The Lyth Bound and the End of Inflation

We derive an extended version of the well-known Lyth Bound on the total variation of the inflaton field, incorporating higher order corrections in slow roll. We connect the field variation ∆φ to both the spectral index of scalar perturbations and the amplitude of tensor modes. We then investigate the implications of this bound for “small field” potentials, where the field rolls off a local maximum of the potential. The total field variation during inflation is generically of order mPl, even for potentials with a suppressed tensor/scalar ratio. Much of the total field excursion arises in the last e-fold of inflation and in single field models this problem can only be avoided via fine-tuning or the imposition of a symmetry. Finally, we discuss the implications of this result for inflationary model building in string theory and supergravity.