Evolution of linear wave dark matter perturbations in the radiation-dominated era

Linear perturbations of the wave dark matter, or psi dark matter (psi DM), of particle mass similar to 10-22 eV in the radiation-dominant era are analyzed, and the matter power spectrum at the photon-matter equality is obtained. We identify four phases of evolution for psi DM perturbations, where the dynamics can be vastly different from the counterparts of cold dark matter (CDM). While in late stages, after mass oscillation, longwave psi DM perturbations are almost identical to CDM perturbations, some subtle differences remain. The differences are even greater with intermediate-to-short waves that bear no resemblance to those of CDM throughout the whole evolutionary history. The dissimilarity is due to quantum mechanical effects which lead to severe mode suppression. We also discuss the axion model with a cosine field potential. The power spectrum of axion models is generally almost identical to that of psi DM, but in the extreme case, when the initial axion angle is near the field potential top, this axion model predicts a power excess over a range of wave numbers and a higher spectral cutoff than psi DM, as if psi DM had a higher particle mass.