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 $\overline{\rm DR}$ scheme the gaugino mass $M_{\tildeλ}$ sets the barrier depth, while the soft scalar mass $m_0$ stabilizes the broken vacuum. For $M_{\tildeλ}/v_X\simeq0.05$--$0.23$, the predicted signal reaches $Ω_{\rm GW}h^2\sim3\times10^{-10}$ near the percolation boundary. The observable amplitude depends sensitively on the portal coupling $δ$ through the hidden-to-visible temperature ratio at percolation: for a cold initial hidden sector the signal rises from the ET floor at $δ=10^{-6}$ to $Ω_{\rm GW}h^2\simeq7\times10^{-11}$ as the sectors approach thermal contact at $δ=10^{-4}$, 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 $Ω_{\rm CDM}h^2=0.12$ through relativistic dark-quark freeze-out followed by entropy dilution from hidden-Higgs decay, with $m_q\simeq30$--$800\;$keV and $N_{\rm eff}\lesssim{\rm few}\times10^{-5}$.