Bounds on Scalar Masses in Theories of Moduli Stabilization

In recent years it has been realised that pre-BBN decays of moduli can be a significant source of dark matter production, giving a `non-thermal WIMP miracle' and substantially reduced fine-tuning in cosmological axion physics. We study moduli masses and sharpen the claim that moduli dominated the pre-BBN Universe. We conjecture that in any string theory with stabilized moduli there will be at least one modulus field whose mass is of order (or less than) the gravitino mass and we prove this for a large class of models based on Calabi-Yau extra dimensions. Cosmology then generically requires the gravitino mass not be less than about 30 TeV and the cosmological history of the Universe is non-thermal prior to BBN. Stable LSP's produced in these decays can account for the observed dark matter if they are `wino-like,' which is consistent with the PAMELA data for positrons and antiprotons. With WIMP dark matter, there is an upper limit on the gravitino mass of order 250 TeV. We briefly consider implications for the LHC, rare decays, and dark matter direct detection and point out that these results could prove challenging for models attempting to realize gauge mediation in string theory.