Last stand of single small field inflation

Abstract

By incorporating both the tensor-to-scalar ratio and the measured value of the spectral index, we set a bound on solo small field inflation of $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}/{m}_{\text{Pl}}\ensuremath{\ge}1.00\sqrt{r/0.1}$. Unlike previous bounds which require monotonic ${\ensuremath{\epsilon}}_{V}$, $|{\ensuremath{\eta}}_{V}|<1$, and 60 $e$-folds of inflation, the bound remains valid for nonmonotonic ${\ensuremath{\epsilon}}_{V}$, $|{\ensuremath{\eta}}_{V}|\ensuremath{\gtrsim}1$, and for inflation which occurs only over the eight $e$-folds which have been observed on the cosmic microwave background. The negative value of the spectral index over the observed eight $e$-folds is what makes the bound strong; we illustrate this by surveying single field models and finding that for $r\ensuremath{\gtrsim}0.1$ and eight $e$-folds of inflation, there is no simple potential which reproduces observed cosmic microwave background perturbations and remains sub-Planckian. Models that are sub-Planckian after eight $e$-folds must be patched together with a second epoch of inflation that fills out the remaining $\ensuremath{\sim}50$ $e$-folds. This second, post--cosmic microwave background epoch is characterized by extremely small ${\ensuremath{\epsilon}}_{V}$ and therefore an increasing scalar power spectrum. Using the fact that large power can overabundantly produce primordial black holes, we bound the maximum energy level of the second phase of inflation.