<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>μ</mml:mi></mml:math>-hybrid inflation, gravitino dark matter, and stochastic gravitational wave background from cosmic strings

We present a successful realization of supersymmetric $\ensuremath{\mu}$-hybrid inflation model based on a gauged $U(1{)}_{B--L}$ extension of the minimal supersymmetric standard model, with the soft supersymmetry breaking terms are playing an important role. Successful nonthermal leptogenesis with gravitino dark matter yields a reheat temperature in the range $2\ifmmode\times\else\texttimes\fi{}{10}^{7}\ensuremath{\lesssim}{T}_{R}\ensuremath{\lesssim}5\ifmmode\times\else\texttimes\fi{}{10}^{9}\text{ }\text{ }\mathrm{GeV}$. This corresponds to the predictions $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}\ensuremath{\lesssim}r\ensuremath{\lesssim}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ for the tensor to scalar ratio, and $\ensuremath{-}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\ensuremath{\lesssim}d{n}_{s}/d\text{ }\mathrm{ln}k\ensuremath{\lesssim}\ensuremath{-}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$ for the running of the scalar spectral index. The $B--L$ breaking scale is estimated as $6\ifmmode\times\else\texttimes\fi{}{10}^{14}\ensuremath{\lesssim}M/\mathrm{GeV}\ensuremath{\lesssim}{10}^{16}$, calculated at the central value of the scalar spectral index, ${n}_{s}=0.9655$, reported by Planck 2018. Finally, in a grand unified theory setup the dimensionless string tension parameter associated with the metastable strings is in the range ${10}^{\ensuremath{-}9}\ensuremath{\lesssim}G{\ensuremath{\mu}}_{\mathrm{cs}}\ensuremath{\lesssim}{10}^{\ensuremath{-}6}$ corresponding to a stochastic gravitational wave background lying within the $2\ensuremath{\sigma}$ bound of the recent NANOGrav 12.5-yr data.