The Lyth Bound Revisited

We investigate the Lyth relationship between the tensor-scalar ratio, r, and the variation of the inflaton field, ∆φ, over the course of inflation. For inflationary models that produce at least 55 e-folds of inflation, there is a correlation between r and ∆φ as anticipated by Lyth, but the scatter around the relationship is huge. However, for inflationary models that satisfy current observational constraints on the scalar spectral index and its first derivative, the Lyth relationship is much tighter. In particular, any inflationary model with r > ∼ 10 −3 must have ∆φ > ∼ mpl. Large field variations are therefore required if a tensor mode signal is to be detected in any foreseeable cosmic microwave background (CMB) polarization experiment.