Non-linear Realizations of Conformal Symmetry and Effective Field Theory for the Pseudo-Conformal Universe

TL;DR

The paper develops a comprehensive EFT for the pseudo-conformal universe by applying the coset construction to the symmetry breaking pattern $so(4,2)\to so(4,1)$, and corroborates the EFT with a curvature-invariants approach. It shows that weight-0 spectator fields acquire a scale-invariant spectrum while the Goldstone mode $\pi$ exhibits a strongly red-tilted spectrum, and it computes key two- and three-point functions to illustrate symmetry constraints. The work highlights the existence of Ward identities tied to the full $so(4,2)$ symmetry that could yield distinguishing relations from inflation, and clarifies the non-inflationary nature of the background despite de Sitter-like behavior for matter fields in the Jordan frame. Overall, this framework generalizes pseudo-conformal realizations beyond specific models and provides a robust platform for exploring observational signatures and consistency relations.

Abstract

The pseudo-conformal scenario is an alternative to inflation in which the early universe is described by an approximate conformal field theory on flat, Minkowski space. Some fields acquire a time-dependent expectation value, which breaks the flat space so(4,2) conformal algebra to its so(4,1) de Sitter subalgebra. As a result, weight-0 fields acquire a scale invariant spectrum of perturbations. The scenario is very general, and its essential features are determined by the symmetry breaking pattern, irrespective of the details of the underlying microphysics. In this paper, we apply the well-known coset technique to derive the most general effective lagrangian describing the Goldstone field and matter fields, consistent with the assumed symmetries. The resulting action captures the low energy dynamics of any pseudo-conformal realization, including the U(1)-invariant quartic model and the Galilean Genesis scenario. We also derive this lagrangian using an alternative method of curvature invariants, consisting of writing down geometric scalars in terms of the conformal mode. Using this general effective action, we compute the two-point function for the Goldstone and a fiducial weight-0 field, as well as some sample three-point functions involving these fields.