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Early Growth of Structure in Warm Wave Dark Matter

This paper investigates the early growth of structure in wave-like dark matter models by deriving scale-dependent evolution of the matter power spectrum, verifying these analytic results with 3+1-dimensional Schrödinger-Poisson simulations, and proposing an analytic formula for the halo mass function that accounts for early halos hosting solitons.

Original authors: Mustafa A. Amin, Simon May, Mehrdad Mirbabayi

Published 2026-04-16

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CC BY 4.0

Original authors: Mustafa A. Amin, Simon May, Mehrdad Mirbabayi

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

The Universe as a Giant, Wobbly Jelly: How Dark Matter Forms

Imagine the universe not as empty space filled with solid stones (like normal matter), but as a giant, invisible jelly. In this jelly, there are tiny, invisible waves moving through space. These waves are the Dark Matter.

Normally, we think of Dark Matter as an accumulation of tiny, invisible beads (like grains of sand) moving through space. But this new study examines a special type of Dark Matter that behaves more like water waves or sound waves. It is called "Wave Dark Matter."

The researchers (Mustafa Amin, Simon May, and Mehrdad Mirbabayi) wanted to find out: How do galaxies and star clusters form in a universe made of these waves?

Here is the story of how they found out:

1. The Starting Gun: Chaotic Chaos

Imagine that shortly after the Big Bang, this jelly was not distributed evenly. Instead, there were two types of disturbances:

The "Rhythm" (Adiabatic Perturbations): This is like a uniform wobble of the entire jelly, coming from the gravity of normal matter.
The "Noise" (Isocurvature Perturbations): This is like random, static noise arising from the way Wave Dark Matter came into existence in the first place. It is like the static noise on an old radio that occurs everywhere simultaneously.

The researchers asked: What happens to this noise and rhythm as the universe ages?

2. The Two Major Obstacles: "Free-Streaming" and the "Jeans Scale"

As the universe expands, two things happen that prevent galaxies from forming immediately everywhere:

The "Free-Streaming":
Imagine the waves are like very fast, small children running around in a large hall. If they are too fast, they simply run through small clusters of children without gathering. They "blur" small structures.
In physics, this is called the Free-Streaming Effect. It erases the small wave patterns (the adiabatic perturbations) before they can grow into large structures. It is like trying to draw a grain-of-sand pattern on the ground, but a strong wind (the speed of the waves) immediately wipes away the fine lines.
The "Jeans Scale":
Now gravity comes into play. When the waves slow down enough, gravity pulls them together. But there is a limit: If the waves are too short (too many wave crests in a tight space), they push each other apart (like water waves that do not want to overlap).
The researchers found that only waves longer than a certain critical length can actually clump together into clumps (halos). Anything smaller remains chaotic noise.

3. The Experiment: The Computer as a Time Machine

Since one cannot recreate the universe in a laboratory, the authors conducted massive computer simulations.
Imagine a digital cube representing the universe. You fill it with these invisible waves and let time run.

What they saw: In the early phases (while radiation dominated), the fine patterns were indeed blown away by the "wind" (free-streaming).
Later: As the universe cooled and gravity became stronger, the waves began to gather. But not uniformly everywhere! Only where the waves were long enough did clumps form.

4. The Surprise: The "Solitons" (The Solid Cores)

The coolest part of this study is a discovery that only waves can make. When this Wave Dark Matter clumps together into a clump (a halo), it does not form a chaotic pile in the center, but a perfect, stable core.
The authors call this a "Soliton".

The Analogy: Imagine a hurricane. Normally, the eye of the storm is chaotic. But with this Dark Matter, a kind of perfect, quiet crystal or solid core forms in the center of the clump, looking like a solid object even though it consists of pure waves.
The simulations showed that almost every one of these early clumps has such a soliton core. It is as if the jelly in the center of every clump turns into a solid stone, while the rest still wobbles.

5. Why is this important?

Previous theories often treated Dark Matter like grains of sand. This study shows that if Dark Matter consists of these light waves, the formation of galaxies proceeds differently:

Small structures are suppressed: Because the waves are too fast to form small clumps (free-streaming).
Large structures grow differently: They grow only if they are large enough to overcome wave repulsion.
The core is different: The galaxies that form have a very specific, wave-like core (the soliton), which differs from what we would expect with normal "sand" Dark Matter.

