The Lyth Bound and the end of inflation

Abstract

We derive an extended version of the well-known Lyth Bound on the total variation of theinflaton field, incorporating higher-order corrections in slow roll. We connect the field variationΔϕ to both the spectral index of scalar perturbations and the amplitude of tensor modes. We theninvestigate the implications of this bound for ‘small field’ potentials, where the field rolls off alocal maximum of the potential. The total field variation during inflation is generically of ordermPl, even for potentials with a suppressed tensor/scalar ratio. Much of the total field excursionarises in the last e-fold of inflation and in single field models this problem can only beavoided via fine-tuning or the imposition of a symmetry. Finally, we discuss theimplications of this result for inflationary model building in string theory and supergravity.