Phase-space analysis of interacting phantom cosmology

TL;DR

This work implements a phase-space (autonomous dynamical system) analysis of flat FRW cosmologies with a phantom dark-energy field interacting with dark matter through four interaction forms. By introducing the variables $x=rac{\kappa\dot\phi}{\sqrt{6}H}$ and $y=rac{\kappa\sqrt{V}}{\sqrt{3}H}$ (and, for Model 4, $v=H_0/H$) and a logarithmic time $M=\ln a$, the authors identify critical points, determine their stability from the eigenvalues of the linearized system, and evaluate $\Omega_\phi$ and $w_{tot}$. They find that all models admit late-time accelerating attractors with $w_{tot}<-1$, but most correspond to $\Omega_\phi=1$ (complete dark-energy domination), so the coincidence problem is not solved; only Interacting Model 1 offers a narrowly allowed region with $\Omega_\phi$ of order unity. The results suggest that solving the coincidence problem within interacting phantom cosmology requires finely-tuned parameters or more elaborate interaction terms.

Abstract

We perform a detailed phase-space analysis of various phantom cosmological models, where the dark energy sector interacts with the dark matter one. We examine whether there exist late-time scaling attractors, corresponding to an accelerating universe and possessing dark energy and dark matter densities of the same order. We find that all the examined models, although accepting stable late-time accelerated solutions, cannot alleviate the coincidence problem, unless one imposes a form of fine-tuning in the model parameters. It seems that interacting phantom cosmology cannot fulfill the basic requirement that led to its construction.

Figures (6)

• Figure 1: The range of the $2D$ parameter space $(\alpha,\lambda)$ that corresponds to an existing and stable critical point B of table \ref{['stability1']}, that is in the case of interacting model 1.
• Figure 2: Phase-space trajectories for interacting model 1. The left panel corresponds to $\alpha=-3.1$ and $\lambda=0.1$, and thus to the stable fixed point B of table \ref{['stability1']} with ($x_{c}$, $y_{c}$)=(-0.41, 0.93). The right panel corresponds to $\alpha=0.5$ and $\lambda=1.0$ and thus to the stable fixed point A of table \ref{['stability1']} with ($x_{c}$, $y_{c}$)=(-0.41, 1.08).
• Figure 3: Phase-space trajectories for interacting model 2 using $\alpha_0=0.5$ and $\lambda=1.0$. The stable fixed point is the critical point $B$ in table \ref{['tab2']}, with ($x_{c2}$, $y_{c2}$)=(-0.41, 1.08).
• Figure 4: The range of the $2D$ parameter space $(\beta,\lambda)$ that corresponds to an existing and stable critical point B of table \ref{['tab3']}, that is in the case of interacting model 3.
• Figure 5: Phase-space trajectories for interacting model 3 with $\beta=0.5$, $\lambda=1.0$ (left panel) and $\beta=-2.0$, $\lambda=1.0$ (right panel). In both cases the stable fixed point is the critical point B of table \ref{['tab3']}, with ($x_{c2}$, $y_{c2}$)=(-0.41, 1.08).