⚛️ phenomenology

Coherent and Stochastic Axion Dark Matter from Thermal Relaxation

This paper proposes a unified framework where thermal relaxation of axions in an expanding plasma partitions dark matter into coherent and stochastic components, resulting in a relic population with lower mean momentum, reduced isocurvature, and distinct structural properties compared to standard equilibrium scenarios.

Original authors: Shahid Hussain Gurmani, Arzu Cilli, Phongpichit Channuie, Ahmadjon Abdujabbarov, Farruh Atamurotov, Ertan Güdekli

Published 2026-08-07

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CC BY 4.0

Original authors: Shahid Hussain Gurmani, Arzu Cilli, Phongpichit Channuie, Ahmadjon Abdujabbarov, Farruh Atamurotov, Ertan Güdekli

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 Invisible Ghosts and the Cosmic Fog

Imagine the universe is filled with invisible, ghostly particles called "axions." Scientists think these particles might be the mysterious "dark matter" that holds galaxies together, acting like the invisible glue of the cosmos. For decades, the leading idea was that these axions were born as a giant, perfectly synchronized wave—a "coherent condensate"—where every single particle moved in lockstep, like a massive school of fish swimming in the exact same direction. This is known as the "misalignment" mechanism.

However, the universe is also a hot, messy place filled with energy and plasma. Just as a swimmer slows down when moving through thick water, these axion waves should interact with the hot plasma of the early universe. This interaction creates friction and randomness. The big question scientists have been asking is: Does this friction just slow the axions down, or does it completely change their nature? Does it turn that perfect, synchronized wave into a chaotic, jumbled crowd of individual particles? Understanding this is crucial because the way these particles move and interact determines how they clump together to form the structures we see in the sky today, and whether they leave behind any "fingerprint" of the very early universe that we could detect.

The Paper's Story: When Waves Break into a Crowd

In this paper, the authors, led by Shahid Hussain Gurmani and colleagues, explore a scenario where axions don't just sit still or move in a perfect wave. Instead, they are pushed toward a "thermal minimum"—a comfortable resting spot determined by the temperature of the universe—while being bombarded by a hot, hidden "bath" of energy.

The authors propose a new way to look at this process. They suggest that as the axion field tries to settle down, it doesn't just stay as one big wave. Instead, the universe acts like a filter that splits the axions into two groups. One group remains as the original, coherent wave (the synchronized swimmers), while the other group gets kicked into random, chaotic motion, becoming a "stochastic" crowd of individual particles.

The key discovery here is a mathematical rule that links these two groups. The authors found that for every possible speed (or momentum) an axion can have, the amount of energy left in the coherent wave plus the number of random particles created must always add up to a perfect whole. Think of it like a pie: if you take a slice for the random particles, the coherent wave gets a smaller slice. If the "friction" from the hot plasma is strong, the wave shrinks, and the random crowd grows. If the friction is weak, the wave stays big.

Crucially, the paper argues that this random crowd isn't just a hot, messy mess. Because the universe has a finite size and the interactions take a tiny bit of time to happen, the random axions produced are actually "cooler" and slower than you would expect from a standard hot gas. The authors show that the "optical depth"—a measure of how much the axions interact with the plasma—decreases as the axions get faster. This means the fast, high-energy particles are less likely to be created than the slow, low-energy ones. As a result, the final population of axions has a lower average speed and travels a shorter distance before stopping (a shorter "free streaming length") than a standard thermal population would.

The paper also rules out the idea that this process creates massive, clumpy structures called "miniclusters" on large scales. Because the random interactions happen on incredibly tiny, microscopic scales, the "noise" they create is smoothed out by the time gravity tries to pull things together. The authors calculate that the "skewness" (a measure of how lumpy the distribution is) drops to almost zero on the scales where galaxies form. So, while the local density might look weird and non-uniform on a microscopic level, the universe looks perfectly smooth and calm on the grand scale of galaxies.

In their specific model, the authors show that the dark matter we see today is mostly made up of these new, random "stochastic" particles, not the old synchronized waves. The remaining coherent wave is so tiny that it barely contributes to the total mass. This is a big deal because it means the "fingerprint" of the early universe (called isocurvature) is almost completely erased, leaving a much cleaner slate for the cosmos to evolve.

