Gravity Waves and Gravitino Dark Matter in $\mu$-Hybrid Inflation

We propose a novel reformulation of supersymmetric (more precisely $\mu$-) hybrid inflation based on a local U(1) or any suitable extension of the minimal supersymmetric standard model (MSSM) which also resolves the $\mu$ problem. We employ a suitable Kahler potential which effectively yields quartic inflation with non-minimal coupling to gravity. Imposing the gravitino Big Bang Nucleosynthesis (BBN) constraint on the reheat temperature ($T_r\lesssim {10}^6$ GeV) and requiring a neutralino LSP, the tensor to scalar ratio ($r$) has a lower bound $r\gtrsim 0.004$. The U(1) symmetry breaking scale lies between ${10}^8$ and ${10}^{12}$ GeV. We also discuss a scenario with gravitino dark matter whose mass is a few GeV.