Supersymmetric extensions of the standard model of particle physics assuming the gravitino to be the lightest supersymmetric particle (LSP), and with the next-to-LSP decaying to the gravitino during Big Bang nucleosynthesis, are analyzed. Particular emphasis is laid on their potential to solve the " 7 Li problem", an apparent factor 2 − 4 overproduction of 7 Li in standard Big Bang nucleosynthesis (BBN), their production of cosmologically important amounts of 6 Li, as well as the resulting gravitino dark matter densities in these models. The study includes several improvements compared to prior studies concerning NLSP hadronic branching ratios, NLSP dark matter densities, the evaluation of hadronic NLSP decays on BBN, BBN catalytic effects, as well as updated nuclear reaction rates. Heavy gravitinos in the constrained minimal supersymmetric standard model (CMMSM) are reanalyzed, whereas light gravitinos in gauge-mediated supersymmetry breaking scenarios (GMSB) are studied for the first time. It is confirmed that decays of NLSP staus to heavy gravitinos, while producing all the dark matter, may at the same time resolve the 7 Li problem. For NLSP decay times ≈ 10 3 sec, such scenarios also lead to cosmologically important 6 Li (and possibly 9 Be) abundances. However, as such scenarios require heavy > ∼ 1 TeV staus they are likely not testable at the LHC. It is found that decays of NLSP staus to light gravitinos may lead to significant 6 Li (and 9 Be) abundances, whereas NLSP neutralinos decaying into light gravitinos may solve the 7 Li problem. Though both scenarios are testable at the LHC they may not lead to the production of the bulk of the dark matter. A section of the paper outlines particle properties required to significantly reduce the 7 Li abundance, and/or enhance the 6 Li (and possibly 9 Be) abundances, by the decay of an arbitrary relic particle.

We re-consider the gravitino as dark matter in the framework of the Constrained MSSM. We include several recently suggested improvements on: (i) the thermal production of gravitinos, (ii) the calculation of the hadronic spectrum from NLSP decay and (iii) the BBN calculation including stau bound-state effects. In most cases we find an upper bound on the reheating temperature T R ∼ < a few × 10 7 GeV from over-production of 6 Li from bound state effects. We also find an upper limit on the stau lifetime of 3 × 10 4 sec, which is nearly an order of magnitude larger than the simple limit 5 × 10 3 sec often used to avoid the effect of bound-state catalysis. The bound on T R is relaxed to ∼ < a few ×10 8 GeV when we use a more conservative bound on 6 Li/ 7 Li, in which case a new region at small stau mass at ∼ 100 GeV and much longer lifetimes opens up. Such a low stau mass region can be easily tested at the LHC.Keywords: Supersymmetric Effective Theories, Cosmology of Theories beyond the SM, Dark Matter, Supersymmetric Standard Model. * On leave of absence from The Andrzej Soltan Institute for Nuclear Studies, Warsaw, Poland. breaking mechanism. Assuming standard big bang cosmology, it was shown early on that cosmological constraints require that the mass of gravitino m e G be much less than 1 keV, or else heavier than some 10 TeV [1,2]. While a primordial gravitino population can efficiently be diluted by inflation [3], subsequently the Universe can be repopulated with gravitinos via thermal production (TP) processes involving scatterings of Standard Model (SM) and SUSY particles in the hot plasma, with the number density proportional to the reheating temperature, T R . If, assuming R-parity, the gravitino is the lightest supersymmetric particle (LSP) and stable, one can also generate gravitinos via non-thermal production (NTP) process of freeze-out and decays of the next-to-lightest superpartner (NLSP). Because of the gravitino's exceedingly weak couplings to ordinary matter, the latter process usually takes place during or after the period of Big Bang Nucleosynthesis (BBN) and involves releasing substantial amounts of electromagneticaly and/or hadronically interacting particles (the importance of the latter shown to be important in [4,5,6]), which could wreck havoc to successful predictions of Big Bang Nucleosynthesis (BBN). In order to avoid this, and assuming gravitinos to be a dominant component of dark matter (DM), one imposes an upper bound on the reheating temperature of T R < 10 6−8 GeV [7,8,9,10,11,12] (for recent updates see, e.g., [13,5,14,15,16,6,17]).Considering NTP processes only, constraints on the parameter space of Minimal Supersymmetric Standard Model (MSSM) were derived in [18,19,20], basically eliminating

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