Cosmic string gravitational waves from global $U(1)_{B-L}$ symmetry breaking as a probe of the type I seesaw scale

In type I seesaw models, the right-handed neutrinos are typically super-heavy, consistent with the generation of baryon asymmetry via standard leptogenesis. Primordial gravitational waves of cosmological origin provides a new window to probe such high scale physics, which would otherwise be inaccessible. By considering a {\em global} $U(1)_{B-L}$ extension of the type I seesaw model, we explore the connection between the heaviest right-handed neutrino mass and primordial gravitational waves arising from the dynamics of global cosmic string network. As a concrete example, we study a global $U(1)_{B-L}$ extension of the Littlest Seesaw model, and show that the inevitable GW signals, if detectable, probe the parameter space that can accommodate neutrino oscillation data and successful leptogenesis, while respecting theoretical constraints like perturbativity of the theory. Including CMB constraints from polarization and dark radiation leaves a large region of parameter space of the model, including the best fit regions, which can be probed by GW detectors like LISA and ET in the near future. In general, the GW detectors can test high scale type I seesaw models with the heaviest right-handed neutrino mass above $2.5 \times 10^{14}$ GeV, assuming the perturbativity, and $7 \times 10^{13}$ GeV assuming that the coupling between the heaviest right-handed neutrino and the $U(1)_{B-L}$ breaking scalar is less than unity.