Abundance of moduli, modulini and gravitinos produced by the vacuum fluctuation

Moduli, modulini and gravitinos have gravitational-strength interactions, and particle collisions after reheating create all of them with roughly the same abundance. With the usual provisos, this leads to the well-known bound TR ∼ < 10 9 GeV on the reheat temperature. The vacuum fluctuation also creates these particles. It is shown that the abundance from this mechanism is lower than the one from particle collisions in the case of moduli and modulini. Using recently-proposed evolution equations for the mode functions, the creation of gravitinos from the vacuum is studied. It is argued that in non-hybrid models of inflation, gravitinos are produced only with the same abundance as moduli and modulini, while in non-hybrid the abundance may be enhanced by a factor 10 GeV/V 1 4 ∼ > 10 . (Here, V is the energy density during inflation.) The usual bound on the reheat temperature is then tightened to TR ∼ < 10 V 1 4 , and the nucleosynthesis requirement TR ∼ > 10MeV implies V 1 4 ∼ > 10 8 GeV. Some new points are made, regarding the presently unclear situation with regard to the equations describing gravitino production. Introduction Long-lived particle species constrain models of the early Universe, because they are cosmologically dangerous. Their abundance is conveniently specified by the ratio n/s, where n is the number density and s is the entropy density, evaluated at the epoch of nucleosynthesis. In order not to upset nucleosynthesis, a particle with a gravitational-strength decay rate, and mass 100MeV to 10GeV must have [1] n s ∼ < 10 . (1) (The precise bound lies between 10 and 10, depending on the mass.) If supersymmetry breaking is gravity-mediated, the masses of the gravitino and certain superstring moduli and modulini are expected to lie in this range. A particle with gravitationalstrength interactions and mass ∼ < 100MeV is stable (lifetime longer than the age of the