Equivalence between Horndeski and beyond Horndeski theories and imperfect fluids

In this paper we approach to Horndeski and beyond Horndeski theories from the effective fluid perspective. We make explicit the formal equivalence between these theories and general relativity with an effective imperfect fluid. It is shown that, for the viable Horndeski theories, in the general case (arbitrary geometry) the nonvanishing contribution from the higher-order derivative terms to the imperfect fluidlike behavior affects only the heat flux vector but not the anisotropic stresses. The only contribution to the anisotropic stress tensor is due to the nonminimal coupling of the scalar field to the curvature as it is in standard scalar-tensor theories. For the viable beyond Horndeski theories the higher-order derivatives contribute both to the heat flux and to the anisotropic stresses. The effective fluid description is applied to several particular cases of interest. It is corroborated that, in Friedmann-Robertson-Walker background space, due to the underlying symmetries, the effective stress-energy tensor of viable Horndeski and beyond Horndeski theories is formally equivalent to that of a perfect fluid. This result might not be true for other less symmetric backgrounds such as the anisotropic Bianchi I space.