Ostrogradsky in Theories with Multiple Fields

TL;DR

This paper reviews how multiple-field theories can avoid Ostrogradsky ghosts despite higher derivatives, with implications for modified gravity models and dark energy stability.

Contribution

It clarifies conditions under which multi-field theories evade Ostrogradsky instabilities and introduces a new Lorentz-violating scalar-tensor model.

Findings

Certain multi-field theories are free of Ostrogradsky ghosts despite higher derivatives.

A new Lorentz-violating scalar-tensor model is identified that cannot be transformed into Horndeski form.

Tensor modes in some theories are either non-dynamical or strongly coupled.

Abstract

We review how the (absence of) Ostrogradsky instability manifests itself in theories with multiple fields. It has recently been appreciated that when multiple fields are present, the existence of higher derivatives may not automatically imply the existence of ghosts. We discuss the connection with gravitational theories like massive gravity and beyond Horndeski which manifest higher derivatives in some formulations and yet are free of Ostrogradsky ghost. We also examine an interesting new class of Extended Scalar-Tensor Theories of gravity which has been recently proposed. We show that for a subclass of these theories, the tensor modes are either not dynamical or are infinitely strongly coupled. Among the remaining theories for which the tensor modes are well-defined one counts one new model that is not field-redefinable to Horndeski via a conformal and disformal transformation but that does require the vacuum to break Lorentz invariance. We discuss the implications for the effective field theory of dark energy and the stability of the theory.