The three phases of self-gravitating scalar field ground states

This paper challenges the conventional assumption that ultralight dark matter halos universally possess spherically symmetric solitonic cores by demonstrating that strong repulsive interactions between multiple scalar fields can cause immiscibility, leading to diverse ground state phases dependent on mass, density, and interaction strength.

The Gist

Imagine the universe is filled with a mysterious, invisible substance called Dark Matter. For a long time, scientists thought this stuff behaved like a giant, fluffy cloud of tiny, ultra-light particles. When these particles clump together to form a galaxy, they were expected to settle into a perfect, round ball at the center, like a smooth, dense marble. This "marble" is called a soliton.

However, this new paper suggests that if there is more than one type of these invisible particles interacting with each other, the story changes completely. The center of a galaxy might not be a single smooth marble at all. Instead, it could look like a hollow shell, a donut, or even two separate blobs floating apart.

Here is a breakdown of the paper's findings using simple analogies:

1. The Setup: Two Types of "Ghost" Particles

The authors are studying a scenario where there are two different species of these ultra-light dark matter particles (let's call them Type A and Type B).

• Gravity acts like a magnet, pulling both types toward the center of the galaxy to form a clump.
• Repulsion acts like a force that pushes them apart. If Type A and Type B don't like each other (they have "repulsive interactions"), they want to get away from one another.

The paper asks: What happens when you mix these two types of particles in a galaxy? Do they mix like milk in coffee, or do they separate like oil and water?

2. The Three "Phases" (The Three Shapes)

The researchers found that depending on how strongly the two particle types push against each other, the galaxy's core can settle into one of three distinct shapes:

Phase 1: The "Mixed Marble" (Solid/Nested)

• The Analogy: Imagine mixing red and blue playdough. If they are friendly, they blend perfectly into a single purple ball.
• The Science: When the repulsive force is weak, the two particle types happily coexist. They form a single, round, dense core at the center of the galaxy, just like scientists previously assumed. Both types of particles peak right at the very center.

Phase 2: The "Hollow Shell" (Nested Hollow)

• The Analogy: Imagine a chocolate truffle with a soft, gooey center and a hard shell. Or think of a fruit with a pit. One type of particle stays in the very center, while the other type is pushed out to form a ring or shell around it.
• The Science: As the repulsion gets stronger, one type of particle is pushed away from the exact center. It creates a "hole" in the middle of its own density. The other type stays in the center. They are still overlapping (nested), but the center is no longer a solid ball of both; it's a core surrounded by a shell.

Phase 3: The "Two Separate Blobs" (Separate/Immiscible)

• The Analogy: Imagine two magnets with the same pole facing each other. They push so hard that they can't sit in the same spot. Instead of one big ball, you get two smaller balls floating side-by-side, or perhaps one orbiting the other, breaking the perfect round shape.
• The Science: If the repulsion is very strong, the two particle types completely separate. They stop sharing the same center. The galaxy's core is no longer a single, perfect sphere. It might look like two distinct clumps or an oval shape. This breaks the "perfect sphere" rule that scientists usually assume.

3. Why This Matters

For years, the standard model of dark matter assumed that every galaxy has a single, smooth, spherical "marble" at its heart. This paper shows that if the universe contains multiple types of these particles, that assumption might be wrong.

• Complexity: The centers of galaxies could be much more complex and diverse than we thought.
• Observation: If we look at real galaxies and see cores that aren't perfect spheres, or if we see "hollow" centers, it might be evidence that our universe has these multiple types of dark matter particles pushing against each other.

Summary

The paper uses computer simulations to show that when you have two types of ultra-light dark matter particles:

1. Weak Push: They mix into one perfect ball.
2. Medium Push: One gets pushed out, creating a hollow center or a shell.
3. Strong Push: They split apart completely, creating two separate blobs and ruining the perfect round shape.

This means the "ground state" (the most stable, resting shape) of dark matter galaxies isn't just one thing; it's a whole family of different shapes depending on how the particles interact.

Technical Summary

Technical Summary: The Three Phases of Self-Gravitating Scalar Field Ground States

Problem Statement
The paper addresses the structure of ground states in self-gravitating scalar field dark matter (SDM) models, specifically focusing on Ultralight Dark Matter (ULDM) scenarios involving multiple interacting species. While single-field ULDM models generally predict that dark matter halos contain spherically symmetric, solitonic cores (ground states), the behavior of multifield systems—motivated by the "string axiverse" and inflationary models with multiple scalar fields—remains less understood. The central problem is determining how repulsive non-gravitational interspecies interactions affect the equilibrium configurations of these systems. Specifically, the authors investigate whether the standard paradigm of a single, nested soliton at the halo center holds when multiple fields with varying masses and interaction strengths are present.

