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Multi-copy Axion Transfer Function and Observational Implications of Effective de Broglie Scales

This paper extends the study of ultra-light axion dark matter from single-copy to multi-copy scenarios motivated by the String Axiverse, demonstrating that gravitational coupling creates an effective mass governing linear structure suppression and non-linear halo dynamics, with a radially varying effective mass in galaxies offering a testable signature for JWST lensing observations.

Original authors: Jiashuo Zhang, Tom Broadhurst, Jeremy Lim, Paloma Morilla, Sung Kei Li

Published 2026-07-29

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CC BY 4.0

Original authors: Jiashuo Zhang, Tom Broadhurst, Jeremy Lim, Paloma Morilla, Sung Kei Li

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 as a giant, invisible ocean. For decades, scientists have been trying to figure out what makes up the "water" of this ocean. We know there's a lot of it—enough to hold galaxies together—but we can't see it, touch it, or taste it. This invisible stuff is called "dark matter." For a long time, the leading theory was that this dark matter is made of heavy, slow-moving particles, like tiny, invisible marbles bouncing around in the dark. But recently, a new idea has been gaining traction: what if the dark matter isn't made of marbles at all, but of a super-light, wave-like substance? Think of it less like a collection of marbles and more like a giant, shimmering fog or a ripple in a pond that stretches across the entire cosmos. This wave-like dark matter is called "axion" dark matter.

The big mystery is: how heavy are these waves? If they are too heavy, they act like normal marbles. If they are incredibly light, they act like giant waves that can't clump together to form small structures, which solves some puzzles about why we don't see as many tiny galaxies as we expected. However, there's a catch. Most of our current theories assume there is only one type of these axion waves, all with the exact same mass. But what if the universe is more like a choir than a solo singer? What if there are many different types of axion waves, all singing at once but with slightly different pitches (masses)? This paper explores that very possibility: a universe filled with a "choir" of axion copies, rather than just a single note.

The Cosmic Choir: When Dark Matter Has Many Voices

In this new study, astronomers Jiashuo Zhang and his team decided to stop looking at dark matter as a single, lonely particle and start treating it like a complex, multi-layered symphony. They asked a simple but profound question: What happens if the dark matter in our universe isn't just one type of ultra-light axion, but a mix of several different copies, each with its own mass?

To understand why this matters, picture the universe's structure as a giant web. In the standard "single-copy" model, this web has a specific texture. The axion waves are so light that they create a "quantum pressure" that stops them from clumping together too tightly on small scales. It's like trying to build a sandcastle with water instead of sand; the waves wash the structure away. This explains why we don't see a zillion tiny dwarf galaxies. However, recent observations have created a bit of a headache. Some data suggests the axions need to be very light to explain the smooth cores of dwarf galaxies, while other data (like the Lyman-α forest) suggests they need to be heavier to match the large-scale structure of the universe. It's as if the universe is giving us two different instructions that don't quite fit together.

The authors propose a clever solution: maybe the universe is using both instructions at once. Perhaps there are multiple copies of axions, some light and some heavy, all contributing to the total dark matter density.

The "Effective Mass" Trick

The team's main discovery is a mathematical shortcut that makes this complex scenario much easier to understand. They found that even if you have a chaotic mix of different axion copies, they don't just act independently. Because they all feel gravity, they talk to each other. On the largest scales of the universe (the "linear regime"), this mutual gravitational coupling makes the whole group behave as if it were a single, "effective" axion.

Think of it like a group of runners in a race. If they are all running at different speeds but are tied together by a rope (gravity), they will eventually settle into a pace that is an average of their individual speeds. The authors calculated that this "average" isn't a simple arithmetic mean. Instead, it's a specific type of average where the lighter axions (the faster runners) have a bigger say in the final speed. They call this the "effective mass."

This is a huge deal because it means we don't need to simulate every single copy of axion to understand the big picture. We can just use this "effective mass" to predict how the universe's large-scale structure forms. If the universe has a mix of axions, the "faint-end turnover" (the point where small galaxies stop forming) will look exactly like it would if there were just one type of axion with this specific effective mass. This suggests that the conflicting data we see might not be a contradiction, but rather a sign that we are looking at a multi-copy universe that looks like a single-copy one from a distance.

