Gravitino-induced baryogenesis, primordial nucleosynthesis, and the Tremaine-Gunn limit

We examine constraints on the gravitino-induced baryogenesis model which arise from primordial nucleosynthesis. Agreement with observed element abundances requires that the gravitino mass satisfy $m_{\tilde{G}}>50-53\mathrm{}$ TeV, corresponding to a lifetime ${\tau }_{\tilde{G}}<0.04-0.05\mathrm{}$ sec. For $50 \mathrm{TeV}<m_{\tilde{G}}<80\mathrm{} \mathrm{TeV}$, differential heating of the neutrinos and photons lowers the $\mu$- and $\tau$-neutrino-to-photon number ratio, increasing the neutrino mass needed to close the Universe and, for a narrow range of $m_{\tilde{G}}$, allowing the Tremaine-Gunn limit to be evaded. For $55 \mathrm{TeV}<m_{\tilde{G}}<80\mathrm{} \mathrm{TeV}$, this differential heating also lowers the primordial helium abundance below the value predicted in the standard model.