Conclusion for Everyday Life

Imagine you are building a city out of sand (normal matter) and a city out of water (this special Dark Matter).

The Sand City builds small towers everywhere, which then slowly grow into large castles.
The Water City first has many small waves that are blown away by the wind. Only when the waves are large enough do they form huge, stable lakes. And in the center of each lake, a perfect, solid block of ice (the soliton) forms, creating the center of the city.

This study helps us understand what the universe looked like in its earliest days and why we might not see certain small galaxies that we would have expected. It also provides a kind of "recipe" to predict how many galaxies and stars we will find in the future, if we know that Dark Matter is more like water waves than grains of sand.

1. Problem Statement

The microscopic nature of Dark Matter (DM) remains unknown. While standard Cold Dark Matter (CDM) assumes massive, collisionless particles, Wave Dark Matter (WDM) models (e.g., ultralight axions, vector fields) treat DM as a coherent classical field governed by the Schrödinger-Poisson system.

A specific scenario arises when such wave-like DM is produced via post-inflationary mechanisms (e.g., the Peccei-Quinn phase transition). Unlike inflationary production, which yields a homogeneous field, post-inflationary production creates:

Sub-horizon field patches: The field is dominated by modes with wavenumbers $k \sim k_*$ (where $k_*$ is the characteristic de Broglie scale at production).
Enhanced Isocurvature: Significant density fluctuations on small scales due to the randomness of field phases.
Free-streaming: A velocity dispersion that suppresses adiabatic perturbations on small scales.

The Gap: Previous analytical work (e.g., Amin et al. [26]) utilized a "quasi-particle" picture (BBGKY hierarchy) to model these effects. However, this approach may miss distinct wave-dynamical effects (interference, phase coherence) near the scale $k \sim k_*$ . Furthermore, there was a lack of rigorous 3+1 dimensional cosmological simulations incorporating both adiabatic and isocurvature initial conditions for this specific "warm" wave scenario to verify analytical predictions and study non-linear structure formation (halos and solitons).

2. Methodology

A. Analytical Derivation

The authors derive the scale-dependent evolution of the matter power spectrum ( $P_\delta$ ) starting directly from the Schrödinger-Poisson (SP) system in an expanding universe, rather than the quasi-particle hierarchy.

Formalism: They define a bilinear variable $\hat{f}_k(p)$ related to the field Fourier modes to track density and velocity perturbations.
Perturbation Types:
- Adiabatic: Treated as a shift in the initial field configuration. The evolution is governed by a transfer function $T^{(ad)}$ involving free-streaming kernels.
- Isocurvature: Treated as the growth of Poisson noise arising from field randomness. The evolution involves a "sourced" solution to the correlation function equations, leading to a transfer function $T^{(c)}$ .
Key Scales Identified:
- Free-streaming scale ( $k_{fs}$ ): Suppresses adiabatic power during radiation domination.
- Jeans scale ( $k_J$ ): Determines the onset of gravitational clustering for isocurvature modes during matter domination.
- Wave parameter ( $\gamma$ ): A dimensionless ratio $\gamma \sim k_J(a_{eq})/k_*$ quantifying the deviation from the quasi-particle limit. When $\gamma \to 0$ , wave effects vanish; when $\gamma \sim 1$ , wave interference is dominant.

B. Numerical Simulations

To verify the analytic results and explore the non-linear regime, the authors performed 3+1 dimensional cosmological simulations using the AxiREPO code.

Initial Conditions (ICs): A novel setup was developed to generate fields with:
1. A power spectrum peaked at $k_*$ (isocurvature).
2. Large-scale adiabatic perturbations ( $\delta_{ad}$ ) and bulk velocities ( $v_{ad}$ ) consistent with inflationary curvature spectra.
Evolution: The SP equations were solved using a pseudo-spectral method with a "kick-drift-kick" leapfrog integrator.
Parameters: Simulations covered masses $m \sim 10^{-19}$ eV, box sizes $L \sim 0.4-0.8$ Mpc, and resolutions up to $N^3 = 2880^3$ . They evolved from the radiation era ( $z \sim 5 \times 10^5$ ) to the matter era ( $z \sim 99$ ).

