Higher Derivative Scalar Tensor Theory in Unitary Gauge

TL;DR

This paper investigates higher derivative scalar-tensor theories, identifying conditions to avoid Ostrogradsky instability through degeneracy, and constructs a stable, GR-like Lagrangian in the unitary gauge with applications to cosmological models.

Contribution

It derives degeneracy conditions for higher derivative scalar-tensor theories to prevent Ostrogradsky instability and constructs a stable, GR-like Lagrangian in the unitary gauge.

Findings

Identified degeneracy conditions avoiding Ostrogradsky instability.

Constructed a stable Lagrangian resembling GR in the unitary gauge.

Derived equations of motion for flat FRW cosmology.

Abstract

Ostrogradsky instability generally appears in nondegenerate higher-order derivative theories and this issue can be resolved by removing any existing degeneracy present in such theories. We consider an action involving terms that are at most quadratic in second derivatives of the scalar field and non-minimally coupled with the curvature tensors. We perform a 3+1 decomposition of the Lagrangian to separate second-order time derivative terms from rest. This decomposition is useful for checking the degeneracy hidden in the Lagrangian and helps us find conditions under which Ostrogradsky instability does not appear. We show that our construction of Lagrangian resembles that of a GR-like theory for a particular case in the unitary gauge. As an example, we calculate the equation of motion for the flat FRW. We also write the action for open and closed cases, free from higher derivatives for a particular choice derived from imposing degeneracy conditions.