This paper studies a type of cosmic phase change that can happen in a “hidden” particle sector governed by supersymmetry. The authors show that when a U(1)_X symmetry in that hidden sector breaks through a strongly supercooled first-order transition, the resulting burst of gravitational waves can fall in the frequency band of planned ground-based detectors such as the Einstein Telescope and the Cosmic Explorer.

The theoretical set-up uses a supersymmetric model with a so‑called D-flat direction. Along that direction the usual tree‑level stabilizing term (the quartic) disappears, so the energy barrier that allows a first‑order transition is produced only by quantum effects (the Coleman–Weinberg, or one‑loop, mechanism) acting on soft supersymmetry‑breaking mass splittings. Two soft parameters control the shape: the gaugino mass M̃λ sets the barrier depth and a soft scalar mass m0 stabilizes the broken phase. Working in the DR‑bar renormalization scheme, the authors scan the ratio M̃λ/vX (vX is the symmetry‑breaking scale) and m0/M̃λ to map where the gravitational‑wave signal could be large.

They estimate the gravitational‑wave energy density and follow the thermal history of the hidden and visible sectors with an 11‑variable Boltzmann system. That system keeps track of a cold exterior where bubbles form and a reheated interior where the new vacuum has taken over. For M̃λ/vX roughly between 0.05 and 0.23 they find peaks near ΩGW h2 ∼ 3×10−10 close to the percolation boundary. The observable amplitude depends strongly on how strongly the hidden sector shares heat with the ordinary sector. A tiny portal coupling δ∼10−6 gives a weak signal, while increasing δ toward 10−4 (stronger thermal contact) can raise the signal to ΩGW h2 ≈ 7×10−11 for a cold initial hidden sector; a hotter initial hidden sector can produce a large signal even for weak portal coupling.