The Conformal Limit of Inflation in the Era of CMB Polarimetry

TL;DR

This paper introduces the conformal limit of inflation, constrained by current CMB data, which predicts specific shapes for primordial correlators and non-Gaussianity, linking them to the spectral index running.

Contribution

It defines the conformal limit of inflation based on observational bounds and derives the resulting constraints on primordial correlators and non-Gaussianity shapes.

Findings

Primordial correlators are constrained by the conformal group.

The shape of non-Gaussianity is predicted to be conformal.

Non-Gaussianity size is related to spectral index running.

Abstract

We argue that the non-detection of primordial tensor modes has taught us a great deal about the primordial universe. In single-field slow-roll inflation, the current upper bound on the tensor-to-scalar ratio, $r < 0.07$ $(95 \% ~CL)$, implies that the Hubble slow-roll parameters obey $\varepsilon \ll \eta$, and therefore establishes the existence of a new hierarchy. We dub this regime the conformal limit of (slow-roll) inflation, and show that it includes Starobinsky-like inflation as well as all viable single-field models with a sub-Planckian field excursion. In this limit, all primordial correlators are constrained by the full conformal group to leading non-trivial order in slow-roll. This fixes the power spectrum and the full bispectrum, and leads to the "conformal" shape of non-Gaussianity. The size of non-Gaussianity is related to the running of the spectral index by a consistency condition, and therefore it is expected to be small. In passing, we clarify the role of boundary terms in the $\zeta$ action, the order to which constraint equations need to be solved, and re-derive our results using the Wheeler-deWitt formalism.