Can the coincidence problem be solved by a cosmological model of coupled dark energy and dark matter?

Understand

Motivated by the cosmological constant and the coincidence problems, we consider a cosmological model where the dark sectors are interacting together through a phenomenological decay law $\dot{\rho}_{\Lambda}=Q\rho_{\Lambda}^n$ in a FRW spacetime with spatial curvature.

• We show that the only value of $n$ for which the late-time matter energy density to dark energy density ratio ($r_m=\rho_m/\rho_{\Lambda}$) is constant (which could provide an explanation to the coincidence problem) is $n=3/2$.
• For each value of $Q$, there are two distinct solutions.
• One of them involves a spatial curvature approaching zero at late times ($\rho_k\approx0$) and is stable when the interaction is weaker than a critical value ${Q_0=-\sqrt{32\pi G/c^2}}$.