The majority of models of inflation in string theory predict an absence of measurable gravitational waves, $r<<{10}^{-3}$. The most promising proposals for making string theoretic models that yield measurable tensor fluctuations involve axion fields with slightly broken shift symmetry. We consider such models in detail, with a particular focus on the $N$-flation scenario and on axion valley/natural inflation models. We find that in Calabi-Yau threefold compactifications with logarithmic Kähler potentials $K$ it appears to be difficult to meet the conditions required for axion inflation in the supergravity regime. However, in supergravities with an (approximately) quadratic shift-symmetric $K$, axion inflation may be viable. Such Kähler potentials do arise in some string models, in specific limits of the moduli space. We describe the most promising classes of models; more detailed study will be required before one can conclude that working models exist.

• Received 31 October 2007

DOI:https://doi.org/10.1103/PhysRevD.77.043501