Phase space analysis of sign-shifting interacting dark energy models

Abstract

The theory of non-gravitational interaction between a pressure-less dark matter (DM) and dark energy (DE) is a phenomenologically rich cosmological domain which has received magnificent attention in the community. In the present article we have considered some interacting scenarios with some novel features: the interaction functions do not depend on the external parameters of the universe, rather, they depend on the intrinsic nature of the dark components; the assumption of unidirectional flow of energy between DM and DE has been extended by allowing the possibility of bidirectional energy flow characterized by some sign shifting interaction functions; and the DE equation of state has been considered to be either constant or dynamical in nature. These altogether add new ingredients in this context, and, we performed the phase space analysis of each interacting scenario in order to understand their global behaviour. According to the existing records in the literature, this combined picture has not been reported elsewhere. From the analyses, we observed that the DE equation of state as well as the coupling parameter(s) of the interaction models can significantly affect the nature of the critical points. It has been found that within these proposed sign shifting interacting scenarios, it is possible to obtain stable late time attractors which may act as global attractors corresponding to an accelerating expansion of the universe. The overall outcomes of this study clearly highlight that the sign shifting interaction functions are quite appealing in the context of cosmological dynamics and they deserve further attention.

Figures (19)

• Figure 1: The phase portrait plot describing Model I (eqn. (\ref{['model1']})) with $w_d\geq-1$ and $\gamma>0$. In this case we have taken $w_d =-0.95$ and $\gamma = 0.4$. We note that one can take any value of $w_d \geq -1$ and any positive value of $\gamma$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e. $q<0$) and the pink shaded region corresponds to the decelerated region (i.e. $q>0$).
• Figure 2: The phase portrait plot describing Model I (eqn. (\ref{['model1']})) for $-2< w_d < -1$ and $\gamma > 0$. In this case we have taken $w_d =-1.3$ and $\gamma = 0.8$. We note that one can take any specific value of $\gamma~(>0)$ to draw the plot, however, as long as $\gamma$ decreases, the regions I and IV become very small and they look indistinguishable from one another.
• Figure 3: We show the evolution of the CDM density parameter ($\Omega_c$), dark energy density parameter ($\Omega_d$) and the total equation of state (EoS) parameter ($w_{\rm tot}$) for Model I (eqn. (\ref{['model1']})) for $-2<w_d<-1$. We have taken $w_d=-1.3$, $\gamma=0.8$ with the initial conditions $x~(N =0) =0.25$, $z~(N =0) = 0.05$ taken from region II of Fig. \ref{['fig2:model-I']}. For the initial condition on $x(N)$ and $z(N)$ from the region I of Fig. \ref{['fig2:model-I']}, again we shall obtain $\Omega_c=0$ and $\Omega_d=1$ at late time. If we choose the initial conditions on $x(N)$ and $y(N)$ from region III and IV of Fig. \ref{['fig2:model-I']}, we shall reach $\Omega_c=\Omega_d=1/2$ in an asymptotic way.
• Figure 4: The phase portrait plot describing Model I of (\ref{['model1']}) with $w_d \leq -2$ and $\gamma>0$. In this case we have taken $w_d =-2.5$ and $\gamma = 0.8$. We note that one can take any value of $w_d \leq -2$ and any positive value of $\gamma$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e. $q<0$) and the pink shaded region corresponds to the decelerated region (i.e. $q>0$)
• Figure 5: Phase portrait plot depicting Model II (eqn. (\ref{['model2']})) with $\alpha>0, \beta>0$ and $w_d\geq-1$. In this case we have taken $w_d =-0.9$, $\alpha = 0.5$ and $\beta=0.6$. We note that one can take any value of $w_d \geq -1$ and any positive value of $\alpha$ and $\beta$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e. $q<0$) and the pink shaded region corresponds to the decelerated region (i.e. $q>0$).