(1+1)-Dimensional Schrödinger-Poisson equation with contact interaction

Abstract

We investigate the role of contact interactions in the dynamics of fuzzy dark matter (FDM) modeled through the Schrödinger-Poisson equation in one spatial dimension. While the $Λ$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.

Figures (5)

• Figure 1: Ground states obtained with the imaginary-time propagation method for attractive and repulsive contact interactions at fixed mass $M = 100$ in a static universe ($a = 1$). (a) Density profiles for different values of $\lambda$, including the purely gravitational case ($\lambda = 0$) for reference. (b) Ground-state spectra for varying contact interaction strengths.
• Figure 2: Ground-state width as a function of the contact interaction strength $\lambda$ for $a = 1$ and mass $M = 100$. Solid lines indicate the points that fit well to a power-law function.
• Figure 3: Comparison between the mean quasistationary states obtained after the relaxation process and the corresponding ground states for (a) no contact interaction, (b) attractive contact interaction, and (c) repulsive contact interaction. Panel (d) shows the mean densities: the peak is more pronounced in the attractive case compared with the pure gravitational and repulsive cases, while the spatial width is larger for the repulsive contact interaction. The mean spectra and densities were obtained by first propagating the initial state up to $t = 90$; during the remaining time interval, 500 wavefunctions were saved. Finally, the densities and spectra were computed and averaged.
• Figure 4: Spatiotemporal representation of the wavepacket for (a) attractive, (b) no, and (c) repulsive contact interaction. The initial interaction strength is $\abs{\lambda/a_\text{ini}} \approx 1.25$. The corresponding shell-crossing times are $t_\text{sc} = 31.5$ ($z_\text{sc} = 89.6$), $t_\text{sc} = 41.5$ ($z_\text{sc} = 28.6$), and $t_\text{sc} = 48.0$ ($z_\text{sc} = 6.7$), respectively. Panels (d)-(f) display the associated Husimi functions at shell crossing. For equal resolution in both directions, the parameter was set to $\sigma_x = 1/\sqrt{2}$. In panel (e), the classical solution is shown as a black solid line.
• Figure 5: Onset of the first shell crossing as a function of the contact interaction strength. Negative values on the horizontal axis correspond to attractive interactions, while positive values correspond to repulsive interactions. The solid line shows the exponential fit.