Transformation of Primordial Cosmological Perturbations Under the General Extended Disformal Transformation

TL;DR

This paper studies how primordial cosmological perturbations transform under a broad class of extended disformal transformations, finding tensor perturbations are invariant and scalar perturbations are mostly unaffected on large scales within Horndeski gravity.

Contribution

It introduces a comprehensive analysis of scalar and tensor perturbations under the most general extended disformal transformation, extending previous invariance results within Horndeski theory.

Findings

Tensor perturbations are invariant under the transformation.

Scalar curvature perturbations receive corrections only from conformal terms.

All correction terms vanish on superhorizon scales, preserving the original scalar perturbation.

Abstract

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.