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., $r\sim {10}^{-15}$. 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 ${\alpha }_s$ 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.

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• Cosmic microwave background
• Inflation
• Quantum fields in curved spacetime
• Axions