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

This paper investigates the impact of attractive and repulsive contact interactions on the nonlinear dynamics of one-dimensional fuzzy dark matter, revealing that while these interactions modify stationary density profiles and collapse stages, they do not enable relaxation to the lowest-energy stationary state.

The Gist

Imagine the universe as a giant, invisible ocean. For decades, scientists have believed this ocean is made of "Cold Dark Matter" (CDM)—tiny, invisible particles that don't bump into each other, just like ghosts drifting through a crowded room. This theory works great for explaining big things like galaxy clusters, but it gets messy when we look at small things, like individual galaxies. It predicts too many tiny satellite galaxies and cores that are too dense, which doesn't match what telescopes actually see.

Enter Fuzzy Dark Matter (FDM). Instead of being tiny particles, FDM suggests dark matter is made of ultra-light waves, like ripples on a pond. These waves can interfere with each other, creating a "fuzzy" texture that smooths out the messy predictions of the old theory.

This paper asks a simple but profound question: What if these dark matter waves could also "talk" to each other?

In the real world, atoms in a cloud can push each other away (repel) or pull each other closer (attract). The authors of this paper decided to add this "conversation" to their mathematical model of Fuzzy Dark Matter to see how it changes the story of how the universe forms.

Here is a breakdown of their findings using everyday analogies:

1. The "Shape-Shifting" Cloud (Ground States)

Imagine a cloud of fog floating in a room.

• Without talking (No interaction): The cloud settles into a specific, natural shape determined only by gravity pulling it down and the quantum "jitter" pushing it apart.
• If the fog pushes away (Repulsive interaction): Imagine the fog particles are like magnets with the same pole facing each other. They hate being close. The cloud spreads out, becoming wider and flatter.
• If the fog pulls together (Attractive interaction): Imagine the fog particles are magnets with opposite poles. They love to hug. The cloud collapses inward, becoming narrower and denser in the center.

The paper confirms that adding this "talking" ability changes the shape of these dark matter clouds exactly as you'd expect: repulsion makes them fluffy, and attraction makes them compact.

2. The "Relaxation" Problem (Why they don't settle down)

In the 3D universe (our real world), if you shake a jar of marbles, they eventually settle into a neat pile at the bottom. Scientists hoped that if you "shook" a fuzzy dark matter wave, it would settle into that perfect, lowest-energy shape (the "ground state" we just talked about).

However, in this 1D model (a simplified, flat version of the universe), the authors found something surprising: The wave never actually settles into that perfect shape.

Think of it like trying to balance a broom on your hand. Even if you try to find the perfect spot, the broom keeps wobbling. In this 1D world, the dark matter wave keeps oscillating and never finds a permanent, stable home, even if you let it sit there for a long time. This is a crucial difference from the 3D world, suggesting that our simplified models might be missing something important about how dark matter behaves in reality.

3. The "Traffic Jam" of the Universe (Gravitational Collapse & Shell-Crossing)

One of the most dramatic moments in the history of the universe is Gravitational Collapse. Imagine a long line of cars (dark matter) driving on a highway. As they get closer to a destination, they speed up and bunch together.

In the old "Ghost" theory (CDM), the cars can drive right through each other. They overlap, creating a chaotic traffic jam called "Shell-Crossing." This is where the math breaks down because the density becomes infinite.

The authors used their "talking" dark matter model to see how this traffic jam happens:

• Attractive Interaction (The "Hug" Force): If the cars want to hug, they rush toward each other faster. The traffic jam (shell-crossing) happens sooner.
• Repulsive Interaction (The "Push" Force): If the cars hate being close, they resist the crush. They slow down the bunching process. The traffic jam happens later.

This is a big deal because the timing of when galaxies form depends on when this "traffic jam" happens. If dark matter has a repulsive "push," it might delay the formation of small structures, potentially fixing the mismatches between theory and observation.

The Big Picture

The authors used a simplified, one-dimensional model to test these ideas. While the universe is 3D, this 1D model acts like a wind tunnel for physicists. It lets them test how "talking" dark matter behaves without the massive computer power needed for a full 3D simulation.

The Takeaway:
If dark matter can "talk" to itself (interact), it acts like a social creature.

• If it's shy and pushes away, it forms wider, flatter galaxies and delays the formation of structure.
• If it's clingy and pulls together, it forms tight, dense cores and speeds up the formation of structure.

This paper suggests that adding these "social interactions" to our models of the universe could be the missing key to solving the mystery of why the universe looks the way it does on small scales. It's a reminder that even the invisible stuff of the universe might have a personality.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the "small-scale crisis" in the standard $\Lambda$CDM cosmological model, which overpredicts the formation of dark matter structures on scales smaller than galaxy separations (e.g., the missing satellites and cusp-core problems). Fuzzy Dark Matter (FDM), modeled as an ultra-light boson field ($m \sim 10^{-22}$ eV), is proposed as an alternative. FDM is governed by the Schrödinger-Poisson (SP) system, which naturally produces solitonic cores.

However, the dynamics of FDM in reduced dimensions (specifically 1D) differ significantly from the 3D case. In 1D, the relaxation of localized states does not converge to the lowest-energy stationary solution (the soliton), unlike in 3D. Furthermore, the role of local self-interactions (contact interactions) in shaping these dynamics remains under-explored in the 1D SP framework. The authors aim to investigate how attractive and repulsive contact interactions influence:

1. The properties of the ground state.
2. The relaxation process of localized initial states.
3. The gravitational collapse and shell-crossing events in an expanding universe.

