Gravitino Thermal Production, Dark Matter, and Reheating of the Universe

We present a full one-loop calculation of the gravitino thermal production rate, beyond the so-called hard thermal loop approximation, using the corresponding thermal spectral functions in numerical form on both sides of the light cone. This framework requires a full numerical evaluation. We interpret our results within the framework of a general supergravity-based model, remaining agnostic about the specifics of supersymmetry breaking. In this context, assuming that gravitinos constitute the entirety of the dark matter in the Universe imposes strict constraints on the reheating temperature. For example, with a gluino mass at the current LHC limit, a maximum reheating temperature of $T_\mathrm{reh} \simeq 10^9$ GeV is compatible with a gravitino mass of $m_{3/2} \simeq 1$ TeV. Additionally, with a reheating temperature an order of magnitude lower at $T_\mathrm{reh} \simeq 10^8$ GeV, the common gaugino mass $M_{1/2}$ can range from $2$ to $4 $ TeV within the same gravitino mass range. For much higher values of $M_{1/2}$, which are favored by current accelerator and cosmological data in the context of supersymmetric models, such as $M_{1/2} = 10$ TeV, and for $m_{3/2} \simeq 1$ TeV the reheating temperature compatible with the gravitino dark matter scenario is $ 10^7$ GeV. If other dark matter particles are considered, the reheating temperature could be much lower.