Global asymptotic dynamics of the cubic galileon interacting with dark matter

In this paper we perform a thorough dynamical systems analysis of the cubic galileon model non-minimally coupled to the dark matter. Three well-known classes of interacting models are considered where the energy exchange between the dark components is a function of the dark matter density and of the dark energy density: $Q_1=3\alpha H\rho_m$, $Q_2=3\beta\rho_m\dot{\phi}$ and $Q_3=3 \epsilon H \rho_\phi$, respectively. We are able to show the global asymptotic dynamics of the model for the exponential potential in a homogeneous and isotropic background. The cosmological implications of the proposed scenarios are explored and it is found that, in addition to the appearance of new equilibrium configurations that do not appear neither in the non-interacting cubic galileon model nor in the interacting quintessence model, there is a significant impact of the non-minimal coupling through modification of the stability properties of the critical points. The resulting cosmological scenario provides a bigbang origin of the cosmic expansion, an early transient stage of inflationary expansion, as well as matter-scaling late time stable state of the universe, among other solutions of lesser cosmological interest. This work extends previous studies of coupled dark energy to a broader class of gravitational theories.