  We propose a novel scenario of Dark Matter production naturally connected
with generation of gravitational waves. Dark Matter is modelled as a real
scalar, which interacts with the hot primordial plasma through a portal
coupling to another scalar field. For a particular sign of the coupling, this
system exhibits an inverse second order phase transition. The latter leads to
an abundant Dark Matter production, even if the portal interaction is so weak
that the freeze-in mechanism is inefficient. The model predicts domain wall
formation in the Universe, long time before the inverse phase transition. These
domain walls have a tension decreasing with time, and completely disappear at
the inverse phase transition, so that the problem of overclosing the Universe
is avoided. The domain wall network emits gravitational waves with
characteristics defined by those of Dark Matter. In particular, the peak
frequency of gravitational waves is determined by the portal coupling constant,
and falls in the observable range for currently planned gravitational wave
detectors.
