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 Δϕ 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.