Most general cubic-order Horndeski Lagrangian allowing for scaling solutions and the application to dark energy

In cubic-order Horndeski theories where a scalar field $\phi$ is coupled to nonrelativistic matter with a field-dependent coupling $Q(\phi)$, we derive the most general Lagrangian having scaling solutions on the isotropic and homogenous cosmological background. For constant $Q$ including the case of vanishing coupling, the corresponding Lagrangian reduces to the form $L=Xg_2(Y)-g_3(Y)\square \phi$, where $X=-\partial_{\mu}\phi\partial^{\mu}\phi/2$ and $g_2, g_3$ are arbitrary functions of $Y=Xe^{\lambda \phi}$ with constant $\lambda$. We obtain the fixed points of the scaling Lagrangian for constant $Q$ and show that the $\phi$-matter-dominated-epoch ($\phi$MDE) is present for the cubic coupling $g_3(Y)$ containing inverse power-law functions of $Y$. The stability analysis around the fixed points indicates that the $\phi$MDE can be followed by a stable critical point responsible for the cosmic acceleration. We propose a concrete dark energy model allowing for such a cosmological sequence and show that the ghost and Laplacian instabilities can be avoided even in the presence of the cubic coupling.