2. Methodology

Theoretical Framework

The authors derive the dynamics from a scalar field action with a quartic self-interaction term. In the non-relativistic, weak-gravity limit within a Friedmann-Lemaître-Robertson-Walker (FLRW) metric, the system reduces to a dimensionless Schrödinger-Poisson system:
$$i\partial_t\psi = \left( -\frac{1}{2}\nabla^2 + a(t)\Phi + \frac{\lambda}{a(t)}|\psi|^2 \right)\psi$$
$$\nabla^2\Phi = |\psi|^2 - 1$$
Where:

• $\psi$ is the wavefunction (normalized such that $\langle |\psi|^2 - 1 \rangle = 0$).
• $\Phi$ is the gravitational potential.
• $a(t)$ is the cosmological scale factor.
• $\lambda$ is the dimensionless contact interaction parameter ($\lambda < 0$ for attractive, $\lambda > 0$ for repulsive).

Numerical Implementation

• Domain: 1D spatial domain $x \in [-x_{max}, x_{max})$ with periodic boundary conditions.
• Discretization: Space and time are discretized using a grid ($N_x, N_t$).
• Solver: The time evolution is solved using a second-order Strang splitting method:
  • Kinetic Step ($U_K$): Computed in momentum space via Fast Fourier Transform (FFT).
  • Potential Step ($U_R$): Includes the gravitational potential and the contact interaction term. The density $|\psi|^2$ is treated as constant during the potential step to simplify integration, with the scale factor $a(t)$ integrated via the midpoint method.
• Cosmology: Simulations are run in both static ($a(t)=1$) and expanding universes (using $\Omega_{m0}=0.3, \Omega_{\Lambda0}=0.7$).

Scenarios Analyzed

1. Ground States: Minimization of the energy functional using imaginary time propagation.
2. Relaxation: Time evolution of an initial Gaussian wavepacket to observe if it converges to the ground state.
3. Gravitational Collapse: Evolution of a sine-wave perturbation in an expanding universe from redshift $z=500$ to $z=0$ to study shell-crossing (the transition from single-stream to multi-stream flow).

3. Key Contributions and Results

A. Ground State Properties

• Density Profile Modification: Contact interactions significantly alter the shape of the lowest-energy stationary solution.
  • Repulsive ($\lambda > 0$): Broadens the density profile and lowers the central density.
  • Attractive ($\lambda < 0$): Narrows the density profile and increases the central density.
• Scaling Laws: The standard deviation ($\sigma$) of the ground state follows power laws with interaction strength:
  • Repulsive: $\sigma \propto \lambda^{0.42}$ (consistent with the Thomas-Fermi approximation where kinetic energy is negligible).
  • Attractive: $\sigma \propto |\lambda|^{-0.15}$ (kinetic and contact terms dominate, resembling bright solitons).
• Spectral Changes: Repulsive interactions introduce additional peaks in the momentum spectrum due to sharper edges in the density profile, whereas attractive interactions maintain smoother spectra.

B. Relaxation Dynamics

• Non-Convergence: The study confirms that in the (1+1)-dimensional model, a localized initial state does not converge to the ground state, even with contact interactions included. This extends previous findings (without interactions) to the interacting case.
• Fidelity Analysis: The fidelity ($F$) between the relaxed state and the ground state oscillates.
  • Repulsive interactions yield more stable states with higher mean fidelity.
  • Attractive interactions lead to more unstable states with lower mean fidelity and larger oscillations, attributed to a smaller Jeans length allowing more unstable modes.
• Qualitative vs. Quantitative: While the relaxed state qualitatively mimics the ground state's dependence on $\lambda$ (broader for repulsive, narrower for attractive), there is no quantitative agreement.

C. Gravitational Collapse and Shell-Crossing

• Shell-Crossing Timing: The contact interaction shifts the redshift ($z_{sc}$) at which the first shell-crossing event occurs (the onset of multi-streaming).
  • Attractive Interactions: Advance shell-crossing (occur at higher $z$) by enhancing gravitational focusing.
  • Repulsive Interactions: Delay shell-crossing (occur at lower $z$) by reinforcing quantum pressure against gravity.
• Empirical Fit: The onset redshift follows an exponential law: $z_{sc} = z_0 e^{-0.97 \lambda / a_{ini}}$.
• Phase Space Analysis: Using the Husimi representation, the authors visualize the transition from single-stream to multi-stream flow. Repulsive interactions result in higher $k$-mode components, indicating faster motion of matter streams during the crossing.

4. Significance and Conclusion

• Role of Interactions: The paper establishes that local self-interactions are a relevant factor in the nonlinear dynamics of FDM, capable of modifying density profiles and the timing of structure formation events.
• Dimensional Constraints: The results reinforce the unique behavior of 1D FDM, where the ground state is not a dynamical attractor for relaxation, contrasting with 3D expectations.
• Cosmological Implications: The ability of contact interactions to delay or advance shell-crossing suggests a potential mechanism to reconcile FDM predictions with small-scale observational data. Repulsive interactions, in particular, could suppress small-scale structure formation by delaying collapse.
• Future Directions: While the 1D model provides qualitative insights (supported by the validity of the Zel'dovich approximation in 1D pre-collapse), the authors emphasize the need for higher-dimensional (3D) simulations to fully assess the impact of these interactions in realistic cosmological settings.

In summary, this work provides a rigorous numerical characterization of how contact interactions alter the fundamental dynamics of Fuzzy Dark Matter in a simplified 1D cosmological setting, offering new perspectives on the interplay between quantum pressure, gravity, and self-interactions.