Understand
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.
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