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We investigate supersymmetric hybrid inflation in a realistic model based on the gauge symmetry $SU(4)_c \times SU(2)_L \times SU(2)_R$. The minimal supersymmetric standard model (MSSM) $\mu$ term arises, following Dvali, Lazarides, and Shafi, from the coupling of the MSSM electroweak doublets to a gauge singlet superfield which plays an essential role in inflation. The primordial monopoles are inflated away by arranging that the $SU(4)_c \times SU(2)_L \times SU(2)_R$ symmetry is broken along the inflationary trajectory. The interplay between the (above) $\mu$ coupling, the gravitino mass, and the reheating following inflation is discussed in detail. We explore regions of the parameter space that yield gravitino dark matter and observable gravity waves with the tensor-to-scalar ratio $r \sim 10^{-4}-10^{-3}$., Comment: 14 pages, 9 figures; version published in Phys. Rev. D

In $\mu$-hybrid inflation a nonzero inflaton vacuum expectation value induced by supersymmetry breaking is proportional to the gravitino mass $m_{3/2}$, which can be exploited to resolve the minimal supersymmetric standard model $\mu$ problem. We show how this scenario can be successfully implemented with $m_{3/2} \sim 1-100$ TeV and reheat temperature as low as $10^6$ GeV by employing a minimal renormalizable superpotential coupled with a well defined non-minimal K\"ahler potential. The tensor-to-scalar ratio $r$, a canonical measure of primordial gravity waves in most cases is less than or of the order of $10^{-6}-10^{-3}$., Comment: 19 pages, 6 figures, published version