$, respectively.

Figure 7

Regions of existence for Point $G$ in the $(w,{\overline{\beta }}_2)$ parameter space. The left and right panels show the case with ${\overline{\beta }}_1=0$ and $-2$, respectively.

Figure 8

Regions of existence for Point $H$ in the $(w,{\overline{\beta }}_2)$ parameter space. The left and right panels show the case with ${\overline{\beta }}_1=0$ and $-2$, respectively.

Figure 9

Regions of stability for Point $E$ in the $(w,{\overline{\beta }}_2)$ parameter space. The left, middle, and right panels show the case with ${\overline{\beta }}_1=2$, 0, and $-2$, respectively.

Figure 10

Regions of stability for Point $G$ in the $(w,{\overline{\beta }}_2)$ parameter space. The left and right panels show the case with ${\overline{\beta }}_1=0$ and $-2$, respectively.

Figure 11

The evolutions of the stable Point $G$ and unstable one $H$ with the decreasing of $w$.