Title: Frame invariance in higher-order gravitational theories
In the first part, we investigate whether the cosmological perturbations are invariant under general disformal transformation with the second-order covariant derivatives of the scalar field. We show that on superhorizon scales the difference in the comoving curvature perturbations between frames is given by the combination of the time derivative of the comoving curvature perturbation itself and the intrinsic entropy perturbation of the scalar field. In the case that the intrinsic adiabaticity of the scalar field is satisfied, the comoving curvature perturbation becomes invariant under generalized disformal transformation on superhorizon scales.
In the second part, we investigate how physical quantities associated with relativistic stars are related by the generalized disformal transformations constructed by the scalar and vector fields within the slow-rotation approximation. By imposing the asymptotic flatness in both frames, the ADM mass becomes frame invariant. We then discuss the disformal transformations of the frame-dragging function, angular velocity, angular momentum, and moment of inertia of the star. While the angular velocity of the star is frame invariant in all the cases, the invariance of the angular momentum and moment of inertia depends on whether the transformation is generated by the scalar or vector field. We also argue the subtlety to determine the surface of the star
in both the frames.