Slowly rotating black hole solutions in Horndeski gravity

We study black hole solutions at first order in the Hartle-Thorne slow-rotation approximation in Horndeski gravity theories. We derive the equations of motion including also cases where the scalar depends linearly on time. In the Hartle-Thorne formalism, all first-order rotational corrections are described by a single frame-dragging function. We show that the frame-dragging function is exactly the same as in general relativity for all known black hole solutions in shift-symmetric Horndeski theories, with the exception of theories with a linear coupling to the Gauss-Bonnet invariant. Our results extend previous no-hair theorems for a broad class of Horndeski gravity theories.