  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 $10^8 - 10^9$
GeV and energy scales of the PT around $10^{2} - 10^3$ 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-$\alpha$, 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.
