Propagating instability for wave dark matter

In the early Universe, large-scale flows were omnipresent, and the flow collisions produced sheets and filaments. This phenomenon occurs for particle dark matter, as it also does for wave dark matter. However, for the latter, these sheets and filaments are the modulations of even finer-scale, large-amplitude interference fringes. This work aims to investigate the instability of the interference fringes arising from colliding waves. Two colliding streams in classical collisionless fluid systems can often produce small-scale unstable oscillations with a finite complex frequency, a situation identified as propagating instabilities. In fact, propagating unstable oscillations have never been observed in the conventional quantum system due to its being Sturm-Liouville property. For example, quantum fluid equations with Madelung variables only exhibit either Jeans instability, a purely growing unstable mode, or stable oscillations, for which the squared frequency ${\ensuremath{\omega}}^{2}$ is real. Despite that, this work discovers that quantum interference fringes can indeed generate propagating unstable oscillations with a complex ${\ensuremath{\omega}}^{2}$ when the gravitational feedback perturbation is included. The presence of local density nulls in the background density is shown to be the necessary condition for such an instability. We establish a phase diagram separating the propagating instability, Jeans instability, and stable oscillation regions, verified by computer simulations. Generally speaking, Jeans instabilities tend to occur for long-wave density perturbations as expected; propagating instabilities, on the other hand, tend to occur for short density waves with wavelengths comparable to the fringe size, i.e., near the center of the Bloch zone. Finally, both instabilities diminish for very low density fringes. The propagating unstable fluctuation may possibly collapse into halos of small sizes, potentially seeding the formation of protoglobular clusters.