Issues in Complex Structure Moduli Inflation

Supersymmetric compactification with moderately large radius (${\rm Re} \sim {\cal O}(10)$ or more) not only accommodates supersymmetric unification, but also provides candidates for an inflaton in the form of geometric moduli; the value of ${\rm Re} > 1$ may be used as a parameter that brings corrections to the inflaton potential under control. Motivated by a bottom-up idea "right-handed sneutrino inflation" scenario, we study whether complex structure moduli can play some role during the slow-roll inflation and/or reheating process in this moderately large radius regime. Even when we allow a tuning introduced by Kallosh and Linde, the barrier of volume stabilization potential from gaugino condensation racetrack superpotential can hardly be as high as $(10^{16} \; {\rm GeV})^4$ for generic choice of parameters in this regime. It is also found that even very small deformation of complex structure during inflation/reheating distorts the volume stabilization potential, so that the volume stabilization imposes tight constraints on large-field inflation scenario involving evolution of complex structure moduli. A few ideas of satisfying those constraints in string theory are also discussed.