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

TL;DR

This paper derives the most general cubic-order Horndeski Lagrangian with scaling solutions, analyzes fixed points including the matter-dominated epoch, and proposes a stable dark energy model avoiding instabilities.

Contribution

It provides a comprehensive form of the cubic Horndeski Lagrangian with scaling solutions and explores its cosmological implications for dark energy models.

Findings

Identified the general form of the Lagrangian with scaling solutions.

Found conditions for the matter-dominated epoch and stable acceleration.

Demonstrated stability and absence of instabilities in the proposed dark energy model.

Abstract

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.