  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. 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.' 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.
