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Gravitino Dark Matter in the CMSSM and Implications for Leptogenesis and the LHC

In the framework of the CMSSM we study the gravitino as the lightest supersymmetric particle and the dominant component of cold dark matter in the Universe. We include both a thermal contribution to its relic abundance from scatterings in the plasma and a non--thermal one from neutralino or stau decays after freeze--out. In general both contributions can be important, although in different regions of the parameter space. We further include constraints from BBN on electromagnetic and hadronic showers, from the CMB blackbody spectrum and from collider and non--collider SUSY searches. The region where the neutralino is the next--to--lightest superpartner is severely constrained by a conservative bound from excessive electromagnetic showers and probably basically excluded by the bound from hadronic showers, while the stau case remains mostly allowed. In both regions the constraint from CMB is often important or even dominant. In the stau case, for the assumed reasonable ranges of soft SUSY breaking parameters, we find regions where the gravitino abundance is in agreement with the range inferred from CMB studies, provided that, in many cases, a reheating temperature $\treh$ is large, $\treh\sim10^{9}\gev$. On the other side, we find an upper bound $\treh\lsim 5\times 10^{9}\gev$. Less conservative bounds from BBN or an improvement in measuring the CMB spectrum would provide a dramatic squeeze on the whole scenario, in particular it would strongly disfavor the largest values of $\treh\sim 10^{9}\gev$. The regions favored by the gravitino dark matter scenario are very different from standard regions corresponding to the neutralino dark matter, and will be partly probed at the LHC.