Natural Supercooling and Reheating along Supersymmetric Flat Directions and Observable Gravitational Waves at the Einstein Telescope and the Cosmic Explorer

We study supercooled first-order phase transitions in a supersymmetric hidden sector with a spontaneously broken $U(1{)}_X$, focusing on the frequency range of the Einstein Telescope and Cosmic Explorer. Along the D-flat direction the tree-level quartic vanishes, so the barrier is generated radiatively by soft SUSY-breaking splittings. In the $\overline{\mathrm{D}\mathrm{R}}$ scheme the gaugino mass $M_{\tilde{\lambda }}$ sets the barrier depth, while the soft scalar mass $m_0$ stabilizes the broken vacuum. For $M_{\tilde{\lambda }}\mathrm{/}{v}_X\simeq 0.05$–$0.23$, the predicted signal reaches ${\mathrm{\Omega }}_{\mathrm{G}\mathrm{W}}{h}^2\sim 3\times {10}^{-10}$ near the percolation boundary. The observable amplitude depends sensitively on the portal coupling $\delta$ through the hidden-to-visible temperature ratio at percolation: for a cold initial hidden sector the signal rises from the ET floor at $\delta ={10}^{-6}$ to ${\mathrm{\Omega }}_{\mathrm{G}\mathrm{W}}{h}^2\simeq 7\times {10}^{-11}$ as the sectors approach thermal contact at $\delta ={10}^{-4}$, while a hotter initial hidden sector gives a large signal already for weak portal coupling. We follow this evolution with an 11-variable Boltzmann system that separates the cold nucleating exterior from the reheated true-vacuum interior; reheating mainly enters through the energy budget and redshift factors. The same hidden sector can reproduce ${\mathrm{\Omega }}_{\mathrm{C}\mathrm{D}\mathrm{M}}{h}^2=0.12$ through relativistic dark-quark freeze-out followed by entropy dilution from hidden-Higgs decay, with $m_q\simeq 30$–$800$keV and $N_{\mathrm{e}\mathrm{f}\mathrm{f}}\lesssim \mathrm{f}\mathrm{e}\mathrm{w}\times {10}^{-5}$.