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Assuming both that our Universe is evolving into a de Sitter space and avanishing cosmological constant, leaves only the option that the observedacceleration is provided by a "kinetic" energy of a scalar field. From aneffective field theory point of view, the absence of Ostrogradsky instabilitiesrestricts the choice to shift-symmetric Horndeski theories. Within thesetheories, we find the conditions for the existence of a de Sitter criticalpoint in a universe filled by matter, radiation and a Horndeski scalar.Moreover, we show that this point is a universal attractor and we provide thetracking trajectory. Therefore, if a de Sitter fixed point exists within thesemodels, our Universe will eventually evolve into a de Sitter space. As anexample, we have discussed the case of the combined Galileon-Slotheon system,in which the Galileon is kinetically non-minimal coupled to the Einsteintensor. Interestingly, we have also found that the tracker trajectory of thissystem does not follow previous literature assumptions.

Tracking our Universe to de Sitter by a Horndeski scalar