Alleviating the $H_0$ and $\sigma_8$ tensions in the interacting cubic covariant Galileon model

The interaction between dark matter and dark energy has become a focal point in contemporary cosmological research, particularly in addressing current cosmological tensions. This study explores the cubic Galileon model's interaction with dark matter, where the interaction potential in the dark sector is proportional to the dark energy density of the Galileon field. By employing dimensionless variables, we transform the field equations into an autonomous dynamical system. We calculate the critical points of the corresponding autonomous systems and demonstrate the existence of a stable de Sitter epoch. Our investigation proceeds in two phases. First, we conduct a detailed analysis of the exact interacting cubic Galileon (ICG) model, derived from the precise solution of the equations of motion. Second, we explore an approximate tracker solution, labeled TICG, assuming a small coupling parameter between dark matter and dark energy. We evaluate the evolution of these models using data from two experiments, aiming to resolve the tensions surrounding $H_0$ and $S_8$. The analysis of the TICG model indicates a preference for a phantom regime and provides a negative coupling parameter in the dark sector at a $68\%$ confidence level. This model also shows that the current tensions regarding $H_0$ and $S_8$ are alleviated. Conversely, the ICG model, despite its preference for the phantom regime, is plagued by an excess in today's matter density and a higher expansion rate, easing only the $H_0$ tension.