The abundance of moduli, modulini and gravitinos produced by the vacuum fluctuation

Moduli, modulini and the gravitino have gravitational-strength interactions, and thermal collisions after reheating create all of them with roughly the same abundance. With their mass of order 100GeV, corresponding to gravity-mediated supersymmetry breaking, this leads to the well-known bound γTR ∼ < 10 GeV on the reheat temperature, where γ ≤ 1 is the entropy dilution factor. The vacuum fluctuation also creates these particles, with abundance determined by the solution of the equation for the mode function. Taking the equation in each case to be the one corresponding to a free field, we consider carefully the behaviour of the effective mass during the crucial era after inflation. It may have a rapid oscillation, which does not however affect the particle abundance. Existing estimates are confirmed; the abundance of modulini and (probably) of moduli created from the vacuum is less than from thermal collisions, but the abundance of gravitinos may be much bigger, leading to a tighter bound on TR if supersymmetry breaking is gravitymediated. It is noted that in the case of gauge-mediated supersymmetry breaking, the abundance of the gravitino may be sufficient to make it a dark matter candidate. Introduction Long-lived particle species constrain models of the early Universe, because they are cosmologically dangerous. In this note we are concerned with species that have gravitational-strength interactions. According to present ideas, these include the gravitino (spin 3/2), and species usually called moduli (spin 0) and modulini (spin 1/2). The abundance of a given species 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. A particle with a gravitational-strength decay rate, and mass 100MeV to 10 GeV, decays before the present but after nucleosynthesis. To avoid interfering with nucleosynthesis its abundance must satisfy [1] n s ∼ < 10 . (1)