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Core-halo scaling relations in self-interacting scalar field dark matter

Using three-dimensional simulations of soliton mergers, this study demonstrates that self-interactions in scalar field dark matter significantly alter core-halo scaling relations and structural properties, with repulsive forces creating extended, lower-density cores and attractive forces driving high-density collapse, thereby offering a mechanism to regulate core evolution and influence supermassive black hole formation.

Original authors: Jessica N. López-Sánchez, Erick Munive-Villa, Tanja Rindler-Daller

Published 2026-05-13

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CC BY 4.0

Original authors: Jessica N. López-Sánchez, Erick Munive-Villa, Tanja Rindler-Daller

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is filled with a mysterious, invisible substance called Dark Matter. For a long time, scientists thought this stuff was just "cold" and lazy, clumping together under gravity to form the scaffolding for galaxies. But a newer, more exotic idea suggests that dark matter might actually be a giant, cosmic wave—a "fuzzy" field of ultra-light particles.

This paper, titled "Core-halo scaling relations in self-interacting scalar field dark matter," explores what happens when these fuzzy waves don't just sit there, but actually talk to each other.

Here is the story of their findings, explained simply:

1. The Setup: A Cosmic Dance Party

The researchers set up a digital simulation (a virtual universe) to watch what happens when many small, fuzzy "solitons" (think of them as tiny, self-contained waves or bubbles of dark matter) crash into each other.

In the real world, these waves usually just merge to form a galaxy. But in this study, the scientists asked: What if these waves push each other away, or pull each other closer?

They tested three scenarios:

The "Free" Crowd: The waves ignore each other (the standard model).
The "Pushy" Crowd: The waves have repulsive self-interactions (they don't like being close and push away).
The "Clingy" Crowd: The waves have attractive self-interactions (they really want to be close and pull together).

2. The Results: How the Galaxy Cores Change

The "Pushy" Scenario (Repulsive Interactions)

Imagine a group of people at a party who really need personal space. If you try to crowd them together, they push back hard.

What happened: When the dark matter waves pushed against each other, the center of the galaxy (the "core") became larger and fluffier.
The Analogy: It's like a puffy cloud. Because the waves are pushing outward, the core spreads out. It becomes less dense (more spread out) but holds more total mass. It's a "soft" core that resists being squished.

The "Clingy" Scenario (Attractive Interactions)

Now, imagine a group of people who are magnetically attracted to each other. If you bring them together, they huddle tight.

What happened: When the waves pulled toward each other, the center of the galaxy became extremely dense and compact.
The Analogy: It's like a black hole forming in slow motion. The waves squeeze themselves into a tiny, super-dense ball. If they get too heavy, they collapse completely. The paper notes that if this attraction is too strong, the core can collapse into a supermassive black hole.

The "Free" Scenario (No Interaction)

This is the middle ground. The waves merge, but they don't push or pull extra hard. They form a core that is somewhere between the "puffy" and the "squeezed" versions.

3. The Rules of the Game (Scaling Relations)

Scientists love finding rules that predict how big a galaxy's core will be based on how big the whole galaxy is. They call these "scaling relations."

The Old Rule: For the "Free" crowd, there was a known rule: If the galaxy gets bigger, the core gets bigger in a specific, predictable way.
The New Discovery: The authors found that this rule isn't universal. It changes depending on whether the dark matter is "pushy" or "clingy."
- If the waves are pushy, the core is bigger than the old rule predicted.
- If the waves are clingy, the core is smaller and denser than the old rule predicted.

Think of it like baking cookies. If you have a standard recipe (Free Dark Matter), you know exactly how big the cookie will be. But if you add a "pushy" ingredient (Repulsion), the cookie spreads out and gets bigger. If you add a "clingy" ingredient (Attraction), the cookie shrinks and gets denser. You can't use the same recipe for all three.

4. The Energy Balance

The paper also looked at the "energy budget" of these galaxies.

In the pushy galaxies, the "pushing" force (self-interaction energy) becomes the dominant player in the center, fighting against gravity to keep the core puffy.
In the clingy galaxies, gravity and the "pulling" force team up to crush the center.

5. Why This Matters (According to the Paper)

The authors conclude that the way dark matter behaves isn't just a simple, one-size-fits-all story.

Repulsive interactions act as a natural "regulator" that keeps galaxy cores from getting too dense, making them bigger and more spread out.
Attractive interactions act as a trigger that can make galaxy cores collapse into black holes.

The paper emphasizes that to understand the galaxies we see in the sky, we need to know exactly how these invisible waves interact with each other. If they push, galaxies look one way; if they pull, they look very different.

In short: Dark matter isn't just a passive background; it has a personality. It can be a "pushy" neighbor that keeps galaxies puffy, or a "clingy" neighbor that squeezes them into black holes. The rules of the universe change depending on which personality it has.

