Helical Phase Inflation via Non-Geometric Flux Compactifications: from
  Natural to Starobinsky-like Inflation
  We show that a new class of helical phase inflation models can be simply
realized in minimal supergravity, wherein the inflaton is the phase component
of a complex field and its potential admits a deformed helicoid structure. We
find a new unique complex-valued index $\chi$ that characterizes almost the
entire region of the $n_s-r$ plane favored by new Planck observations.
Continuously varying the index $\chi$, predictions interpolate from
quadratic/natural inflation parameterized by a phase/axion decay constant to
Starobinsky-like inflation parameterized by the $\alpha$-parameter. We
demonstrate that the simple supergravity construction realizing
Starobinsky-like inflation can be obtained from a more microscopic model by
integrating out heavy fields, and that the flat phase direction for slow-roll
inflation is protected by a mildly broken global $U(1)$ symmetry. %, which is
mildly broken at the inflation energy scale. We study the geometrical origin of
the index $\chi$, and find that it corresponds to a linear constraint relating
\kah moduli. We argue that such a linear constraint is a natural result of
moduli stabilization in Type \MyRoman{2} orientifold compactifications on
Calabi-Yau threefolds with geometric and non-geometric fluxes. Possible choices
for the index $\chi$ are discrete points on the complex plane that relate to
the distribution of supersymmetric Minkowski vacua on moduli space. More
precise observations of the inflationary epoch in the future may provide a
better estimation of the index $\chi$. Since $\chi$ is determined by the fluxes
and vacuum expectation values of complex structure moduli, such observations
would characterize the geometry of the internal space as well.