Methodology
The authors employ a combination of analytical energy minimization arguments and numerical simulations to explore the ground states of a two-species system governed by coupled Gross-Pitaevskii-Poisson (GPP) equations.

1. Analytical Approach: The authors derive analytic expressions for the system's total energy (gravitational, kinetic, self-interaction, and interspecies interaction) using a spherically symmetric Gaussian ansatz. By comparing the total energy at zero separation ($d=0$) versus infinite separation ($d \to \infty$), they estimate a critical interaction strength ($\lambda^*_{12}$) at which the system transitions from a miscible (nested) phase to an immiscible (separated) phase.
2. Numerical Simulations:
   • 1D Solver: The primary numerical tool is an extension of the nSPIRal Mathematica module, which evolves the spherically symmetric Schrödinger-Poisson system in imaginary time. This method allows the system to relax from an initial superposition of eigenstates to the lowest energy ground state. The authors test various initial conditions, including symmetric ($\psi_1 = \psi_2$) and perturbed ($\psi_1 \neq \psi_2$) profiles, to explore the parameter space of interaction strengths ($\lambda_{ij}$), mass ratios ($m_2/m_1$), and total mass ratios ($M_2/M_1$).
   • 3D Verification: To verify that observed non-coincident density maxima are not artifacts of the 1D spherical symmetry constraint, the authors perform preliminary 3D simulations using UltraDark.jl. They compare the energy of 1D "hollow" solutions against configurations where two solitons are placed at varying separations on a 3D grid.

Key Contributions and Results
The study identifies three distinct classes of ground state configurations, dependent on the strength of repulsive interspecies interactions and the relative properties of the fields:

1. Nested Solid Phase: At low interaction strengths, the two fields form a "nested" configuration where their density maxima are coincident at the center ($r=0$). This resembles the standard single-field soliton, though the individual profiles are modulated by the presence of the other field.
2. Nested Hollow Phase: As repulsive interactions increase, a phase transition occurs where one field develops a local density minimum at the center, forming a "shell" around the other field's core. In this phase, the centers of mass remain coincident ($d=0$), but the density profiles are no longer co-located. The authors distinguish between "hollow 1" and "hollow 2" depending on which species forms the shell. This phase was found to be sensitive to initial conditions; symmetric initializations preserved the solid phase, while perturbed initializations revealed the hollow phase.
3. Separate (Immiscible) Phase: At sufficiently high interaction strengths, the system favors a configuration where the two solitons separate spatially ($d > 0$). This phase breaks spherical symmetry, resulting in axisymmetric configurations. Analytical arguments suggest this occurs when the energy cost of gravitational binding is outweighed by the reduction in repulsive interaction energy. Preliminary 3D results support the existence of this phase, showing that separated configurations can have lower energy than hollow nested ones at high interaction strengths.

The authors construct a phase diagram mapping these transitions as a function of interaction strength and mass ratios. They note that the transition from "solid" to "hollow" occurs at a critical interaction strength consistent with their analytical estimates ($\lambda^*_{12} \approx 0.1 \Lambda_0$).

Significance and Claims
The paper claims that the assumption of a universally spherically symmetric solitonic core in ULDM halos is insufficient for multifield scenarios. The existence of these three phases implies that the inner regions of dark matter halos could be significantly more complex and diverse than previously assumed.

• Phenomenological Diversity: The results suggest that the "soliton" is not a single, universal structure in the axiverse. Depending on interaction strengths, halos could exhibit hollow cores or completely separated, non-spherical structures.
• Constraints on the Axiverse: The diversity of predicted ground states offers a new avenue for constraining the parameter space of the axiverse. Observational data on halo density profiles (e.g., from dwarf galaxies or sub-halos) could potentially rule out specific combinations of particle masses and interaction strengths.
• Cosmological Implications: The authors note that these findings have implications for inflationary models where multiple scalar fields drive reheating, potentially altering the formation of inflaton halos and stars.
• Limitations: The authors modestly acknowledge that full 3D simulations of the symmetry-breaking "separate" phase are computationally challenging due to resolution requirements. Consequently, the precise boundaries of the separate phase and the behavior of these ground states within dynamic, realistic halos (including random walk motions and baryonic feedback) remain subjects for future work. The current work establishes the existence of these phases and provides a qualitative phase diagram, deferring a complete quantitative 3D phase diagram to future research.