The Upper Bound of the Second Higgs Boson Mass in Minimal Gauge
  Mediation with the Gravitino Warm Dark Matter
  A keV-scale gravitino arsing from a minimal supersymmetric (SUSY) Standard
Model (MSSM) is an interesting possibility since the small scale problems that
$\Lambda$CDM 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 upperbounds 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 upperbound
on the mass of $H/A$ is found to be $\sim 4$ TeV for the gravitino mass of
$\mathcal{O}(10$-$100)$ keV. Interestingly, the mass of $H/A$ can be as small
as 2-3 TeV and the predicted $\tan\beta$ is as large as 55-60 for the gravitino
mass of $\mathcal{O}(10)$ keV. This will be tested in the near future Large
Hadron Collider (LHC) experiments.
