Spatially covariant theories of gravity: disformal transformation,
  cosmological perturbations and the Einstein frame
  We investigate the cosmological background evolution and perturbations in a
general class of spatially covariant theories of gravity, which propagates two
tensor modes and one scalar mode. We show that the structure of the theory is
preserved under the disformal transformation. We also evaluate the primordial
spectra for both the gravitational waves and the curvature perturbation, which
are invariant under the disformal transformation. Due to the existence of
higher spatial derivatives, the quadratic Lagrangian for the tensor modes
itself cannot be transformed to the form in the Einstein frame. Nevertheless,
there exists a one-parameter family of frames in which the spectrum of the
gravitational waves takes the standard form in the Einstein frame.
