Natural inflation with exponentially small tensor-to-scalar ratio

We demonstrate that ``natural inflation,'' also known as ``axion inflation,'' can be compatible with Planck 2018 measurements of the cosmic microwave background while predicting an exponentially small tensor-to-scalar ratio, e.g., $r\ensuremath{\sim}{10}^{\ensuremath{-}15}$. The strong suppression of $r$ arises from dynamics of the radial component of the complex scalar field, whose phase is the axion. Such tiny values of $r$ remain well below the threshold for detection by CMB-S4 or Simons Observatory B-mode searches. The model is testable with the running ${\ensuremath{\alpha}}_{s}$ of the spectral index, which is within reach of next-generation cosmic microwave background and large-scale structure experiments, motivating the running as a primary science goal for future experiments.