Phenomenological Classification of Inflationary Potentials

We investigate the Lyth relationship between the tensor‐scalar ratio, r, and the variation of the inflaton field, Δφ, over the course of inflation. We compute r and Δφ for 10⁶ single‐field inflationary models generated using the the Monte Carlo reconstruction method of Easther and Kinney. Despite a large scatter, our results show that the Lyth relation provides a useful classification of inflationary models since a detectable value of the tenser‐scalar ratio necessarily requires an inflaton field variation of the order of the Planck mass. Inflationary models that satisfy current observational constraints on the scalar spectral index and its first derivative satisfy a much tighter relation between r and Δφ. For example, all models with r ≳ 0.01 (roughly the limit at which lensing of E‐modes produces a B‐mode signal comparable to those generated by inflation), require a field variation Δφ ≳ 1.5m_Pl. Such a large variation probably requires that the underlying theory of inflation possesses a shift symmetry φ → φ + constant.