Horndeski's theory

Definition

Let $g$ be a Lorentzian metric on a four-dimensional manifold and $\phi$ a real scalar field. Write $\nabla$ for the covariant derivative of $g$, $\square\phi = \nabla^a\nabla_a\phi$, $X = -\frac12 g^{ab}\partial_a\phi\,\partial_b\phi$, $R$ for the scalar curvature and $G_{ab}$ for the Einstein tensor. Let $G_2, G_3, G_4, G_5$ be smooth functions of $(\phi, X)$, and write $G_{iX} = \partial G_i/\partial X$. Horndeski's theory is the scalar–tensor theory with action $S = \int (L_2 + L_3 + L_4 + L_5)\sqrt{-g}\,d^4x + S_m[g, \psi]$, where

1. $L_2 = G_2(\phi, X)$;
2. $L_3 = -G_3(\phi, X)\,\square\phi$;
3. $L_4 = G_4(\phi, X)\,R + G_{4X}\left[(\square\phi)^2 - \nabla_a\nabla_b\phi\,\nabla^a\nabla^b\phi\right]$;
4. $L_5 = G_5(\phi, X)\,G_{ab}\nabla^a\nabla^b\phi - \frac{1}{6} G_{5X}\left[(\square\phi)^3 - 3\,\square\phi\,\nabla_a\nabla_b\phi\,\nabla^a\nabla^b\phi + 2\,\nabla_a\nabla^b\phi\,\nabla_b\nabla^c\phi\,\nabla_c\nabla^a\phi\right]$;

and the matter action $S_m$ depends on $g$ and the matter fields $\psi$ but not on $\phi$.