Gravitino Decay into a Neutrino and a Sneutrino in the Inflationary Universe

Gravitino produced in the inflationary universe are studied. When the gravitino decays into a neutrino and a sneutrino, the emitted high energy neutrinos scatter off the background neutrinos and produce charged leptons (mainly electrons and positrons), which cause the electro-magnetic cascades and produce many soft photons. We obtain the spectra of the high energy neutrinos as well as the spectrum of the high energy photon by integrating a set of Boltzmann equations. Requiring these photons should not alter the abundances of the light elements (D, $^3$He, $^4$He) in the universe, we can set the stringent upperbound on the reheating temperature after the inflation. We find that $T_R \lesssim (10^{10}-10^{12})$GeV for $m_{3/2}\sim (100\GEV - 1\TEV)$, which is more stringent than the constraints in the previous works.