Gravitational Wave Constraints on the Bouncing Energy Scale of Big Bounce Cosmology

Big bounce cosmology provides a solution to the Universe's initial singularity, and stochastic gravitational wave background (SGWB) searches offer a promising avenue for testing this paradigm. In this work, we establish an analytical relation between the bouncing energy scale, $\rho_{s\downarrow}^{1/4}$, and SGWB spectrum, $\Omega_\mathrm{GW}(f)h^2$, for big bounce cosmology. By combining sensitivities from major GW detectors (e.g., Planck/BICEP, PTA, and LIGO/Virgo across low, medium, and high frequencies, respectively), we provide the first systematic GW constraint on $\rho_{s\downarrow}^{1/4}$. Our results show that the region $-\tfrac{1}{3} < w_1 < -0.17$ is excluded by current SGWB searches, given the constraint $\rho_{s\downarrow}^{1/4} > 1~\mathrm{TeV}$, where $w_1$ is the contraction-phase equation of state parameter. Additionally, no detectable SGWB can be generated for $0.038 < w_1 < \infty$ with $\rho_{s\downarrow}^{1/4} < 10^{16}~\mathrm{TeV}$. We identify a window, $-0.17 < w_1 < 0.038$, in which a detectable SGWB can be produced, disfavoring nearly all big bounce models except for the matter-dominated contraction model ($w_1 \simeq 0$).