Conformal Symmetries of Adiabatic Modes in Cosmology

TL;DR

The paper shows that scalar adiabatic perturbations in any FRW background exhibit non-linearly realized conformal symmetries on spatial slices, including dilatations and special conformal transformations, which act non-trivially on the curvature perturbation $\zeta$ via $\delta_\lambda \zeta$ and $\delta_{\vec{b}}\zeta$; these symmetries can be extended to adiabatic modes and give rise to Ward identities that recover known consistency relations and predict new SCT-based relations. It further analyzes the tensor sector, finding that dilatations extend to tensor perturbations while SCTs are incompatible with the transverse-traceless gauge, and uncover an infinite set of large gauge transformations that non-linearly shift the tensor modes. The results provide a unifying conformal-structure perspective on scalar and tensor perturbations, connect to the decoupling-limit EFT of inflation where de Sitter isometries emerge as exact symmetries, and set the stage for future derivations of explicit Ward identities and tensor-related consistency relations. Overall, the work highlights how conformal symmetry considerations constrain cosmological correlators beyond slow-roll and quasi-de Sitter regimes, with potential observational implications via soft limits.

Abstract

We remark on the existence of non-linearly realized conformal symmetries for scalar adiabatic perturbations in cosmology. These conformal symmetries are present for any cosmological background, beyond any slow-roll or quasi-de Sitter approximation. The dilatation transformation shifts the curvature perturbation by a constant, and corresponds to the well-known symmetry under spatial rescaling. We argue that the scalar sector is also invariant under special conformal transformations, which shift the curvature perturbation by a term linear in the spatial coordinates. We discuss whether these conformal symmetries can be extended to include tensor perturbations. Tensor modes introduce their own set of non-linearly realized symmetries. We identify an infinite set of large gauge transformations which maintain the transverse, traceless gauge condition, while shifting the tensor mode non-trivially.