3. Key Contributions

Unified Analytic Framework: Derived a complete set of evolution equations for the power spectrum in warm wave DM, explicitly separating adiabatic and isocurvature components and identifying the distinct roles of free-streaming and Jeans scales.
Wave-Dynamical Corrections: Identified specific wave effects (oscillatory terms in transfer functions) that modify the growth of isocurvature perturbations at scales $k \sim k_*$ , which are absent in the quasi-particle approximation.
Novel Simulation Setup: Successfully implemented initial conditions that combine sub-horizon isocurvature noise (peaked at $k_*$ ) with large-scale adiabatic modes, bridging the gap between inflationary cosmology and wave DM phenomenology.
Halo and Soliton Formation: Provided the first quantitative estimates for the formation times and mass functions of halos and central solitons in this specific warm wave scenario.

4. Key Results

A. Linear Regime (Power Spectrum)

Adiabatic Suppression: During radiation domination, adiabatic perturbations with $k > k_{fs}$ are exponentially suppressed (erased) due to free-streaming.
Isocurvature Growth: During matter domination, the initially white-noise isocurvature spectrum grows.
- For $k < k_J(a_{eq})$ , growth follows the standard CDM-like $\propto a^2$ .
- For $k_J(a_{eq}) < k < k_J(a)$ , growth is scale-dependent and suppressed.
- For $k > k_J(a)$ , growth is halted.
Agreement: The analytical predictions match the 3+1D simulation results with high precision in the linear regime, validating the derived transfer functions.
Wave Effects: Deviations from the quasi-particle limit (approx. 10%) are observed at $k \sim k_*$ due to wave interference terms ( $\sin, \cos$ in kernels), even when $\gamma$ is small.

B. Non-Linear Regime (Structure Formation)

Non-linear Onset: The isocurvature part of the spectrum becomes non-linear first at a scale factor $a_{nl} \propto \gamma^{-3/2}$ .
Halo Mass Function: The authors propose an analytic formula for the halo mass function based on the linear power spectrum (using excursion set theory).
- The formula matches simulation results well at the high-mass end.
- Discrepancies exist at the low-mass end, likely due to the specific dynamics of soliton formation not captured by standard Press-Schechter arguments.
Soliton-Halo Structure:
- Simulations confirm that early halos typically host a soliton (a stable, localized ground-state solution) at their center.
- Mass Fraction: At the end of the simulations ( $a/a_{eq} \approx 34$ ), solitons constitute $\sim 50\%$ of the mass in collapsed halos (though only $\sim 0.1\%$ of the total volume mass).
- Resolution: Low-mass halos ( $M \sim 10^3 M_\odot$ ) are well-resolved, showing clear soliton cores surrounded by interference granules. Higher mass halos show deviations from the standard NFW profile, hinting at unresolved solitons.

5. Significance and Implications

Observational Constraints: The derived power spectrum shapes (free-streaming cutoffs and isocurvature enhancements) provide specific targets for upcoming observations (Lyman- $\alpha$ forest, stellar streams, 21cm intensity mapping, and high-redshift galaxy counts). This allows for tighter constraints on the mass ( $m$ ) and production mechanism of wave DM.
Beyond Quasi-Particles: The work demonstrates that while the quasi-particle picture is a good approximation for $k \ll k_*$ , wave interference effects are non-negligible near the characteristic scale $k_*$ , necessitating full SP simulations for accurate predictions.
Early Universe Structure: The paper establishes that in warm wave DM scenarios, structure formation is driven primarily by the growth of isocurvature modes, leading to a population of early, soliton-rich halos that differ significantly from standard CDM predictions.
Future Probes: The identification of solitons as a dominant component of early halos suggests that observational signatures of wave DM (e.g., specific density profiles, dynamical heating) may be detectable in the early universe or in dwarf galaxies.

In summary, this paper provides a rigorous theoretical and numerical foundation for understanding structure formation in warm wave dark matter produced after inflation, bridging the gap between linear perturbation theory and non-linear halo formation while highlighting unique wave-mechanical signatures.