by a single scalar field $\phi$ described by the Horndeski Lagrangian (HL). The HL is defined as the sum of four terms $\mathcal{L}_{2}$ to $\mathcal{L}_{5}$ that are fully specified by a non-canonical kinetic term $K(\phi,X)$ and three in principle arbitrary coupling functions $G_{3,4,5}(\phi,X)$, where $X=-g_{\mu\nu}\phi^{,\mu}\phi^{,\nu}/2$ is the canonical kinetic term,

$$\begin{aligned}
\begin{aligned}\mathcal{L}_{2}= & K(\phi,X)\,,\\
\mathcal{L}_{3}= & -G_{3}(\phi,X)\Box\phi\,,\\
\mathcal{L}_{4}= & G_{4}(\phi,X)R+G_{4,X}\left[\left(\Box\phi\right)^{2}-\left(\nabla_{\mu}\nabla_{\nu}\phi\right)^{2}\right]\,,\\
\mathcal{L}_{5}= & G_{5}(\phi,X)G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi-\frac{G_{5,X}}{6}\Bigl[\left(\Box\phi\right)^{3}-\\
 & -3\left(\Box\phi\right)\left(\nabla_{\mu}\nabla_{\nu}\phi\right)^{2}+2\left(\nabla_{\mu}\nabla_{\nu}\phi\right)^{3}\Bigr]\,.
\end{aligned}
\label{eq:HL}\end{aligned}$$

The Horndeski Lagrangian is the most general Lagrangian for a single scalar which gives second-order equations of motion for both the scalar and the metric on an arbitrary background. This is a necessary, but not sufficient, condition for the absence of ghosts.[^3] In general, the equation of motion for the scalar will couple it to the matter energy density. The metric potentials $\Phi$ and $\Psi$ are as usual determined by the Poisson and anisotropy equations, which are constraints, and therefore do not have independent dynamics. We note that the generalization to the case of multiple scalar fields has been discussed in Ref. [@Padilla:2012dx].

In what follows, we will assume that the so-called *quasi-static limit* is valid for the evolution of perturbations. This implies that we are observing scales significantly inside the cosmological horizon, 