Early Growth of Structure in Warm Wave Dark Matter

We explore the growth of structure in wave-like dark matter models, where the field and density spectra are peaked at sub-horizon wavenumbers. Starting with the Schr\"odinger-Poisson system, we derive the scale-dependent evolution of the matter power spectrum during radiation and matter domination. We find a suppression of adiabatic perturbations during radiation domination, controlled by a free-streaming length, and scale-dependent growth of the initially white-noise isocurvature power, controlled by a Jeans scale during matter domination. The results are in qualitative, and in some regimes quantitative, agreement with the quasi-particle picture. We verify the analytic results of the power spectrum with 3+1-dimensional cosmological Schr\"odinger-Poisson simulations. We propose an analytic formula for the halo mass function, which is in rough agreement with the simulation results at early times after matter-radiation equality. Our simulations show that early halos typically host a soliton.