Twin-peaked gravitational wave signal from a dark sector phase transition

We compute the gravitational wave spectrum from a dark sector phase transition driven by spontaneous $\text{ZDW}$ breaking. If the transition is second-order, the only source of gravitational waves is the annihilation of domain walls (biased by quantum gravity). However, if the transition is first-order, this yields a twin-peaked signal from both the transition itself and the biased domain wall annihilation. Both scenarios originate when a scalar singlet odd under the $\text{ZDW}$ obtains a non-zero vacuum expectation value. An additional $\text{ZDM}$ odd scalar doublet strengthens the transition and produces fermionic dark matter via freeze-in, matching observed dark matter relic density.