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In the framework of the CMSSM we study the gravitino as the lightestsupersymmetric particle and the dominant component of cold dark matter in theUniverse. We include both a thermal contribution to its relic abundance fromscatterings in the plasma and a non--thermal one from neutralino or stau decaysafter freeze--out. In general both contributions can be important, although indifferent regions of the parameter space. We further include constraints fromBBN on electromagnetic and hadronic showers, from the CMB blackbody spectrumand from collider and non--collider SUSY searches. The region where theneutralino is the next--to--lightest superpartner is severely constrained by aconservative bound from excessive electromagnetic showers and probablybasically excluded by the bound from hadronic showers, while the stau caseremains mostly allowed. In both regions the constraint from CMB is oftenimportant or even dominant. In the stau case, for the assumed reasonable rangesof soft SUSY breaking parameters, we find regions where the gravitino abundanceis in agreement with the range inferred from CMB studies, provided that, inmany cases, a reheating temperature $\treh$ is large, $\treh\sim10^{9}\gev$. Onthe other side, we find an upper bound $\treh\lsim 5\times 10^{9}\gev$. Lessconservative bounds from BBN or an improvement in measuring the CMB spectrumwould provide a dramatic squeeze on the whole scenario, in particular it wouldstrongly disfavor the largest values of $\treh\sim 10^{9}\gev$. The regionsfavored by the gravitino dark matter scenario are very different from standardregions corresponding to the neutralino dark matter, and will be partly probedat the LHC.Comment: JHEP version, several improvements and update

Gravitino dark matter in the CMSSM and implications for leptogenesis and the LHC