(1+1)-Dimensional Schr\"odinger-Poisson equation with contact interaction

TL;DR

This study examines how contact interactions influence the nonlinear dynamics of fuzzy dark matter modeled by the Schr"odinger-Poisson equation in one dimension, revealing their impact on stationary states and gravitational collapse.

Contribution

It provides the first detailed numerical analysis of contact interactions in the (1+1)D Schr"odinger-Poisson model, highlighting their effects on structure formation and collapse processes.

Findings

Contact interactions alter the density profiles of stationary solutions.

They influence the timing and nature of gravitational shell-crossing.

Relaxation to the lowest-energy state remains elusive even with interactions.

Abstract

We investigate the role of contact interactions in the dynamics of fuzzy dark matter (FDM) modeled through the Schr\"odinger-Poisson equation in one spatial dimension. While the $\Lambda$CDM paradigm successfully explains structure formation on large scales, its small-scale predictions remain in tension with observations. FDM offers an alternative framework, where local self-interactions can further influence the formation and evolution of structures. We explore both attractive and repulsive contact interactions in static and expanding backgrounds. Using numerical simulations, we examine their impact on three key scenarios: the properties of the lowest-energy stationary solution, the relaxation of localized initial states, and the gravitational collapse of nonlocalized states. Our results show that contact interactions modify the density profile of the stationary solution and affect the onset of characteristic stages of gravitational collapse, particularly the shell-crossing event. In the (1+1) model, we confirm that relaxation does not converge to the lowest-energy stationary solution, even when local self-interactions are included. Taken together, local self-interactions play a relevant role in shaping the nonlinear dynamics of FDM and motivate further studies in higher-dimensional and cosmologically realistic settings.