  The thermal freeze-out mechanism in its classical form is tightly connected
to physics beyond the Standard Model around the electroweak scale, which has
been the target of enormous experimental efforts. In this work we study a dark
matter model in which freeze-out is triggered by a strong first-order phase
transition in a dark sector, and show that this phase transition must also
happen close to the electroweak scale, i.e. in the temperature range relevant
for gravitational wave searches with the LISA mission. Specifically, we
consider the spontaneous breaking of a $U(1)^\prime$ gauge symmetry through the
vacuum expectation value of a scalar field, which generates the mass of a
fermionic dark matter candidate that subsequently annihilates into dark Higgs
and gauge bosons. In this set-up the peak frequency of the gravitational wave
background is tightly correlated with the dark matter relic abundance, and
imposing the observed value for the latter implies that the former must lie in
the milli-Hertz range. A peculiar feature of our set-up is that the dark sector
is not necessarily in thermal equilibrium with the Standard Model during the
phase transition, and hence the temperatures of the two sectors evolve
independently. Nevertheless, the requirement that the universe does not enter
an extended period of matter domination after the phase transition, which would
strongly dilute any gravitational wave signal, places a lower bound on the
portal coupling that governs the entropy transfer between the two sectors. As a
result, the predictions for the peak frequency of gravitational waves in the
LISA band are robust, while the amplitude can change depending on the initial
dark sector temperature.
