Higher derivative scalar-tensor theory through a non-dynamical scalar field

TL;DR

This paper introduces a novel class of higher derivative scalar-tensor theories that avoid Ostrogradsky instabilities by fixing a non-dynamical scalar field through a spatial gauge, resulting in a theory with only three dynamical degrees of freedom.

Contribution

The authors develop a new scalar-tensor theory framework with higher derivatives that maintains stability by fixing a scalar field non-dynamically, breaking time diffeomorphism invariance but preserving spatial covariance.

Findings

The theory has at most three dynamical degrees of freedom.

It avoids Ostrogradsky ghost instabilities.

The construction generalizes scalar-tensor theories with higher derivatives.

Abstract

We propose a new class of higher derivative scalar-tensor theories without the Ostrogradsky's ghost instabilities. The construction of our theory is originally motivated by a scalar field with spacelike gradient, which enables us to fix a gauge in which the scalar field appears to be non-dynamical. We dub such a gauge as the spatial gauge. Though the scalar field loses its dynamics, the spatial gauge fixing breaks the time diffeomorphism invariance and thus excites a scalar mode in the gravity sector. We generalize this idea and construct a general class of scalar-tensor theories through a non-dynamical scalar field, which preserves only spatial covariance. We perform a Hamiltonian analysis and confirm that there are at most three (two tensors and one scalar) dynamical degrees of freedom, which ensures the absence of a degree of freedom due to higher derivatives. Our construction opens a new branch of scalar-tensor theories with higher derivatives.