Tracking our Universe to de Sitter by a Horndeski scalar

Assuming both that our Universe is evolving into a de Sitter space and a vanishing cosmological constant, leaves only the option that the observed acceleration is provided by a "kinetic" energy of a scalar field. From an effective field theory point of view, the absence of Ostrogradsky instabilities restricts the choice to shift-symmetric Horndeski theories. Within these theories, we find the conditions for the existence of a de Sitter critical point in a universe filled by matter, radiation and a Horndeski scalar. Moreover, we show that this point is a universal attractor and we provide the tracking trajectory. Therefore, if a de Sitter fixed point exists within these models, our Universe will eventually evolve into a de Sitter space. As an example, we have discussed the case of the combined Galileon-Slotheon system, in which the Galileon is kinetically non-minimal coupled to the Einstein tensor. Interestingly, we have also found that the tracker trajectory of this system does not follow previous literature assumptions.