Formation of cosmic structures in a light gravitino-dominated universe

We analyze the formation of cosmic structures in models where dark matter is dominated by light gravitinos with a mass of 100 eV – 1 keV, as predicted by gauge-mediated supersymmetry (SUSY) breaking models. After evaluating the number of degrees of freedom at the gravitino decoupling ${(g}_{*})$, we compute the transfer function for matter fluctuations and show that gravitinos behave like warm dark matter (WDM) with a free-streaming scale comparable to the galaxy mass scale. We consider different low-density variants of the WDM model, both with and without a cosmological constant, and compare the predictions on the abundances of neutral hydrogen within high-redshift damped Ly-$\alpha$ systems and on the number density of local galaxy clusters with the corresponding observational constraints. We find that none of the models satisfy both constraints at the same time, unless a rather small ${\Omega }_0$ value $(\lesssim 0.4)$ and a rather large Hubble parameter $(\gtrsim 0.9)$ is assumed. Furthermore, in a model with warm + hot dark matter, with the hot component provided by massive neutrinos, the strong suppression of fluctuation on scales of $\sim 1h^{-1}\mathrm{Mpc}$ precludes the formation of high-redshift objects, when the low-$z$ cluster abundance is required. We conclude that all different variants of a light gravitino DM dominated model show strong difficulties for what concerns cosmic structure formation. This gives a severe cosmological constraint on the gauge-mediated SUSY breaking scheme.