The Galaxy as a Ripple Pool

But the story gets even more interesting when we zoom in on individual galaxies. Here, the axion copies start to "decouple." It's like the runners in our race finally untie the rope and start running on their own paths. In the dense center of a galaxy, the different axion copies interfere with each other in complex ways, creating a "corrugated" or bumpy surface density.

The authors show that this interference creates a new kind of "effective mass" for the galaxy, which they call $m'_{eff}$ . Unlike the large-scale effective mass, this one can change depending on where you are in the galaxy. Near the center, the heavier axion copies might dominate, while on the outskirts, the lighter copies take over.

This has a fascinating consequence for how we see the universe. When light from a distant galaxy passes through a massive galaxy in front of it (a process called gravitational lensing), it gets bent. In a standard dark matter universe, this creates a sharp, clean ring of light. But in this multi-copy axion universe, the "bumpy" interference pattern of the dark matter waves smears out that ring. It turns a sharp line into a fuzzy, corrugated band.

The paper suggests that we might already be seeing this! Observations of the "Dragon Arc" (a lensed galaxy seen by the James Webb Space Telescope) show a spread of bright, transient stars that is wider than expected. The authors argue that this spread matches the predictions of a multi-copy axion universe with an effective mass of about $2.40 \times 10^{-22}$ eV. If this is true, it means the dark matter in that galaxy isn't uniform; it's a mix of different axion types, with the lighter ones dominating the outer regions and the heavier ones clustering in the center.

What This Means for the Future

The paper doesn't claim to have solved the mystery of dark matter. Instead, it offers a new lens through which to view the data. It suggests that the "missing satellite problem" (why we don't see enough small galaxies) and the "core-cusp problem" (why galaxy centers are smoother than expected) might both be solved if we accept that dark matter is a multi-copy entity.

The authors are careful to note that their results are based on mathematical models and simulations, not direct measurements of the axions themselves. They propose that future observations with the James Webb Space Telescope could test this idea. If we look at different parts of a lensed galaxy and find that the "fuzziness" of the light changes depending on the distance from the center, it would be a smoking gun for multiple axion copies.

In short, this paper suggests that the dark matter universe might be a lot more diverse than we thought. Instead of a single, uniform fog, it could be a rich, layered tapestry of different wave-like particles, each playing its own part in the cosmic symphony. By understanding how these different "voices" mix together, we might finally be able to tune in to the true nature of the invisible universe.

Technical Summary: Multi-copy axion transfer function and observational implications of effective de Broglie scales

Problem Statement
Ultra-light axions (ψDM) are viable fuzzy dark matter candidates predicted by the string axiverse, which generically suggests the existence of multiple axion species with a discrete mass spectrum spanning many decades. However, current astronomical constraints on ψDM—derived from Lyman-α forest data, galaxy luminosity functions, and lensing observations—typically assume a single-particle copy model. This assumption creates tension, as single-copy models with masses $\sim 10^{-22}$ eV (favored by dwarf galaxy cores and lensing anomalies) are often excluded by large-scale structure constraints requiring masses $> 2 \times 10^{-20}$ eV. The paper addresses the gap in understanding how a multi-copy axion scenario modifies linear transfer functions and non-linear observational signatures, specifically regarding effective de Broglie scales.

Methodology
The authors employ a two-pronged approach covering linear and non-linear regimes:

Linear Regime (Large-Scale Structure):
- The authors derive linear perturbation equations for an arbitrary number of axion fields ( $\phi_i$ ) in a FLRW universe using the Schrödinger-Poisson system.
- They perform a Madelung decomposition to express the system in terms of density and velocity fields, leading to a matrix equation for density contrasts ( $\delta_i$ ).
- By summing the equations for all copies, they derive an equation for the total density contrast ( $\delta_{tot}$ ) that resembles the single-axion case but is governed by an effective mass scale ( $m_{eff}$ ).
- To obtain individual transfer functions ( $T_i$ ) for each copy, they propose a simplified "interpolating sum ansatz." This method combines the single-copy solution (dominant at high wavenumbers $k$ ) and the total solution (dominant at low $k$ ) using a frequency-dependent coefficient $A(k)$ .
- The validity of this ansatz is tested against full numerical integrations of the double-copy (2FDM) perturbation equations from Luu et al. (2025).
Non-Linear Regime (Galaxy Haloes):
- The authors analyze the superposition of Gaussian random fields (GRFs) generated by individual axion copies within galaxy haloes.
- They treat the projected surface density perturbations as a weighted sum of independent GRFs, justified by the lack of strong phase correlations between copies with significantly different masses.
- They derive expressions for an effective mass governing lensing ( $m'_{eff}$ ) and stellar heating ( $m''_{eff}$ ) based on the local relative surface density contributions ( $w'_i$ ) and mass fractions of the copies.

