The Lyth Bound Revisited

TL;DR

Problem addressed: robustness of the Lyth bound relating $r$ to inflaton excursion $\Delta \phi$ across broad inflationary models. Approach: generate a large ensemble of inflationary-flow realizations with $N \ge 55$ and compute observables, applying current constraints on $n_s$ and $dn_s/d\ln k$. Key findings: without priors there is large scatter, but restricting to observationally allowed models yields a tight relation $\Delta \phi/m_{pl} \approx 6 r^{1/4}$ for $r \gtrsim 10^{-3}$, implying large-field excursions are needed for detectable tensors. Significance: near-term CMB B-mode polarization experiments will primarily probe high-field inflation, with implications for model-building and EFT embeddings (e.g., shift-symmetric constructions).

Abstract

We investigate the Lyth relationship between the tensor-scalar ratio, r, and the variation of the inflaton field, Delta phi, over the course of inflation. For inflationary models that produce at least 55 e-folds of inflation, there is a correlation between r and Delta phi 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 Delta phi > m_pl. Large field variations are therefore required if a tensor mode signal is to be detected in any foreseeable cosmic microwave background (CMB) polarization experiment.

Figures (2)

• Figure 1: The tensor-scalar ratio, $r$, scalar spectral index, $n_s$, and run in the spectral index, $dn_s/d{\rm ln} k$, plotted against each other for realizations of the inflationary flow equations described in the text. The colour coding is as follows: the majority of models (colour coded red) inflate forever with $\epsilon = 0$ and have negligible tensor component. In the remaining models, inflation ends when $\epsilon = 1$ after at least 55 e-folds of inflation. These models have been colour coded blue if $r \le 0.36$ (the observational upper limit derived by Seljak 2004) and green if $r >0.36$.
• Figure 2: The absolute value of $\Delta \phi$ over the last $55$ e-folds of inflation plotted against the tensor-scalar ratio, $r$. Figure 2a shows all models colour coded so that hybrid-type models are shown in red and the rest are shown in blue. Figure 2b shows the subset of models that satisfy the observational constraints on $n_s$ and $dn_s/d{\rm ln} k$ discussed in the text. The colour coding in this figure is as follows: hybrid-type models are shown in orange and the rest of the models are shown in magenta.