Observational constraints on models of inflation from the density   perturbation and gravitino production
  Present data require a spectral index $n\gsim 0.95$ at something like 1-$\sigma$ level. If this lower bound survives it will constrain `new' and `modular' inflation models, while raising it to 1.00 would rule out all of these models plus many others.   After inflation, gravitinos are created by the oscillating field until the `intermediate' epoch when the Hubble parameter falls below the gravitino mass, or reheating, whichever is earlier. In a wide range of parameter space, these gravitinos are more abundant than those from thermal collisions, leading to stronger cosmological constraints. 
David H. Lyth (Lancaster)