Quartic Horndeski-Cartan theories in a FLRW universe

We consider the Quartic Horndeski theory with torsion on a Friedmann-Lema\^{\i}tre-Robertson-Walker background in second-order formalism. show that there is one parameter family of Cartan Lagrangians and all such theories only modify dispersion relations graviton scalar perturbation are usually found standard torsionless spacetime. In other words, for this class, does not induce new degrees freedom, but it modifies propagation. This holds first-order perturbations spite kinetic mixing between field action. also most within Horndeski-Cartan relation mode radically modified. find whose has regular wavelike relation.