BICEP2 implications for single-field slow-roll inflation revisited

TL;DR

The paper investigates whether single-field slow-roll inflation can generate a tensor-to-scalar ratio $r \gtrsim 0.1$ with inflaton excursions $\Delta \phi$ well below the Planck scale. It derives a model-independent bound $\Delta \phi/M_{\rm Pl} \gtrsim \frac{0.11}{\langle \eta - 2\varepsilon\rangle}\sqrt{r/0.1}$ that ties large $r$ to Planck-scale field ranges, and supports this with a local potential reconstruction showing the potential must be nearly linear near $\phi_*$; higher-order slow-roll terms are suppressed by $\sqrt{r}$. The analysis also corrects a prior claim that $r>0.15$ could occur with $\Delta \phi<0.1 M_{\rm Pl}$ by pointing out a faulty momentum-space integral approximation. The results imply that a confirmed BICEP2-like signal would place strong pressure on small-field models of single-field slow-roll inflation and help distinguish inflationary scenarios.

Abstract

It is generally believed that in single-field slow-roll inflation, a large tensor-to-scalar ratio $r > 0.1$ requires inflaton field values close to or above the Planck scale. Recently, it has been claimed that $r > 0.15$ can be achieved with much smaller inflaton field values $Δφ< M_{Pl}/10$. We show that in single-field slow-roll inflation, it is impossible to reconcile $r > 0.1$ with such small field values, independently of the form of the potential, and that the recent claim to the contrary is based on an invalid approximation. We conclude that the result of the BICEP2 measurement of $r > 0.1$, if confirmed, truly has the potential to rule out small-field models of single-field slow-roll inflation.

Figures (2)

• Figure 1: Schematic potential shape which would be optimal for evading the Lyth bound. The slope is very big at the horizon crossing scale $\phi_*$, thereby generating the required tensor-to-scalar ratio $r=16\varepsilon_*$. It then quickly flattens until a point $\phi_{\rm min}$ where the slow-roll parameter $\varepsilon$ has a local minimum. This flat region is necessary to generate sufficiently many e-folds with small field excursion, as $\operatorname{dN/d\phi} \simeq (2\varepsilon)^{-1/2}$. Afterwards, inflation ends at some point $\phi_{\rm e} \leq \phi_{\rm min}$, either because the potential becomes to steep for slow-roll inflation or because of a waterfall transition like in hybrid inflation.
• Figure 2: Left: local reconstruction of the inflaton potential around $\phi_*$ for $r=0.1$ (blue band) and $r=0.2$ (red band), see eq. \ref{['eq:VAnsatz']}. The width of the bands (upper and lower bound of the potential) is due to the uncertainty in the slow-roll parameters $\eta$, $\xi^2$ and $\sigma^3$, which must not be larger than $O(10^{-2})$ in order to satisfy the Planck constraints on the observed approximate scale invariance of the scalar spectrum. For this plot, we have allowed these slow-roll parameters to vary within the range $[-0.05,0.05]$. Due to the large value of $r$, the potential is then forced to be nearly linear near $\phi_*$. Right: Slow-roll parameter $\varepsilon$ for this potential. Because the potential is nearly linear near $\phi_*$, $\varepsilon$ cannot decrease significantly until $\lvert \delta \phi \rvert > M_{\rm Pl}/2$. This makes it impossible to generate enough e-foldings of inflation with small field values. As a reference, the maximum $\varepsilon$ for which $\Delta N \sim 8$ e-folds could be realized within $\Delta \phi \leq M_{\rm Pl}/2$ is shown as a dashed horizontal line.