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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 $ρ_x \propto ρ_d^α$, where $ρ_x$ and $ρ_d$ are the energy densities of dark energy and dark matter components, respectively, and $α$ is a free parameter. For a dark energy equation of state parameter $w=-1$ we found that, if $α=0$, the standard $Λ$CDM model is recovered, where the coincidence problem is unsolved. For $0<α<1$, the CP would be alleviated and for $α\sim 1$, 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 $α=0.109^{+0.062}_{-0.072}$ at 68% C.L., and the CP is alleviated. For $w\neq -1$, a degeneracy arises on the $w$ - $α$ plane. In order to break such degeneracy we add cosmic microwave background distance priors and baryonic acoustic oscillations data to the constraints, yielding $α=-0.075\pm 0.046$ at 68% C.L.. In this case we find that the CP is not alleviated even for 2$σ$ interval for $α$. Furthermore, this last model is discarded against flat $Λ$CDM according to BIC analysis.