  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 nonperturbative 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
compactifications, even D-term models with shift symmetries, will require a
certain amount of fine-tuning to avoid this new contribution to the eta
problem.
