Coupling Hybrid Inflation to Moduli
  Hybrid inflation can be realised in low-energy effective string theory, as described using supergravity. We find that the coupling of moduli to F-term hybrid inflation in supergravity leads to a slope and a curvature for the inflaton potential. The epsilon and eta parameters receive contributions at tree and one loop level which are not compatible with slow roll inflation. Furthermore the coupling to the moduli sector can even prevent inflation from ending at all. We show that introducing shift symmetries in the inflationary sector and taking the moduli sector to be no-scale removes most of these problems. If the moduli fields are fixed during inflation, as is usually assumed, it appears that viable slow-roll inflation can then be obtained with just one fine-tuning of the moduli sector parameters. However, we show this is not a reasonable assumption, and that the small variation of the moduli fields during inflation gives a significant contribution to the effective inflaton potential. This typically implies that eta is approximately -6, although it may be possible to obtain smaller values with heavy fine-tuning. 
Ph. Brax, C. van de Bruck, A. C. Davis and Stephen C. Davis