Symmetric teleparallel Horndeski gravity

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 analog 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.

Automated summary

In this paper, an analog version of Horndeski gravity is formulated in a symmetric teleparallel geometry assuming vanishing curvature and torsion. The theory can be recast as a sum of the Riemannian-Horndeski theory and purely teleparallel terms. The most general k-essence extension of symmetric teleparallel Horndeski gravity is found, as well as a novel theory containing higher-order derivatives acting on nonmetricity while respecting the second-order conditions. The FLRW cosmological equations for the model are also presented.