Key Contributions and Results

Linear Regime Equivalence: The study demonstrates that due to mutual coupling to the gravitational potential, all axion copies exhibit an equivalence on scales larger than the effective Jeans scale. Consequently, the suppression of large-scale structure (LSS) and the faint-end turnover of galaxy luminosity functions are governed by a single effective mass defined as:
$\frac{1}{m_{eff}^2} = \sum_i w_i \frac{1}{m_i^2}$
where $w_i$ are the fractional contributions of different copies to the total cosmic dark matter density.
Transfer Function Prescription: The authors provide a practical, simplified prescription (Eq. 21) to calculate multi-copy transfer functions by interpolating between single-copy solutions. This ansatz captures the overall power spectrum suppression with high accuracy (differences in $\sigma^2(M)$ are $< 0.063\%$ compared to full 2FDM integrations), though it does not reproduce high-frequency oscillatory features dominated by the heaviest copy.
Non-Linear Effective Masses:
- Lensing: The net surface-density perturbations in galaxy haloes are governed by a lensing effective mass $m'_{eff}$ , defined as:
  $\frac{1}{m'_{eff}} = \sum_i \left( \frac{\Sigma_i}{\Sigma_{tot}} \right)^2 \frac{1}{m_i}$
  Unlike the linear case, $m'_{eff}$ depends on the local surface density distribution. The authors show that in a multi-copy halo, $m'_{eff}$ varies radially because heavier copies tend to be more centrally concentrated.
- Stellar Heating: Similarly, the net stellar heating rate scales with an effective mass $m''_{eff}$ defined by:
  $\frac{1}{(m''_{eff})^3} = \sum_i \left( \frac{\rho_{h,i}}{\rho_{h,tot}} \right)^2 \frac{1}{m_i^3}$
  This suggests that the lightest copy dominates the heating rate if copies have equal density contributions.

Observational Implications
The paper argues that the radial variation of $m'_{eff}$ within a single halo is a distinctive signature of the multi-copy scenario, absent in single-copy models.

JWST Transients: The spread of microlensed transients (e.g., along the Dragon Arc) is determined by the local de Broglie scale. In a multi-copy universe, this spread scale should vary with radial position or source redshift. The authors suggest that future JWST observations of transients at different locations (e.g., near tangential vs. radial arcs in systems like the Cosmic Horseshoe) could test for this variation, thereby inferring the presence of multiple axion copies.
Consistency Checks: A discrepancy between the soliton size inferred from stellar kinematics and the size predicted by the lensing-derived $m'_{eff}$ could also indicate a multi-copy composition.

Significance and Claims
The authors claim that their work bridges the gap between the string axiverse prediction of multiple axion species and current observational constraints. By establishing that large-scale structure is governed by an effective mass $m_{eff}$ , they suggest that the tension between small-scale observations (favoring $\sim 10^{-22}$ eV) and large-scale constraints (favoring heavier masses) may be resolved if the dark matter sector contains multiple copies with a hierarchical mass spectrum.

The paper modestly notes that while the linear interpolation ansatz is highly accurate for the power spectrum, full numerical simulations are still required to determine precise halo density profiles and the detailed effects of radial variations in $m'_{eff}$ and $m''_{eff}$ on stellar heating and disc thickening. The proposed framework offers a pathway to extend current single-copy simulations and interpret future high-resolution lensing data from JWST without requiring finely tuned subhalo populations.

The Cosmic Choir: When Dark Matter Has Many Voices

The "Effective Mass" Trick

The Galaxy as a Ripple Pool

What This Means for the Future

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