Gravitino Dark Matter in Tree Level Gauge Mediation with and without R-parity

We investigate the cosmological aspects of Tree Level Gauge Mediation, a recently proposed mechanism in which the breaking of supersymmetry is communicated to the soft scalar masses by extra gauge interactions at the tree level. Embedding the mechanism in a Grand Unified Theory and requiring the observability of sfermion masses at the Large Hadron Collider, it follows that the Lightest Supersymmetric Particle is a gravitino with a mass of the order of 10 GeV. The analysis in the presence of R-parity shows that a typical Tree Level Gauge Mediation spectrum leads to an overabundance of the Dark Matter relic density and a tension with the constraints from Big Bang Nucleosynthesis. This suggests to relax the exact conservation of the R-parity. The underlying SO(10) Grand Unified Theory together with the bounds from proton decay provide a rationale for considering only bilinear R-parity violating operators. We finally analyze the cosmological implications of this setup by identifying the phenomenologically viable regions of the parameter space.