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-Lemaître-Robertson-Walker background. In addition to a dark energy field $\chi$ associated with the gravitational sector, we take into account multiple scalar fields ${\varphi }_I$ ($I=1,2,\dots ,N-1$) characterized by the Lagrangians $P^{(I)}(X_I)$ with $X_I={\partial }_{\mu }{\varphi }_I{\partial }^{\mu }{\varphi }_I$. These additional scalar fields can model the perfect fluids of radiation and nonrelativistic matter. We derive propagation speeds of scalar and tensor perturbations as well as conditions for the absence of ghosts. The theories beyond Horndeski induce nontrivial modifications to all the propagation speeds of $N$ scalar fields, but the modifications to those for the matter fields ${\varphi }_I$ 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.

Figure 1

Evolution of the scalar propagation speed squares $c_{s1}^2,c_{s2}^2,c_{s3}^2$ and the dark energy equation of state $w_{\text{DE}}$ versus the redshift $z=1/a-1$ in Model (B). We choose the model parameters $\alpha =0.3$ and $\beta =0.14$ with the initial conditions $r_1=5\times 10^{-11}$, $r_2=8\times 10^{-12}$, ${\mathrm{\Omega }}_r=0.999995$, and $w_m=10^{-3}$ at $z=6.0\times 10^8$. This case corresponds to the late-time tracking solution that approaches the tracker ($r_1=1$) at low redshifts. During most of the radiation and matter eras the solution is in the regime $r_1\ll 1$ and $r_2\ll 1$, in which case the first propagation speed square is given by $c_{s1}^2\simeq (3{\mathrm{\Omega }}_r-1)/40$.