The EFT of dark energy is extended to include time derivatives of the lapse, which had previously been ommitted because they generically propagate an additional scalar dof, but here they are included in such a way that the resulting Lagrangian is degenerate (c.f. DHOST theories). Linear perturbations about Minkowski and FRW can only obey a Poisson-like equation in the static limit in the beyond Horndeski theories.

This new EFT encapsulates not only Horndeski, but also the ghost-free higher derivative scalar field theories, known as DHOST (which includes the `beyond Horndeski' theories). Elsewhere it has been shown that Lorentz invariance and no strong coupling in the tensor sector restricts one to theories which are field redefinable to Horndeski. This work suggests that to reproduce Poisson-like cosmological behaviour, one is similarly restricted to Horndeski (or beyond-Horndeski) theory.