  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 "Li7
problem", an apparent factor 2-4 overproduction of Li7 in standard Big Bang
nucleosynthesis (BBN), their production of cosmologically important amounts of
Li6, as well as the resulting gravitino dark matter densities in these models.
The study includes several improvements compared to prior studies. 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 Li7 problem. For NLSP decay times ~ 1000 sec, such
scenarios also lead to cosmologically important Li6 (and possibly Be9)
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 Li6 (and Be9) abundances, whereas NLSP
neutralinos decaying into light gravitinos may solve the Li7 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 Li7 abundance, and/or enhance
the Li6 (and possibly Be9) abundances, by the decay of an arbitrary relic
particle.
