A Permutation on Hybrid Natural Inflation

We analyze a model of hybrid natural inflation based on the smallest non-Abelian discrete group S_3. Leading invariant terms in the scalar potential have an accidental global symmetry that is spontaneously broken, providing a pseudo-Goldstone boson that is identified as the inflaton. The S_3 symmetry restricts both the form of the inflaton potential and the couplings of the inflaton field to the waterfall fields responsible for the end of inflation. We identify viable points in the model parameter space. Although the power in tensor modes is small in most of the parameter space of the model, we identify parameter choices that yield potentially observable values of r without super-Planckian initial values of the inflaton field.