Generalized disformal invariance of cosmological perturbations with second-order field derivatives

We investigate how the comoving curvature and tensor perturbations are transformed under generalized disformal transformation with second-order covariant derivatives of scalar field, where free functions depend on fundamental elements constructed field at most quadratic order derivatives. Our analysis reveals that superhorizon scales difference between in original new frames is given by combination time derivative perturbation, intrinsic entropy perturbation its frame. Thus, case (1) vanish (2) frame, Rc, conserved, becomes invariant scales. also show disformally invariant, frame conserved time.