The Lyth Bound and the end of inflation

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Abstract
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 Δ 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.