We propose an extension of natural inflation , where the inflaton potential is a general periodic function .
Specifically , we study elliptic inflation where the inflaton potential is given by Jacobi elliptic functions , Jacobi theta functions or the Dedekind eta function , which appear in gauge and Yukawa couplings in the string theories compactified on toroidal backgrounds .
We show that in the first two cases the predicted values of the spectral index and the tensor-to-scalar ratio interpolate from natural inflation to exponential inflation such as R ^ { 2 } - and Higgs inflation and brane inflation , where the spectral index asymptotes to n _ { s } = 1 - 2 / N \simeq 0.967 for the e-folding number N = 60 .
We also show that a model with the Dedekind eta function gives a sizable running of the spectral index due to modulations in the inflaton potential .
Such elliptic inflation can be thought of as a specific realization of multi-natural inflation , where the inflaton potential consists of multiple sinusoidal functions .
We also discuss examples in string theory where Jacobi theta functions and the Dedekind eta function appear in the inflaton potential .