$\mu$-Term Hybrid Inflation and Split Supersymmetry

We consider $\mu$-term hybrid inflation which, in its minimal format with gravity mediated supersymmetry breaking, leads to split supersymmetry. The MSSM $\mu$-term in this framework is larger than the gravitino mass $m_G$, and successful inflation requires $m_G$ (and hence also $\mathrm{\mid }\mu \mathrm{\mid }$) $\gtrsim 5 \times 10^7$ GeV, such that the gravitino decays before the LSP neutralino freezes out. Assuming universal scalar masses of the same order as $m_G$, this leads to split supersymmetry. The LSP wino with mass $\simeq$ 2 TeV is a plausible dark matter candidate, the gluino may be accessible at the LHC, and the MSSM parameter $\tan \beta \simeq 1.7$ in order to be compatible with the measured Higgs boson mass. The tensor-to-scalar ratio $r$, a canonical measure of gravity waves, can be as high as $0.001$.