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.