  We propose to search for a new type of gravitational wave signature relevant
for particle physics models with symmetries broken at vastly different energy
scales. The spectrum contains a characteristic double-peak structure consisting
of a sharp peak from domain walls and a smooth bump from a first order phase
transition in the early Universe. We demonstrate how such a gravitational wave
signal arises in a new theory unifying baryon number and color into an SU(4)
gauge group broken at the multi-TeV scale, and with lepton number promoted to
an SU(2) gauge symmetry broken at the multi-EeV scale. The model contains two
types of dark matter particles, explains the observed domination of matter over
antimatter in the Universe, and accommodates nonzero neutrino masses. We
discuss how future gravitational wave experiments, such as LISA, Big Bang
Observer, DECIGO, Einstein Telescope, and Cosmic Explorer, can be utilized to
look for this novel signature.
