Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations

TLDR

Degenerate scalar‑tensor theories quadratic in second‑order derivatives of a scalar field, previously identified, include the quartic Horndeski Lagrangian, its beyond‑Horndeski extension, and other Lagrangians, and are designed to avoid Ostrogradsky instability. The authors investigate how these degenerate scalar‑tensor theories transform under general disformal transformations, showing they can be grouped into three stable classes. The study yields a complete classification of these theories modulo disformal transformations, revealing that they encompass mimetic gravity, certain khronometric theories, and novel models not previously identified even after accounting for disformal equivalence.

Abstract

We consider all degenerate scalar-tensor theories that depend quadratically on second-order derivatives of a scalar field, which we have identified in a previous work. These theories, whose degeneracy, in general, ensures the absence of Ostrogradsky's instability, include the quartic Horndeski Lagrangian and its quartic extension beyond Horndeski, as well as other Lagrangians. We study how all these theories transform under general disformal transformations and find that they can be separated into three main classes that are stable under these transformations. This leads to a complete classification modulo disformal transformations. Finally, we show that these higher order theories include mimetic gravity and some particular khronometric theories. They also contain theories that do not correspond, to our knowledge, to already studied theories, even up to disformal transformations.