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

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 \({Q_0=-\sqrt{32\pi G/c^2}}\). The other one allows for a non-negligible spatial curvature (\(\rho_k\napprox0\)) at late times and is stable when the interaction is stronger than \(Q_0\). We constrain the model parameters using various observational data (SNeIa, GRB, CMB, BAO, OHD). The limits obtained on the parameters exclude the regions where the cosmological constant problem is significantly ameliorated and do not allow for a completely satisfying explanation for the coincidence problem.