Slowly rotating black holes in alternative theories of gravity

We present, in closed analytic form, a general stationary, slowly rotating black hole, which is a solution to a large class of alternative theories of gravity in four dimensions. In these theories, the Einstein-Hilbert action is supplemented by all possible quadratic, algebraic curvature invariants coupled to a scalar field. The solution is found as a deformation of the Schwarzschild metric in general relativity. We explicitly derive the changes to the orbital frequency at the innermost stable circular orbit and at the light ring in closed form. These results could be useful when comparing general relativity against alternative theories by (say) measurements of x-ray emission in accretion disks, or by stellar motion around supermassive black holes. When gravitational-wave astronomy comes into force, strong constraints on the coupling parameters can in principle be made.