(\mu\)-hybrid Inflation, Gravitino Dark Matter and Stochastic Gravitational Wave Background from Cosmic Strings

We present a successful realization of supersymmetric \(\mu\)-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 non-thermal leptogenesis with gravitino dark matter yields a reheat temperature in the range \(2 \times 10^{7} \lesssim T_R \lesssim 5 \times 10^{9}\) GeV. This corresponds to the predictions \(2 \times 10^{-18} \lesssim r\lesssim 4 \times 10^{-13}\) for the tensor to scalar ratio, and \(-2 \times 10^{-6} \lesssim dn_s/d\ln k \lesssim -5 \times 10^{-11}\) for the running of the scalar spectral index. The \(B-L\) breaking scale is estimated as \( 6 \times 10^{14}\lesssim M/ \text{GeV}\lesssim 10^{16}\), 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 \( 10^{-9} \lesssim G\mu_\text{cs} \lesssim 10^{-6}\) corresponding to a stochastic gravitational wave background lying within the 2\(\sigma\) bound of the recent NANOGrav 12.5-yr data.