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Primordial cosmological perturbations are the seeds that were cultivated byinflation and the succeeding dynamical processes, eventually leading to thecurrent Universe. In this work, we investigate the behavior of thegauge-invariant scalar and tensor perturbations under the general extendeddisformal 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 $\Phi_\mu \equivC\phi_{;\mu} + DX_{;\mu}$, with $C$ and $D$ being a general functional of$(\phi,X,Y,Z)$. We find that the tensor perturbation is invariant under thistransformation. On the other hand, the scalar curvature perturbation receives acorrection due the conformal term only; it is independent of the disformal termat least up to linear order. Within the framework of the full Horndeski theory,the correction terms turn out to depend linearly on the gauge-invariantcomoving density perturbation and the first time-derivative thereof. In thesuperhorizon limit, all these correction terms vanish, leaving only theoriginal scalar curvature perturbation. In other words, it is invariant underthe general extended disformal transformation in the superhorizon limit, in thecontext of full Horndeski theory. Our work encompasses a chain of researchstudies on the transformation or invariance of the primordial cosmologicalperturbations, generalizing their results under our general extended disformaltransformation.

Transformation of Primordial Cosmological Perturbations Under the General Extended Disformal Transformation