Ballistic Dark Matter oscillates above $\Lambda$CDM

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.