Finding Horndeski theories with Einstein gravity limits

TL;DR

This paper develops a simple scaling method to identify when Horndeski scalar-tensor theories recover Einstein gravity through screening mechanisms, aiding the analysis of modified gravity models.

Contribution

It introduces a new scaling technique to determine Einstein gravity limits in Horndeski theories, applicable to various models including galileons and Brans-Dicke.

Findings

The method effectively identifies dominant terms in equations of motion.

It helps evaluate the presence of screening effects in models.

Applicable to post-Newtonian expansions in screened regimes.

Abstract

The Horndeski action is the most general scalar-tensor theory with at most second-order derivatives in the equations of motion, thus evading Ostrogradsky instabilities and making it of interest when modifying gravity at large scales. To pass local tests of gravity, these modifications predominantly rely on nonlinear screening mechanisms that recover Einstein's Theory of General Relativity in regions of high density. We derive a set of conditions on the four free functions of the Horndeski action that examine whether a specific model embedded in the action possesses an Einstein gravity limit or not. For this purpose, we develop a new and surprisingly simple scaling method that identifies dominant terms in the equations of motion by considering formal limits of the couplings that enter through the new terms in the modified action. This enables us to find regimes where nonlinear terms dominate and Einstein's field equations are recovered to leading order. Together with an efficient approximation of the scalar field profile, one can then further evaluate whether these limits can be attributed to a genuine screening effect. For illustration, we apply the analysis to both a cubic galileon and a chameleon model as well as to Brans-Dicke theory. Finally, we emphasise that the scaling method also provides a natural approach for performing post-Newtonian expansions in screened regimes.