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

Abstract

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 $\overline{\mathrm{DR}}$ scheme the gaugino mass $M_{\tilde{\lambda }}$ sets the barrier depth, while the soft scalar mass $m_0$ stabilizes the broken vacuum. For $M_{\tilde{\lambda }}/v_X\simeq 0.05$--$0.23$, the predicted signal reaches ${\Omega }_{\mathrm{GW}}{h}^2\sim 3\times 10^{-10}$ 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 ${\Omega }_{\mathrm{GW}}{h}^2\simeq 7\times 10^{-11}$ 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 ${\Omega }_{\mathrm{CDM}}{h}^2=0.12$ through relativistic dark-quark freeze-out followed by entropy dilution from hidden-Higgs decay, with $m_q\simeq 30$--$800$keV and $N_{\mathrm{eff}}\lesssim \mathrm{few}\times 10^{-5}$.