Elliptic Inflation: Interpolating from natural inflation to
  $R^2$-inflation
  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.
