Constraint on cosmic density of the string moduli field in gauge-mediated supersymmetry-breaking theories

We derive a constraint on the cosmic density of string moduli fields in gauge-mediated supersymmetry-breaking theories by requiring that photons emitted from the unstable moduli fields should not exceed the observed X-ray backgrounds. Since the mass of the moduli field lies in the range between O(0.1) keV and O(1) MeV and the decay occurs through a gravitational interaction, the lifetime of the moduli field is much longer than the age of the present universe. The obtained upper bound on their cosmic density becomes more stringent than that from the enclosure condition for the present universe for a mass greater than about 100 keV.