| Summary: | Abstract A keV-scale gravitino arising from a minimal supersymmetric (SUSY) Standard Model (MSSM) is an interesting possibility since the small scale problems that the ΛCDM model encounters in the modern cosmology could be alleviated with the keV-scale gravitino serving as the warm dark matter (WDM). Such a light gravitino asks for a low scale supersymmetry (SUSY) breaking for which the gauge mediation (GM) is required as a consistent SUSY-breaking mediation mechanism. In this paper, we show upper bounds of the masses of the second CP-even Higgs boson H and the CP-odd Higgs boson A, assuming the keV-scale gravitino to be responsible for the current DM relic abundance: the upper bound on the mass of H/A is found to be ∼ 4 TeV for the gravitino mass of O 10 − 100 $$ \mathcal{O}\left(10-100\right) $$ keV. Interestingly, the mass of H/A can be as small as 2–3 TeV and the predicted tan β is as large as 55–60 for the gravitino mass of O 10 $$ \mathcal{O}(10) $$ keV. This will be tested in the near future Large Hadron Collider (LHC) experiments.
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