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We argue that the non-detection of primordial tensor modes has taught us agreat 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 thisregime the conformal limit of (slow-roll) inflation, and show that it includesStarobinsky-like inflation as well as all viable single-field models with asub-Planckian field excursion. In this limit, all primordial correlators areconstrained by the full conformal group to leading non-trivial order inslow-roll. This fixes the power spectrum and the full bispectrum, and leads tothe "conformal" shape of non-Gaussianity. The size of non-Gaussianity isrelated to the running of the spectral index by a consistency condition, andtherefore it is expected to be small. In passing, we clarify the role ofboundary terms in the $\zeta$ action, the order to which constraint equationsneed to be solved, and re-derive our results using the Wheeler-deWittformalism.

The Conformal Limit of Inflation in the Era of CMB Polarimetry