We point out that dark matter which is produced non-adiabatically in
a phase transition (PT) with fast bubble walls receives a boost in
velocity which leads to long free-streaming lengths. We find that this
could be observed via the suppressed matter power spectrum for dark
matter masses around \mathbf{ 10^8 - 10^9}108-109
GeV and energy scales of the PT around \mathbf{ 10^{2} - 10^3}102-103 GeV.
The PT should take place at the border of the supercooled regime,
i.e. approximately when the Universe becomes vacuum dominated. This work
offers novel physics goals for galaxy surveys,
Lyman-\alphaɑ,
stellar stream, lensing, and 21-cm observations, and connects these to
the gravitational waves from such phase transitions, and more
speculatively to possible telescope signals of heavy dark matter
decay.