Transformation of Primordial Cosmological Perturbations Under the
  General Extended Disformal Transformation
  Primordial cosmological perturbations are the seeds that were cultivated by
inflation and the succeeding dynamical processes, eventually leading to the
current Universe. In this work, we investigate the behavior of the
gauge-invariant scalar and tensor perturbations under the general extended
disformal transformation, namely, $g_{\mu\nu} \rightarrow A(X,Y,Z)g_{\mu\nu} +
\Phi_\mu\Phi_\nu$, where $X \equiv -\tfrac{1}{2}\phi^{;\mu}\phi_{;\mu}, Y
\equiv \phi^{;\mu}X_{;\mu}, Z \equiv X^{;\mu}X_{;\mu} $ and $\Phi_\mu \equiv
C\phi_{;\mu} + DX_{;\mu}$, with $C$ and $D$ being a general functional of
$(\phi,X,Y,Z)$. We find that the tensor perturbation is invariant under this
transformation. On the other hand, the scalar curvature perturbation receives a
correction due the conformal term only; it is independent of the disformal term
at least up to linear order. Within the framework of the full Horndeski theory,
the correction terms turn out to depend linearly on the gauge-invariant
comoving density perturbation and the first time-derivative thereof. In the
superhorizon limit, all these correction terms vanish, leaving only the
original scalar curvature perturbation. In other words, it is invariant under
the general extended disformal transformation in the superhorizon limit, in the
context of full Horndeski theory. Our work encompasses a chain of research
studies on the transformation or invariance of the primordial cosmological
perturbations, generalizing their results under our general extended disformal
transformation.
