Stability of the Einstein static universe in open cosmological models

The stability properties of the Einstein static solution of general relativity are altered when corrective terms arising from modification of the underlying gravitational theory appear in the cosmological equations. In this paper the existence and stability of static solutions are considered in the framework of two recently proposed quantum gravity models. The previously known analysis of the Einstein static solutions in the semiclassical regime of loop quantum cosmology with modifications to the gravitational sector is extended to open cosmological models where a static neutrally stable solution is found. A similar analysis is also performed in the framework of Hořava-Lifshitz gravity under detailed balance and projectability conditions. In the case of open cosmological models the two solutions found can be either unstable or neutrally stable according to the admitted values of the parameters.

Figure 1

Dynamical behavior of the system around the GR fixed point for the case $k=-1$, $\Lambda <0$, $w<-1$ with $\Lambda =-100$, $w=-2$, $\kappa =25.13274123$.

Figure 2

Dynamical behavior of the system around the LQ fixed point for the case $k=-1$, $\Lambda <\kappa {\rho }_c$, $w<-1$ with $\Lambda =10$, $w=-2$, $\kappa =25.13274123$.

Figure 3

Dynamical behavior of the system around the $\mathrm{HL}1$ fixed point for the case $k=-1$ with $ϵ=1$, $\lambda >1/3$, $\Lambda <0$, $w>1/3$.

Figure 4

Dynamical behavior of the system around the $\mathrm{HL}2$ fixed point for the case $k=-1$ with $ϵ=1$, $\lambda >0$, $\Lambda <0$, $w>1/3$.