Wave Interference in Self-Interacting Fuzzy Dark Matter

In the Fuzzy Dark Matter (FDM) scenario, the dark matter is composed of an ultra-light scalar field with coherence length and wave interference on astrophysical scales. Scalar fields generically have quartic self-interactions that modify their dispersion relation and the associated evolution of density perturbations. We perform the first dedicated analysis of the role of wave interference on this evolution due to self-interactions in FDM and vice versa, developing a perturbative treatment applicable at early times and then comparing against a suite of fully nonlinear benchmark simulations, varying the dark matter density, interaction strength, and fiducial momentum scale. We explicitly simulate the limit where this momentum scale is relatively high compared with the scale of the simulation volume, applicable to cases where the dark matter is initially ``warm" due to causal constraints on a post-inflationary production or in virialized halos and other ``thermalized" cases with initially cold production. We find that in such scenarios, density perturbations are unable to grow on the expected self interaction time scale because of interference effects, instead saturating on the much shorter de Broglie crossing time, with a dependence on the sign of the interaction. Finally, we comment on the implications of our results for astrophysical systems such as high-density ultra-faint dwarf galaxies where wave interference plays an important role.