The Lyth Bound and the End of Inflation

TL;DR

This work extends the Lyth bound to second order in slow-roll, deriving a relation between the total inflaton excursion $Δφ$ and both $r$ and $n$ via $Δφ/m_Pl ≈ sqrt{r/(4π)} [1 - (n-1) - r/8]$, while emphasizing how the end-stage dynamics can alter the bound. It shows that, in generic single-field models, most of the field variation accumulates during the last e-fold, yielding $Δφ ∼ m_Pl$ even when observable-scale perturbations imply a small tensor amplitude. To maintain sub-Planckian excursions, the inflaton mass term must be suppressed, either by tuning or a symmetry, suggesting that viable models may rely on hybrid-type dynamics or symmetry-protected potentials. The results constrain inflationary model-building in string theory and supergravity and inform expectations for tensor mode detections in the CMB, since measuring $r$ fixes the inflationary energy scale and V′ behavior under slow-roll.

Abstract

We derive an extended version of the well-known Lyth Bound on the total variation of the inflaton field, incorporating higher order corrections in slow roll. We connect the field variation $Δφ$ to both the spectral index of scalar perturbations and the amplitude of tensor modes. We then investigate the implications of this bound for ``small field'' potentials, where the field rolls off a local maximum of the potential. The total field variation during inflation is {\em generically} of order $m_{\rm Pl}$, even for potentials with a suppressed tensor/scalar ratio. Much of the total field excursion arises in the last e-fold of inflation and in single field models this problem can only be avoided via fine-tuning or the imposition of a symmetry. Finally, we discuss the implications of this result for inflationary model building in string theory and supergravity.

Figures (2)

• Figure 1: Histogram of $\Delta \phi/m_{\rm Pl}$ over the last e-fold for 12,000 inflation models satisfying the constraint $0.9 < n < 1.1$ on the spectral index. Red ($n < 1$) and blue ($n > 1$) spectra are plotted separately.
• Figure 2: $\sqrt{\epsilon}$ vs. $N$ over the last e-fold for the potentials $V(\phi) \ \lambda \phi^4$ (dotted line) and $V(\phi) =\Lambda^4[1-(\phi/\mu)^4]$ (solid line). The symmetry breaking scale for the inverted potential is $\mu = 0.1m_{\rm Pl}$.