Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability (1510.06930v2)

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.

• The paper classifies and derives conditions for degenerate Lagrangians that eliminate ghostly degrees of freedom.
• It extends Horndeski theory by exploring higher derivative terms while ensuring a bounded Hamiltonian and stability.
• The Hamiltonian framework links degeneracy to the absence of extra polarizations, paving the way for advanced cosmological models.

Degenerate Higher Derivative Theories Beyond Horndeski: Evading the Ostrogradski Instability

The paper by Langlois and Noui investigates the degeneracy in higher derivative scalar-tensor theories, with a specific focus on avoiding the infamous Ostrogradski instability. Scalar-tensor theories have been central to exploring modifications to general relativity, particularly to explain cosmic acceleration, and they often incorporate higher-order derivative terms in the Lagrangian. These terms risk introducing ghost-like instabilities, but the authors propose a systematic framework to avoid such issues by exploiting the degeneracy characteristics of these theories.

Context and Objective

Higher derivative theories often succumb to Ostrogradski instability, leading to unbounded Hamiltonians. A straightforward way to prevent this instability is to employ degenerate Lagrangians, resulting in constraints that limit the number of physical degrees of freedom. The paper, therefore, aims to classify scalar-tensor theories that are quadratically dependent on second-order scalar derivatives and identify conditions under which they remain degenerate.

Key Contributions

1. Classification of Degenerate Theories: The paper presents a classification of scalar-tensor theories according to their degeneracy conditions. The authors derive systematic conditions to identify degenerate Lagrangians that restrict ghostly degrees of freedom.
2. Extension Beyond Horndeski: It expands on Horndeski theories, which inherently avoid Ostrogradski instability by construction of lower-order derivatives. The paper identifies additional degenerate families beyond these known frameworks, emphasizing scenarios where the degeneracy is not apparent in the individual scalar components but arises out of the full theory's structure.
3. Hamiltonian Framework: An in-depth Hamiltonian formulation provides a distinguishing approach to assess the number of degrees of freedom, linking degeneracy directly with the absence of extra polarizations associated with instability. This analysis forms an insightful complement to the Lagrangian analysis.

Implications and Future Directions

The exploration provided by Langlois and Noui forms a critical pathway in understanding viable modifications to gravity that naturally circumvent instabilities. The presented framework abrogates the necessity for exclusively second-order equations by broadening the spectrum of potential ghost-free theories.

Potential future developments will focus on extending this classification to theories with cubic dependencies on the scalar field's second derivatives, such as the quintic Lagrangian beyond Horndeski. Additionally, the application of these theories in cosmological modeling, including dark energy interpretations, and the theoretical groundwork towards full Hamiltonian developments, offer significant areas for research. The work challenges previously held constraints in modified gravity, potentially leading to a richer exploration of gravitational theories discontinuous with traditional paradigms like General Relativity yet stable under physical scrutiny.

Conclusion

The research by Langlois and Noui contributes substantially to the scalar-tensor theory landscape by systematizing the approach to evade Ostrogradski instability through novel degenerate Lagrangians. This not only reinforces understanding of existing theories like Horndeski but also paves the way for new theoretical models that could solve prevailing cosmological challenges.