A metric-affine version of the Horndeski theory

International audienceWe study the metric-affine versions of scalar-tensor theories whose connection enters the action only algebraically. We show that the connection can be integrated out, resulting in an equivalent metric theory. Specifically, we consider the metric-affine generalisations of the subset of the Horndeski theory whose action is linear in second derivatives of the scalar field. We determine the connection and find that it can describe a scalar-tensor Weyl geometry without a Riemannian frame. Still, as this connection enters the action algebraically, the theory admits the dynamically equivalent (pseudo)-Riemannian formulation in the form of an effective metric theory with an extra K-essence term. This may have interesting phenomenological applications