A very simple way to obtain comparable baryon and DM densities in the early Universe is through their contemporary production from the out-of-equilibrium decay of a mother particle, if both populations are suppressed by comparably small numbers, i.e.
the CP violation in the decay and the branching fraction respectively. We present a detailed study of this kind of scenario in the context of a R-parity violating realization of the MSSM in which the baryon asymmetry and the gravitino Dark Matter are produced by the decay of a Bino. The implementation of this simple picture in a realistic particle framework results, however, quite involving, due to the non trivial determination of the abundance of the decaying Bino, as well as due to the impact of wash-out processes and of additional sources both for the baryon asymmetry and the DM relic density. In order to achieve a quantitative determination of the baryon and Dark Matter abundances, we have implemented and solved a system of coupled Boltzmann equations for the particle species involved in their generation, including all the relevant processes. In the most simple, but still general, limit, in which the processes determining the abundance and the decay rate of the Bino are mediated by degenerate right-handed squarks, the correct values of the DM and baryon relic densities are achieved for a Bino mass between 50 and 100 TeV, Gluino NLSP mass in the range 15-60 TeV and a gravitino mass between 100 GeV and few TeV. These high masses are unfortunately beyond the kinematical reach of LHC. On the contrary, an antiproton signal from the decays of the gravitino LSP might be within the sensitivity of AMS-02 and gamma-ray telescopes.
