The munuSSM is a supersymmetric model that has been proposed to solve the problems generated by other supersymmetric extensions of the standard model of particle physics. Given that R-parity is broken in the munuSSM, the gravitino is a natural candidate for decaying dark matter since its lifetime becomes much longer than the age of the Universe. In this model, gravitino dark matter could be detectable through the emission of a monochromatic gamma ray in a two-body decay. We study the prospects of the Fermi-LAT telescope to detect such monochromatic lines in 5 years of observations of the most massive nearby extragalactic objects. The dark matter halo around the Virgo galaxy cluster is selected as a reference case, since it is associated to a particularly high signal-to-noise ratio and is located in a region scarcely affected by the astrophysical diffuse emission from the galactic plane. The simulation of both signal and background gamma-ray events is carried out with the Fermi Science Tools, and the dark matter distribution around Virgo is taken from a N-body simulation of the nearby extragalactic Universe, with constrained initial conditions provided by the CLUES project. We find that a gravitino with a mass range of 0.6 to 2 GeV, and with a lifetime range of about 3x10^27 to 2x10^28 s would be detectable by the Fermi-LAT with a signal-to-noise ratio larger than 3. We also obtain that gravitino masses larger than about 4 GeV are already excluded in the munuSSM by Fermi-LAT data of the galactic halo
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