  Dark matter may have been relativistic and collisional until relatively late
times and become cold and collisionless after a phase transition before the
matter-radiation equality of the standard $\Lambda$CDM cosmology. We show that
such a dark matter has large peculiar velocities due to acoustic oscillations
before the phase transition, and evolves ballistically after the phase
transition in the collisionless phase until the initial acoustic velocities are
redshifted away. We show that this Ballistic Dark Matter (BDM) results in new
non-trivial interesting features in the cosmological observables. In
particular, the linear matter power spectrum exhibits acoustic oscillations on
scales smaller than the Hubble scale at the time of phase transition, and for
fast transitions the power at the acoustic peaks in the matter power spectrum
exceeds that in a $\Lambda$CDM cosmology. If BDM only forms a part of the total
dark matter, an odd vs. even acoustic peak asymmetry becomes prominent. We give
an approximate analytical treatment of the linear perturbations in BDM,
explaining these features. We also discuss the possibility to constrain BDM
using cosmic microwave background and large scale structure data.
