Supersymmetric theories with gravitino dark matter generally do not allow the high reheating temperature required by thermal leptogenesis without running afoul of relic abundance or big bang nucleosynthesis constraints. We report on a successful sear
ch for parameter space that does satisfy these requirements. The main implication is the near degeneracy of the gluino with the other neutralinos in the spectrum. The leading discovery channel at the LHC for this scenario is through monojet plus missing energy events.
Several models of inflection point inflation with the volume modulus as the inflaton are investigated. Non-perturbative superpotentials containing two gaugino condensation terms or one such term with threshold corrections are considered. It is shown
that the gravitino mass may be much smaller than the Hubble scale during inflation if at least one of the non-perturbative terms has a positive exponent. Higher order corrections to the Kahler potential have to be taken into account in such models. Those corrections are used to stabilize the potential in the axion direction in the vicinity of the inflection point. Models with only negative exponents require uplifting and in consequence have the supersymmetry breaking scale higher than the inflation scale. Fine-tuning of parameters and initial conditions is analyzed in some detail for both types of models. It is found that fine-tuning of parameters in models with heavy gravitino is much stronger than in models with light gravitino. It is shown that recently proposed time dependent potentials can provide a solution to the problem of the initial conditions only in models with heavy gravitino. Such potentials can not be used to relax fine tuning of parameters in any model because this would lead to values of the spectral index well outside the experimental bounds.
