Cosmology in generalized Horndeski theories with second-order equations of motion

We study the cosmology of an extended version of Horndeski theories with second-order equations of motion on the flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) background. In addition to a dark energy field $\chi$ associated with the gravitational sector, we take into account multiple scalar fields $\phi_I$ ($I=1,2\cdots,N-1$) characterized by the Lagrangians $P^{(I)}(X_I)$ with $X_I=\partial_{\mu}\phi_I\partial^{\mu}\phi_I$. These additional scalar fields can model the perfect fluids of radiation and non-relativistic matter. We derive propagation speeds of scalar and tensor perturbations as well as conditions for the absence of ghosts. The theories beyond Horndeski induce non-trivial modifications to all the propagation speeds of $N$ scalar fields, but the modifications to those for the matter fields $\phi_I$ are generally suppressed relative to that for the dark energy field $\chi$. We apply our results to the covariantized Galileon with an Einstein-Hilbert term in which partial derivatives of the Minkowski Galileon are replaced by covariant derivatives. Unlike the covariant Galileon with second-order equations of motion in general space-time, the scalar propagation speed square $c_{s1}^2$ associated with the field $\chi$ becomes negative during the matter era for late-time tracking solutions, so the two Galileon theories can be clearly distinguished at the level of linear cosmological perturbations.