Analytic Gravitational Wave Spectrum in Next-to-Minimal Bouncing Cosmology

Bouncing cosmology offers a singularity-free alternative to inflation, but its minimal realization-comprising only four cosmic phases-predicts a simple power-law stochastic gravitational-wave background (SGWB) with a narrow observational window. We introduce the next-to-minimal bouncing cosmology (NMBC), which adds an extra early contraction phase that imprints a broken power-law feature in the SGWB spectrum, enhancing detectability. Using our matrix-representation method grounded in an inequality algebra, we derive a closed-form expression for the NMBC SGWB spectrum. From this analytical result, we show that all NMBC models satisfying the current $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ bound ${\mathrm{\Omega }}_{\mathrm{G}\mathrm{W}}{h}^2(f)<1.7\times {10}^{-6}$ automatically avoid the trans-Planckian problem, ${\rho }_{s\downarrow }^{1\mathrm{/}4}<0.79m_{\mathrm{p}\mathrm{l}}$. These findings establish the NMBC as a self-consistent, self-contained framework capable of generating a potentially detectable SGWB in both astrophysical and laboratory searches, and demonstrate the broad utility of our matrix-representation method for future SGWB analyses in multi-phase cosmologies.