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We investigate the cosmological background evolution and perturbations in ageneral class of spatially covariant theories of gravity, which propagates twotensor modes and one scalar mode. We show that the structure of the theory ispreserved under the disformal transformation. We also evaluate the primordialspectra for both the gravitational waves and the curvature perturbation, whichare invariant under the disformal transformation. Due to the existence ofhigher spatial derivatives, the quadratic Lagrangian for the tensor modesitself cannot be transformed to the form in the Einstein frame. Nevertheless,there exists a one-parameter family of frames in which the spectrum of thegravitational waves takes the standard form in the Einstein frame.

Spatially covariant theories of gravity: disformal transformation, cosmological perturbations and the Einstein frame