Dynamics of an interacting dark energy model in Einstein and loop quantum cosmology

We investigate the background dynamics when dark energy is coupled to dark matter in the Universe described by Einstein cosmology and loop quantum cosmology. We introduce a new general form of dark sector coupling, which presents a more complicated dynamical phase space. Differences in the phase space in obtaining the accelerated scaling attractor in Einstein cosmology and loop quantum cosmology are disclosed.

Figure 1

The stable regions in the $({\omega }_x,c)$ parameter space for fixed $\alpha$ (the left panel is for $\alpha =0.5$ and the right panel is for $\alpha =2$). In Einstein cosmology the critical points $A_2$ ($\alpha =0.5$) and $B_4$ ($\alpha =2$) are late-time attractors in the region $\mathrm{I}+\mathrm{II}$. But in LQC, $A_2$ ($\alpha =0.5$) and $B_4$ are late-time attractors only in region II. Region III represents the region of the solution without physical meaning.

Figure 2

The phase diagram of interacting dark energy (the left panel is for Einstein cosmology and the right is for LQC) with ${\omega }_x=-1.2$, $c=0.5$, and $\alpha =0.5$. Point $A_2$ is the critical point.

Figure 3

The evolution of total cosmic energy $\omega$ for ${\omega }_x=-0.6$, $c=0.18$ and fixed $\alpha$ (the left panel is for Einstein cosmology and the right is for LQC).

Figure 4

The evolution of $\rho$ with $t$ in LQC for fixed $\alpha =-1$ and ${\rho }_c=0.82$ (the left panel is for ${\omega }_x=-1.6$ and $c=0.1$, the middle panel is for ${\omega }_x=-1.6$ and $c=0.3$, and the right panel is for ${\omega }_x=-1.8$ and $c=0.3$). The solid, dotted, and dashed curves correspond to ${\rho }_x+{\rho }_m$, ${\rho }_x$, and ${\rho }_m$, respectively.