The Lyth Bound of Inflation with a Tilt

Abstract

We provide strong evidence for universality of the inflationary field range: given an accurate measurement of $(n_s,r)$, one can infer $Δφ$ in a model-independent way in the sub-Planckian regime for a range of universality classes of inflationary models. Both the tensor-to-scalar ratio as well as the spectral tilt are essential for the field range. Given the Planck constraints on $n_s$, the Lyth bound is strengthened by two orders of magnitude: whereas the original bound gives a sub-Planckian field range for $r \lesssim 2 \cdot 10^{-3}$, we find that $n=0.96$ brings this down to $r \lesssim 2 \cdot 10^{-5}$.

Figures (3)

• Figure 1: Two curves indicating $\sqrt{r(N)/8}$ with identical areas $\Delta \phi=1$. The flat curve depicts the Lyth bound, while the tilted curve indicates the improvement when taking the spectral index into account.
• Figure 2: Field ranges corresponding to $\Delta \phi = (0.1,1,10)$ in the plane ($n_s,~\log_{10}{(r)}$). The green straight dashed lines represent the asymptotic behaviour for large $p$.
• Figure 3: The range of field values corresponding to $r = 0.2,\,0.1,\,004,\,0.01,\,0.001,\,0.00001$ in the plane ($n_s,~\Delta\phi$).