  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.
