Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability

TL;DR

This work addresses the Ostrogradski instability in scalar–tensor theories by developing a systematic degeneracy framework for Lagrangians quadratic in second derivatives of a scalar field. It introduces a toy model to illustrate how a degenerate kinetic matrix can eliminate the ghost and then applies a covariant 3+1 analysis to a broad class of theories, unifying Horndeski quartic terms with extensions beyond Horndeski. The authors classify all degenerate theories within this class, identifying that combinations like $L_4^{\rm H}+L_4^{\rm bH}$ are degenerate, while most general mixes including $L_5^{\rm bH}$ are not; they also reveal new degenerate families where the scalar sector is nondegenerate in isolation but becomes degenerate when coupled to gravity. The results provide a practical tool to locate ghost-free theories and clarify the role of gravity in stabilizing higher-derivative scalar–tensor models, with implications for constructing viable modified gravity theories. A thorough Hamiltonian treatment and further cubic-derivative analyses are suggested for future work.

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.