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The Horndeski Lagrangian brings together all possible interactions betweengravity and a scalar field that yield second-order field equations infour-dimensional spacetime. As originally proposed, it only addressesphenomenology without torsion, which is a non-Riemannian feature of geometry.Since torsion can potentially affect interesting phenomena such asgravitational waves and early Universe inflation, in this paper we allowtorsion to exist and propagate within the Horndeski framework. To achieve thisgoal, we cast the Horndeski Lagrangian in Cartan's first-order formalism, andintroduce wave operators designed to act covariantly on p-form fields thatcarry Lorentz indices. We find that nonminimal couplings and second-orderderivatives of the scalar field in the Lagrangian are indeed generic sources oftorsion. Metric perturbations couple to the background torsion and newtorsional modes appear. These may be detected via gravitational waves but notthrough Yang-Mills gauge bosons.

Nonminimal couplings, gravitational waves, and torsion in Horndeski's theory