Framing the dark: theory and phenomenology of a non-minimally coupled dark matter fluid

The standard Lambda Cold Dark Matter (ΛCDM) model has proven to be extremely powerful in predicting the evolution and formation of the structure we see in the Universe. However, this success comes at the price of accepting that the energy/matter content of the Universe is dominated by two unknown components: dark matter and dark energy. Moreover, in the case of dark matter some tension between simulations and observations exists at galactic scales, while in the case of dark energy long-standing theoretical issues are still unresolved, thus making the investigation of alternative models a powerful and necessary tool to improve the understanding of the Universe dynamics. In this thesis we study a modified gravity model for dark matter, investigating both its theoretical and phenomenological consequences. In particular, we consider a CDM model in which a dark fluid becomes non-minimally coupled to gravity at recent times and investigate and discuss how this modified picture may be able to address some of the issues the CDM paradigm is facing at small scales, while preserving the successes of the model at large scales. On a more formal level we investigate the properties of the Horndeski action under disformal metric transformation, providing the form of the transformations that leaves it invariant. More than a mathematical curiosity, this invariance represents a first step to formalize some recently notices relations between different models of dark energy, which may be seen as equivalent representation of the same fundamental theory.