Cosmological Moduli Problem in Gauge-mediated Supersymmetry Breaking Theories

A generic class of string theories predicts the existence of light moduli fields, and they are expected to have masses $m_{\varphi }$ comparable to the gravitino mass $m_{3\mathrm{/}2}$ which is in a range of ${10}^{-2}$keV–1GeV in gauge-mediated supersymmetry breaking theories. Such light fields with weak interactions suppressed by the Planck scale can not avoid some stringent cosmological constraints, that is, they suffer from `cosmological moduli problems’. We show that all the gravitino mass region ${10}^{-2}$keV $\lesssim m_{3/2} \lesssim$ 1GeV is excluded by the constraints even if we incorporate a late-time mini-inflation (thermal inflation). However, a modification of the original thermal inflation model enables the region ${10}^{-2}$keV $\lesssim m_{3/2} \lesssim$ 500keV to survive the constraints. It is also stressed that the moduli can be dark matter in our universe for the mass region ${10}^{-2}$keV $\lesssim m_{\varphi }\lesssim$ 100keV.