  Inflationary cosmology proposes that the early Universe undergoes accelerated
expansion, driven, in simple scenarios, by a single scalar field, or inflaton.
The form of the inflaton potential determines the initial spectra of density
perturbations and gravitational waves. We show that constraints on the duration
of inflation together with the BICEP3/Keck bounds on the gravitational wave
background imply that higher derivatives of the potential are nontrivial with a
confidence of 99%. Such terms contribute to the scale-dependence, or running,
of the density perturbation spectrum. We clarify the ``universality classes''
of inflation in this limit showing that a very small gravitational wave
background can be correlated with a larger running. If pending experiments do
not observe a gravitational wave background the running will be at the
threshold of detectability if inflation is well-described at third-order in the
slow roll expansion.
