$w_d=-1$ in interacting quintessence model

TL;DR

The work investigates whether a quintessence scalar field interacting with cold dark matter can reach the phantom divide $w_d = -1$ without solving the full dynamics. By deriving a general expression for the time derivative of the dark-energy EoS, $\dot{w}_d$, in the presence of an interaction term $Q$, and imposing $\dot{w}_d(t_0)=0$ at the crossing, the authors connect the possibility of crossing to the form of $Q$ and the scalar potential $V(\phi)$. For the common coupling $Q=H(\lambda_m\rho_m+\lambda_d\rho_d)$, they show the crossing is tied to the density-ratio value $r(t_0)= -\lambda_d/\lambda_m$, and they derive inequalities that constrain the potential and model parameters (e.g., $32\pi \lambda_d(1+r)\rho_d^2(t_0)\le (V'(\phi))^2$). They further analyze a Taylor expansion near $t_0$ and obtain potential-specific conditions: for a quadratic potential $V(\phi)=\tfrac{1}{2} m^2\phi^2$, the crossing requires $8\pi\lambda_d(1-\lambda_d/\lambda_m)\phi^2(t_0)\le 1$ (equivalently $16\pi\lambda_d\rho_d(t_0)\le m^2$); for an exponential potential $V(\phi)=v_0 e^{\lambda\phi}$, a bound $32\pi\lambda_d(1+r_0)<\lambda^2$ arises. Together, these results show that the reachability of $w_d=-1$ depends on a delicate interplay between the interaction and the potential, with concrete parameter constraints identified for the two example potentials.

Abstract

A model consisting of quintessence scalar field interacting with cold dark matter is considered. Conditions required to reach $w_d=-1$ are discussed. It is shown that depending on the potential considered for the quintessence, reaching the phantom divide line puts some constraints on the interaction between dark energy and dark matter. This also may determine the ratio of dark matter to dark energy density at $w_d=-1$.