In cubic-order Horndeski theories where a scalar field $\varphi$ is coupled to nonrelativistic matter with a field-dependent coupling $Q(\varphi )$, 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 $L=X{g}_2(Y)-g_3(Y)\square \varphi$, where $X=-{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi /2$ and $g_2$, $g_3$ are arbitrary functions of $Y=X{e}^{\lambda \varphi }$ with constant $\lambda$. We obtain the fixed points of the scaling Lagrangian for constant $Q$ and show that the $\varphi$-matter-dominated epoch ($\varphi \mathrm{MDE}$) 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 $\varphi \mathrm{MDE}$ 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.

• Received 18 October 2018

DOI:https://doi.org/10.1103/PhysRevD.98.123517