The Conformal Limit of Inflation in the Era of CMB Polarimetry
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 η, 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 ζ action, the order to which constraint equations need to be solved, and re-derive our results using the Wheeler-deWitt formalism.