Interacting dark sector from Horndeski theories and beyond: Mapping fields and fluids

In Cosmology, when dissipative effects are minimal, the energy content of the Universe can be effectively described as a sum of perfect fluids. Perfect fluid descriptions ensure thermal equilibrium since they equilibrate immediately. However, interactions among different energy content of the Universe might prevent such rapid equilibration. This limitation calls for a more fundamental framework that incorporates these interactions directly at the level of the action. In earlier work, two of the authors demonstrated that an interacting dark energy (DE)-dark matter (DM) field theory action could be derived from a modified gravity action via a conformal transformation, establishing a one-to-one correspondence between the field theory action and fluid for a unique interaction term [arXiv:2006.04618]. In this work, we extend that analysis by considering quadratic order Horndeski gravity, identifying two classes of models: field coupling and field-kinetic coupling. Our approach generalizes the coupling function for DE-DM interactions by incorporating an additional dependence on kinetic terms. We establish a field-to-fluid mapping for dark matter and find that this mapping only holds for a specific form of the interaction strength. Interestingly, we show that this interaction strength excludes non-gravitational interactions between dark energy and dark radiation. Numerical analysis reveals that purely kinetic interactions within the dark sector can significantly alter cosmological evolution compared to non-interacting scenarios, highlighting the strong dependence of cosmological dynamics on coupling strength. A preliminary examination of linear scalar perturbations indicates that the field-kinetic coupling results in a non-zero gravitational slip parameter and momentum exchange.