Natural inflation with multiple sub-Planckian axions

We extend the Kim-Nilles-Peloso (KNP) alignment mechanism for natural inflation to models with $N>2$ axions, which obtains a super-Planckian effective axion decay constant $f_{\textrm{eff}}\gg M_{Pl}$ through an alignment of the anomaly coefficients of multiple axions having sub-Planckian fundamental decay constants $f_0\ll M_{Pl}$. The original version of the KNP mechanism realized with two axions requires that some of the anomaly coefficients should be of the order of $f_{\textrm{eff}}/f_0$, which would be uncomfortably large if $f_{\rm eff}/f_0 \gtrsim {\cal O}(100)$ as suggested by the recent BICEP2 results. We note that the KNP mechanism can be realized with the anomaly coefficients of $\mathcal{O}(1)$ if the number of axions $N$ is large as $N\ln N\gtrsim 2\ln (f_{\textrm{eff}}/f_0)$, in which case the effective decay constant can be enhanced as $f_{\rm eff}/f_0 \sim \sqrt{N !}\,n^{N-1}$ for $n$ denoting the typical size of the integer-valued anomaly coefficients. Comparing to the other multiple axion scenario, the N-flation scenario which requires $N \sim f_{\textrm{eff}}^2/f_0^2$, the KNP mechanism has a virtue of not invoking to a too large number of axions, although it requires a specific alignment of the anomaly coefficients, which can be achieved with a probability of ${\cal O}(f_0/f_{\rm eff})$ under a random choice of the anomaly coefficients. We also present a simple model realizing a multiple axion monodromy along the inflaton direction.