An inflation mass problem in string inflation from threshold corrections to volume stabilization

Inflationary models whose vacuum energy arises from a D-term are believed not to suffer from the supergravity eta problem of F-term inflation. That is, D-term models have the desirable property that the inflaton mass can naturally remain much smaller than the Hubble scale. We observe that this advantage is lost in models based on string compactifications whose volume is stabilized by a non-perturbative superpotential: the F-term energy associated with volume stabilization causes the eta problem to reappear. Moreover, any shift symmetries introduced to protect the inflaton mass will typically be lifted by threshold corrections to the volume-stabilizing superpotential. Using threshold corrections computed by Berg, Haack, and Kors, we illustrate this point in the example of the D3 - D7 inflationary model, and conclude that inflation is possible, but only for fine-tuned values of the stabilized moduli. More generally, we conclude that inflationary models in stable string compacti. cations, even D-term models with shift symmetries, will require a certain amount of fine-tuning to avoid this new contribution to the eta problem.