  We uncover a new gravitational-wave production mechanism in cosmological,
first-order, thermal phase transitions. These are usually assumed to proceed
via the nucleation of bubbles of the stable phase inside the metastable phase.
However, if the nucleation rate is sufficiently suppressed, then the Universe
may supercool all the way down the metastable branch and enter the spinodal
region. In this case the transition proceeds via the exponential growth of
unstable modes and the subsequent formation, merging and relaxation of phase
domains. We use holography to follow the real-time evolution of this process in
a strongly coupled, four-dimensional gauge theory. The resulting gravitational
wave spectrum differs qualitatively from that in transitions mediated by bubble
nucleation. We discuss the possibility that the spinodal dynamics may be
preceded by a period of thermal inflation.
