The $\mu \nu$SSM and gravitino dark matter

We consider the phenomenological implications of gravitino dark matter in the context of the $\mu \nu$SSM. The latter is an R-parity breaking model which provides a solution to the $\mu$-problem of the MSSM and explains the origin of neutrino masses by simply using right-handed neutrino superfields. In particular, we analyze the prospects for detecting gamma rays from decaying gravitinos. Gravitino masses larger than 20 GeV are disfavored by the isotropic diffuse photon background measurements, but a gravitino with a mass range between $0.1-20$ GeV gives rise to a signal that might easily be observed by the FERMI satellite. Through this kind of analysis important regions of the parameter space of the $\mu \nu$SSM can be checked.