Can dark matter-dark energy interaction alleviate the Cosmic Coincidence Problem?

In this paper we study a model of interacting dark energy - dark matter where the ratio between these components is not constant, changing from early to late times in such a way that the model can solve or alleviate the cosmic coincidence problem (CP). The interaction arises from an assumed relation of the form $\rho_x \propto \rho_d^\alpha$, where $\rho_x$ and $\rho_d$ are the energy densities of dark energy and dark matter components, respectively, and $\alpha$ is a free parameter. For a dark energy equation of state parameter $w=-1$ we found that, if $\alpha=0$, the standard $\Lambda$CDM model is recovered, where the coincidence problem is unsolved. For $0<\alpha<1$, the CP would be alleviated and for $\alpha\sim 1$, the CP would be solved. The dark energy component is analyzed with both $w=-1$ and $w \neq -1$. Using Supernovae type Ia and Hubble parameter data constraints, in the case $w=-1$ we find $\alpha=0.109^{+0.062}_{-0.072}$ at 68% C.L., and the CP is alleviated. For $w\neq -1$, a degeneracy arises on the $w$ - $\alpha$ plane. In order to break such degeneracy we add cosmic microwave background distance priors and baryonic acoustic oscillations data to the constraints, yielding $\alpha=-0.075\pm 0.046$ at 68% C.L.. In this case we find that the CP is not alleviated even for 2$\sigma$ interval for $\alpha$. Furthermore, this last model is discarded against flat $\Lambda$CDM according to BIC analysis.