Inflation and high-scale supersymmetry with an EeV gravitino

We consider inflation and supersymmetry breaking in the context of a minimal model of supersymmetry in which the only "low" energy remnant of supersymmetry is the gravitino with a mass of order an EeV. In this theory, the supersymmetry breaking scale is above the inflaton mass, m≃3×1013 GeV, as are all sfermion and gaugino masses. In particular, for a no-scale formulation of Starobinsky-like inflation using the volume modulus T, we show that inflation can be accommodated even when the supersymmetry breaking scale is very large. Reheating is driven through a gravitational coupling to the two Higgs doublets and is enhanced by the large μ-parameter. This leads to gravitino cold dark matter where the mass is constrained to be in the range 0.1 EeVm3/21000 EeV.