Horndeski: beyond, or not beyond?

TL;DR

This paper investigates the degrees of freedom in beyond Horndeski scalar-tensor theories, showing it is a healthy but isolated theory that cannot be transformed into Horndeski, and discussing implications for gravity models.

Contribution

It provides a gauge-invariant analysis of beyond Horndeski, revealing its invariance under disformal transformations and its status as an isolated, potentially ghost-free theory.

Findings

Beyond Horndeski is gauge-invariant and isolated.

Mixing beyond Horndeski with different order theories obstructs primary constraint construction.

Disformal transformations cannot map beyond Horndeski to Horndeski alone.

Abstract

Determining the most general, consistent scalar tensor theory of gravity is important for building models of inflation and dark energy. In this work we investigate the number of degrees of freedom present in the theory of beyond Horndeski. We discuss how to construct the theory from the extrinsic curvature of the constant scalar field hypersurface, and find a simple expression for the action which guarantees the existence of the primary constraint necessary to avoid the Ostrogradsky instability. Our analysis is completely gauge-invariant. However we confirm that, mixing together beyond Horndeski with a different order of Horndeski, obstructs the construction of this primary constraint. Instead, when the mixing is between actions of the same order, the theory can be mapped to Horndeski through a generalised disformal transformation. This mapping however is impossible with beyond Horndeski alone, since we find that the theory is invariant under such a transformation. The picture that emerges is that beyond Horndeski is a healthy but isolated theory: combined with Horndeski, it either becomes Horndeski, or likely propagates a ghost.