Symmetric Teleparallel Horndeski Gravity

TL;DR

This paper develops a symmetric teleparallel version of Horndeski gravity, exploring new second-order scalar-tensor theories based on nonmetricity, and applies it to cosmological models.

Contribution

It formulates a symmetric teleparallel analogue of Horndeski gravity, introducing new second-order theories based on nonmetricity and extending Kinetic Gravity Braiding.

Findings

Recast Horndeski as sum of Riemannian and teleparallel terms

Derived the most general k-essence extension under assumptions

Presented FLRW cosmological equations for the new model

Abstract

Horndeski gravity is the most general scalar-tensor theory with one scalar field leading to second-order Euler-Lagrange field equations for the metric and scalar field, and it is based on Riemannian geometry. In this paper, we formulate an analogue version of Horndeski gravity in a symmetric teleparallel geometry which assumes that both the curvature (general) and torsion are vanishing and gravity is only related to nonmetricity. Our setup requires that the Euler-Lagrange equations for not only metric and scalar field but also connection should be at most second order. We find that the theory can be always recast as a sum of the Riemannian Horndeski theory and new terms that are purely teleparallel. Due to the nature of nonmetricity, there are many more possible ways of constructing second-order theories of gravity. In this regard, up to some assumptions, we find the most general $k$-essence extension of Symmetric Teleparallel Horndeski gravity. We also formulate a novel theory containing higher-order derivatives acting on nonmetricity while still respecting the second-order conditions, which can be recast as an extension of Kinetic Gravity Braiding. We finish our study by presenting the FLRW cosmological equations for our model.