Problem Statement
Scalar Field Dark Matter (SFDM), often referred to as Fuzzy Dark Matter (FDM) when self-interactions are neglected, posits that dark matter consists of ultralight bosons ( $m \sim 10^{-22}$ eV/ $c^2$ ). While standard FDM models successfully explain the formation of solitonic cores in galactic halos, the impact of self-interactions (SI) on the structure and evolution of these halos remains an open question. Previous literature has established scaling relations for non-interacting FDM (e.g., core-halo mass relations), but it is unclear how repulsive or attractive self-interactions modify these relations, particularly regarding the stability of cores, the mass-radius relationship, and the potential for gravitational collapse into supermassive black holes. Furthermore, while the Thomas-Fermi (TF) regime of strongly repulsive SFDM has been studied using fluid approximations, direct numerical simulations of soliton mergers in this regime without fluid approximations have been scarce.

Methodology
The authors perform three-dimensional numerical simulations of multiple soliton mergers to investigate SFDM halos across a wide range of self-interaction strengths and signs.

Governing Equations: The study solves the non-relativistic Gross-Pitaevskii-Poisson (GPP) equations, which describe the evolution of the scalar field wave function $\psi$ coupled to gravity. The equations include a cubic self-interaction term (proportional to scattering length $a_s$ ) and a higher-order quintic term.
Numerical Scheme: The authors utilize a pseudo-spectral time-splitting method optimized for GPUs. This approach treats the self-interaction terms as an effective potential, allowing the use of a kick-drift-kick (KDK) scheme. For attractive interactions, a higher-order (O(4)) KDK method is implemented to ensure stability.
Simulation Setup: Simulations involve the merger of $N_{sol}$ solitons (ranging from 10 to 50) within a static, periodic simulation box of $L=100$ kpc. The particle mass is fixed at $m = 10^{-22}$ eV/ $c^2$ .
Parameter Space: The study explores three distinct regimes:
1. Free (FDM): $a_s = 0$ .
2. Repulsive SI: $a_s > 0$ , including the strongly repulsive Thomas-Fermi regime ( $\Lambda \gg 1$ ).
3. Attractive SI: $a_s < 0$ , up to values where the system approaches gravitational collapse.
Analysis: The authors analyze density profiles, core mass ( $M_c$ ), core radius ( $r_c$ ), and total energy ( $E$ ) to derive scaling relations. They compare simulation results against analytical predictions derived from energy minimization and self-similar solutions.

Key Contributions and Results

Core Structure and Density:
- Repulsive SI: Leads to the formation of more massive and extended cores with lower central densities compared to the free FDM case. The repulsive interaction provides additional pressure support, preventing the core from becoming as compact as in the non-interacting case.
- Attractive SI: Enhances central densities. For sufficiently strong attractive interactions, the core density diverges, indicating a gravitational collapse. The authors identify a critical mass scale ( $M_{max}$ ) for attractive models; if the core mass exceeds this threshold, the equilibrium is lost, potentially leading to the formation of supermassive black holes.
- Evolution: Repulsive cores remain stable and extended throughout the simulation ( $t = 10\tau_{dyn}$ ), whereas attractive cores show a monotonic increase in central density until collapse.
Mass-Radius and Core-Halo Scaling Relations:
- Mass-Radius Relation: The simulation results for the core mass-radius relation ( $M_c$ vs. $r_c$ ) align well with analytical predictions (Chavanis 2011; Padilla et al. 2021) even in the presence of strong self-interactions. Repulsive interactions shift the relation upward (allowing more mass for a given radius), while attractive interactions shift it downward.
- Core-Halo Mass Relation: The authors investigate the relation between the core-to-halo mass ratio ( $M_c/M$ ) and the invariant quantity $\Xi = |E| / [M^3(Gm/\hbar^2)]$ . They find that while the slope of this relation remains relatively universal (similar to the FDM case $M_c/M \propto \Xi^{1/3}$ ), the normalization depends on the self-interaction strength. This suggests that the mechanism feeding mass to the core is primarily gravitational and universal once halos are virialized, while SI locally determines the core's pressure support and radius.
- Size-Energy Relation: The relation between core size ( $r_c$ ) and total energy ( $E$ ) is sensitive to SI. Repulsive interactions result in shallower slopes (larger cores for a given energy), whereas attractive interactions result in steeper slopes.
Energy Decomposition:
- In repulsive cases, the self-interaction energy dominates the core region, explaining the lower density and larger size.
- In attractive and free cases, the gravitational potential energy dominates the core.
- The authors confirm that scaling relations are not strictly universal but depend on the evolutionary stage of the halo and the specific SI parameters.

Significance and Claims
The paper claims that self-interactions provide a natural mechanism to regulate the properties of SFDM halo cores. The study demonstrates that:

The core-halo scaling relations derived for free FDM are limiting cases within a broader family of SFDM models that include self-interactions.
Repulsive interactions can stabilize cores against collapse and produce larger, more diffuse structures, which may have implications for the "core-cusp" problem in galaxy formation.
Attractive interactions can drive cores toward instability and collapse, offering a potential formation channel for supermassive black hole seeds in galactic centers.
The pseudo-spectral method is validated as a robust tool for simulating SFDM in the Thomas-Fermi regime without resorting to fluid approximations, successfully capturing the transition from solitonic to fluid-like behavior.

The authors conclude that while the fundamental gravitational dynamics of core growth are universal, the specific structural properties (density, size, stability) are highly sensitive to the sign and magnitude of self-interactions and the evolutionary history of the halo. These findings suggest that observational constraints on galactic cores could be used to constrain the scattering length and particle mass of SFDM.