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We consider super-gravity models in which the lightest supersymmetricparticle (LSP) is a stable gravitino. The next-to-lightest supersymmetricparticle (NLSP) freezes out with its thermal relic density and then decaysafter $(10^5-10^{10})$ sec, injecting high-energy photons into the cosmicplasma. These photons heat up the electron plasma which then thermalizes withthe cosmic microwave background (CMB) via Compton scattering, bremsstrahlungand double-Compton scattering. Contrary to previous studies which assumeinstantaneous energy injection, we solve the full kinetic equation for thephoton number density with a source term describing the decay of the NLSP. Thissource term is based on the requirement that the injected energy be almostinstantaneously redistributed by Compton scattering, hence leading to atime-dependent chemical potential. We investigate the case of a stau NLSP anddetermine the constraints on the gravitino and stau masses from observations ofthe CMB spectrum by assuming that all gravitino LSPs come from stau NLSPdecays. Unlike the analytical approximations, we find that there may be a staumass below which the constraint from the CMB spectrum vanishes.Comment: 8 pages, 6 figures, revtex4, replaced to match published versio

Constraining gravitino dark matter with the cosmic microwave background