The Lyth bound revisited

We investigate the Lyth bound relating the tensor scalar ratio, r, to the variation of the inflaton field, Δphi, over the course of inflation. For inflationary models that produce at least 55 e-folds of inflation, there is a correlation between r and Δphi 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, we find that inflationary models with r \gtrsim 10^{-3} must have \Delta \phi \gtrsim m_{\mathrm {pl}} , unless the parameters of the models are finely tuned. Large field variations are generally required if a tensor mode signal is to be detected in any foreseeable cosmic microwave background (CMB) polarization experiment.