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.
