Supersymmetry, Axions and Cosmology
  Various authors have noted that in particular models, the upper bound on the axion decay constant may not hold. We point out that within supersymmetry, this is a generic issue. For large decay constants, the cosmological problems associated with the axion's scalar partner are far more severe than those of the axion. We survey a variety of models, both for the axion multiplet and for cosmology, and find that in many cases where the cosmological problems of the saxion are solved, the usual upper bound on the axion is significantly relaxed. We discuss, more generally, the cosmological issues raised by the pseudoscalar members of moduli multiplets, and find that they are potentially quite severe. 
T. Banks, M. Dine and M. Graesser