Broken R-parity in the sky and at the LHC

Supersymmetric extensions of the Standard Model with small R-parity and lepton number violating couplings are naturally consistent with primordial nucleosynthesis, thermal leptogenesis and gravitino dark matter. We consider supergravity models with universal boundary conditions at the grand unification scale, and scalar tau-lepton or bino-like neutralino as next-to-lightest superparticle (NLSP). Recent Fermi-LAT data on the isotropic diffuse gamma-ray flux yield a lower bound on the gravitino lifetime. Comparing two-body gravitino and neutralino decays we find a lower bound on a neutralino NLSP decay length, c tau(0)(chi 1) greater than or similar to 30 cm. Together with gravitino and neutralino masses one obtains a microscopic determination of the Planck mass. For a e (tau) over tilde -NLSP there exists no model-independent lower bound on the decay length. Here the strongest bound comes from the requirement that the cosmological baryon asymmetry is not washed out, which yields c tau((tau) over tilde1) greater than or similar to 4mm. However, without fine-tuning of parameters, one finds much larger decay lengths. For typical masses, m(3/2) similar to 100GeV and m(NLSP) similar to 150GeV, the discovery of a photon line with an intensity close to the Fermi-LAT limit would imply a decay length c tau(NLSP) of several hundred meters, which can be measured at the LHC.