Emergent scenario in the Einstein-Cartan theory

We study the emergent scenario, which is proposed to avoid the big bang singularity, in the Einstein-Cartan (EC) theory with a positive cosmological constant and a perfect fluid by analyzing the existence and stability of the Einstein static (ES) solutions. We find that there is no stable ES solution for a spatially flat or open universe. However, for a spatially closed universe, the stable ES solution does exist, and in the same existence parameter regions, there also exists an unstable one. With the slow decrease of the equation of state $w$ of the perfect fluid, the stable and unstable critical points move close gradually and coincide once $w$ reaches a critical value, so that the stable critical point becomes an unstable one. As a result, if $w$ approaches a constant at $t\to -\infty$, the universe can stay at the stable ES state past eternally, and furthermore it can naturally exit from this state and evolve into an inflationary era if $w$ decreases slowly as time goes forward. Therefore, the emergent scenario that avoids the big bang singularity can be successfully implemented in the EC theory of gravity.

Figure 1

Regions of stability for point $G$ in the $(w,D_0)$ parameter space.

Figure 2

Evolutionary curve of the scale factor with time and the corresponding phase diagram in space $(a,\dot{a})$ with $w=0.6$, $\mathrm{\Lambda }=0.1$ and $D_0=2$.

Figure 3

The evolutions of the stable point $G$ and unstable point $H$ with $D_0=2$ and the decrease of $w$ are shown in the left panel, while the right panel shows the phase transition from a stable state to an inflationary one with $D_0=2$, $\mathrm{\Lambda }=0.1$ and a slowly decreasing $w$.