Horndeski theories and beyond from higher dimensions

TL;DR

This paper explores how higher-dimensional gravity theories can be dimensionally reduced to generate four-dimensional scalar-tensor theories, including Horndeski and beyond-Horndeski models, with implications for healthy modified gravity theories.

Contribution

It introduces a method to derive four-dimensional scalar-tensor theories from higher-dimensional actions, encompassing Horndeski and beyond-Horndeski terms, using a generalized metric decomposition.

Findings

Recover Horndeski theories from higher dimensions

Generate beyond-Horndeski terms via higher-dimensional scalar curvature

Include Gauss-Bonnet contributions leading to higher-derivative terms

Abstract

The Einstein-Hilbert action with a cosmological constant is the most general local four-dimensional action leading to second-order derivative equations of motion that are symmetric and divergence free. In higher dimensions, additional terms can appear. We investigate a generalised metric decomposition involving a scalar degree of freedom to express the higher-dimensional action as an effective four-dimensional scalar-tensor theory. From the higher-dimensional Ricci scalar alone and a subclass of our metric ansatz, we recover the subset of Horndeski theories with luminal speed of gravitational waves. More generally, beyond-Horndeski terms appear. When including a Gauss-Bonnet scalar in the higher-dimensional action, we generate contributions to all cubic-order second-derivative terms present in the degenerate higher-order scalar-tensor theory as well as higher-derivative terms beyond that. We discuss this technique as a way to generate healthy four-dimensional gravity theories with an extra scalar degree of freedom and outline further generalisations of our method.