Disformal invariance of cosmological perturbations in a generalized
  class of Horndeski theories
  It is known that Horndeski theories can be transformed to a sub-class of
Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories under the disformal
transformation of the metric $g_{\mu \nu} \to \Omega^2(\phi)g_{\mu \nu}+\Gamma
(\phi,X) \nabla_{\mu} \phi \nabla_{\nu} \phi$, where $\Omega$ is a function of
a scalar field $\phi$ and $\Gamma$ is another function depending on both $\phi$
and $X=g^{\mu \nu}\nabla_{\mu} \phi \nabla_{\nu} \phi$. We show that, with the
choice of unitary gauge, both curvature and tensor perturbations on the flat
isotropic cosmological background are generally invariant under the disformal
transformation. By means of the effective field theories encompassing Horndeski
and GLPV theories, we obtain the second-order actions of scalar/tensor
perturbations and present the relations for physical quantities between the two
frames. The invariance of the inflationary power spectra under the disformal
transformation is explicitly proved up to next-to-leading order in slow-roll.
In particular, we identify the existence of the Einstein frame in which the
tensor power spectrum is of the same form as that in General Relativity and
derive the condition under which the spectrum of gravitational waves in GLPV
theories is red-tilted.
