Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability

TL;DR

This paper systematically classifies degenerate scalar-tensor theories with second derivatives, extending beyond Horndeski, to evade Ostrogradski instabilities and identify new stable theories.

Contribution

It derives degeneracy conditions for quadratic second derivative scalar-tensor theories and classifies all such degenerate models, including extensions beyond Horndeski.

Findings

Quartic Horndeski and beyond Horndeski are degenerate theories.

New families of degenerate theories with nondegenerate scalar parts.

Systematic classification of degenerate scalar-tensor theories.

Abstract

Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering "degenerate" Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of scalar-tensor theories. The quartic Horndeski Lagrangian and its extension beyond Horndeski belong to these degenerate cases. We also identify new families of scalar-tensor theories with the intriguing property that they are degenerate despite the nondegeneracy of the purely scalar part of their Lagrangian.