Gravitational Waves from $\textit{Type-I}$ Strings in a Neutrino Mass Model

In this work, we propose a novel realization of $\textit{type-I}$ cosmic strings arising from the spontaneous breaking of an extended gauge symmetry $SU(2)_R\times U(1)_{B-L}$ in the context of a low-scale split seesaw mechanism for neutrino mass generation. We demonstrate that the split seesaw framework, which explains the smallness of neutrino masses, naturally motivates a small scalar self-coupling $\lambda$. This intrinsically links the neutrino mass generation mechanism to the formation of $\textit{type-I}$ cosmic strings, where the gauge coupling dominates over the scalar self-coupling ($\beta\equiv\lambda/2g^2<1$). We explore the cosmological implications of these strings, including their gravitational wave signatures that are testable in current and future experiments. Our findings establish a compelling and testable connection between neutrino mass generation and cosmic string phenomenology in an underexplored region of parameter space.