Dynamics of interacting quintessence

TL;DR

This work investigates three locally coupled quintessence models, each with a distinct form of the interaction between dark energy and dark matter, within a flat FLRW framework. By recasting the cosmology into an autonomous dynamical system using variables $X$, $Y$, and $\lambda$, the authors perform a phase-space analysis to identify fixed points and their stability, focusing on late-time attractors that yield acceleration and a balanced dark-energy to dark-matter ratio. For each coupling form $A=\alpha\dot{\rho_m}$, $A=\beta\dot{\rho_\phi}$, and $A=\sigma(\dot{\rho_m}+\dot{\rho_\phi})$, stable scaling attractors are found under specific parameter ranges (e.g., $\alpha< -7/2$, $\lambda>\sqrt{2/3}$ for model I; $\lambda\neq 0$ for model II; $\sigma<0.2$, $\lambda \le \sqrt{6(1+\sigma)}/(1-\sigma)$ for model III). Numerical examples show consistent $\Omega_{\phi}$ of order unity and $W_{tot}< -1/3$, indicating sustained acceleration and potential alleviation of the coincidence problem. Overall, the study demonstrates that interacting quintessence with these local couplings can produce viable late-time accelerated scaling attractors, offering a dynamical route to address the dark-energy/dark-matter coincidence without invoking a pure cosmological constant.

Abstract

In this paper, we investigate coupled quintessence with scaling potential assuming specific forms of the coupling as $A$ namely, $α\dot{ρ_m}$, $β\dot{ρ_φ}$ and $σ(\dot{ρ_m}+\dot{ρ_φ})$, and present phase space analysis for three different interacting models. We focus on the attractor solutions that can give rise to late time acceleration with $Ω_{DE}/Ω_{DM}$ of order unity in order to alleviate the coincidence problem.

Figures (3)

• Figure 1: This figure shows the phase portrait, evolution of energy density and density parameter for model I for the stable fixed point 5 which is an attractive node. The panel (a) corresponds to $\alpha= -3.6$ and $\lambda= 0.94$, for these values of the parameters we obtain $\Omega_{\phi}=0.73$, $W_{tot}=-0.78$, $W_{\phi}=-1.06$ and an accelerating attractor solution. The panel (b) corresponds to $\alpha= -4.6$ and $\lambda= 0.94$, correspondingly we get $\Omega_{\phi}=0.60$, $W_{tot}=-0.82$, $W_{\phi}=-1.36$ and an accelerated attractor solution. In both the panels, black dots represent attractor stable point. The panels (c) and (d) have same values of the parameters as panel (a), and show scaling behaviours that provide an accelerated expansion.
• Figure 2: The figure represents phase space trajectories, evolution of energy density and density parameter for interacting model II for the stable fixed point 3. The panel (a) corresponds to $\beta= 0.7$ and $\lambda= 1.3$, for these values of the parameters we obtain $\Omega_{\phi}=0.76$, $W_{tot}=-0.57$, $W_{\phi}=-0.75$ and an accelerating attractor solution. The panel (b) corresponds to $\beta= 0.3$ and $\lambda= 1.3$, correspondingly we get $\Omega_{\phi}=0.83$, $W_{tot}=-0.53$, $W_{\phi}=-0.64$ and an accelerated attractor solution. In both the panels, black dots designate attractor stable point, and the stable point behaves as an attractive focus under the chosen parameters. The panels (c) and (d) are plotted the same values of the parameters as panel (a), and show scaling behaviours that gives an accelerated expansion.
• Figure 3: The figure displays phase space trajectories, evolution of energy density and density parameter for interacting model III for the stable fixed point 3. The panel (a) corresponds to $\sigma= -0.3$ and $\lambda= 1.57$, for these values of the parameters we get $\Omega_{\phi}=0.65$, $W_{tot}=-0.46$, $W_{\phi}=-0.71$ and an accelerating attractor solution. The panel (b) corresponds to $\sigma= 0.1$ and $\lambda= 2.85$, and correspondingly we obtain $\Omega_{\phi}=0.45$, $W_{tot}=0.22$, $W_{\phi}=0.49$, since total equation of state is positive therefore, attracting solution but not accelerating. In both the panels, black dots represent attractor stable point, and the stable point acts as an attractive focus under the chosen parameters. For panels (c) and (d), we use same values of the parameters as panel (a). Both panels are showing attractor behaviour that corresponds to scaling solutions.