Linearization of a warped $f(R)$ theory in the higher-order frame

The linearization of a type of $f(R)$ gravity is studied directly in the higher-order frame for an arbitrary five-dimensional warped space-time background. The quadratic actions of the normal modes of the scalar, vector, and tensor perturbations are derived by taking the curvature gauge, under which the linear perturbation of the scalar curvature is zero, and all the perturbation equations reduce to second order. By comparing our results to those obtained in the Einstein frame, we find that the quadratic actions of the normal modes are equivalent for these two frames.