The authors are quite confident in these results within the framework of their calculations. They derived these relationships using standard physics equations for how fields interact with heat and friction. They didn't just guess; they showed that if you follow the math of a "weakly coupled" system (where the interactions are gentle but present), the universe naturally produces a dark matter population that is dominated by these random, slow-moving particles. They also checked that their model fits with the known amount of dark matter and doesn't break any rules about how the universe expands.

So, the main takeaway is that thermal relaxation—the process of axions settling down in a hot universe—doesn't just dampen a wave; it fundamentally reshapes the dark matter. It turns a coherent, synchronized entity into a momentum-resolved, stochastic relic that is colder, slower, and smoother on large scales than previously thought. This changes how we think about the composition of the dark matter halo surrounding our galaxy and how it might have influenced the formation of the first stars and galaxies.

Technical Summary: Coherent and Stochastic Axion Dark Matter from Thermal Relaxation

Problem Statement
Standard axion dark matter cosmology typically relies on the misalignment mechanism, where a coherent field displacement evolves into a non-relativistic condensate. However, this framework often leaves the statistical composition of the relic unresolved, treating the axion as purely coherent or purely thermal. The paper addresses the gap in understanding how a thermal environment simultaneously drives the attenuation of the coherent field mean and the production of stochastic momentum modes. Specifically, it seeks to determine the dynamical relation between coherent suppression, stochastic occupation, and the resulting momentum distribution, which standard peak-mass comparisons (e.g., $m_T \sim H$ ) fail to resolve. The authors argue that a thermal minimum determines the force on the field but does not inherently specify the statistical composition (coherent fraction vs. Bose occupation) or the spatial power retained after decoupling.

Methodology
The authors derive a modewise relation for axion evolution in an expanding plasma, treating the system as an open quantum system coupled to a thermal bath.

Thermal Realization: The model employs a scalar multiplet $\chi_A$ to generate a temperature-dependent curvature in the axion potential $V_T(\theta, T)$ , driving the field toward a thermal minimum. A hidden Yang-Mills bath provides dissipative response and stochastic forcing via topological transport.
Dynamical Operations: The evolution is decomposed into four sequential operations acting on different field moments:
1. Restoring Strength: Determines the angular displacement before thermal curvature vanishes.
2. Coherent Survival: Governed by the least damped local eigenvalue, determining the survival of the coherent amplitude.
3. Stochastic Occupation: Determined by a momentum-dependent optical depth $\tau_x$ arising from the bath's finite temporal and spatial correlation scales.
4. Collisionless Transfer: Determines which spatial correlations survive to gravitational scales.
Modewise Relation: The core derivation establishes that for any momentum mode $x$ , the fractional Bose occupation ( $n_x/n_B$ ) and the retained coherent power ( $|A_x/A_{x,i}|^2$ ) sum to unity:
$\frac{n_x}{n_B(x)} + \left| \frac{A_x}{A_{x,i}} \right|^2 = 1$
This identity holds before momentum integration, linking the one-point function (coherent mean) and the connected two-point function (stochastic variance) through a single optical depth.
Response Functions: The authors utilize finite-range spatial and temporal response functions (Gaussian and exponential profiles) to model the bath's correlation scales, ensuring the optical depth decreases monotonically with momentum.

Key Contributions

Unified Dynamical Relation: The paper derives a single relation that partitions every momentum mode between coherent power and stochastic Bose occupation. This unifies standard misalignment, partial relaxation, and stochastic production into one framework.
Kinematic Ordering: By demonstrating that the optical depth decreases with momentum due to finite spatial and temporal response, the authors prove analytically that the generated stochastic population has a lower mean momentum and shorter free-streaming length than a Bose equilibrium population at the same mass and bath temperature.
Statistical Distinction: The work distinguishes between the local non-Gaussian density statistics (driven by microscopic bath correlations) and the cosmological density power (determined by free-streaming). It shows that while the local distribution may be non-Gaussian, the microscopic correlation length prevents the formation of miniclusters on gravitational scales.
Isocurvature Suppression: The model demonstrates that a weakly coupled realization can produce the full dark matter abundance primarily through stochastic modes, thereby strongly reducing the inherited coherent isocurvature component.

Results

Benchmark Parameters: Using a specific benchmark (Table I), the authors show that the dark matter abundance is dominated by the stochastic population ( $f_{coh} \approx 1.21 \times 10^{-5}$ ), unlike standard misalignment where the coherent fraction is dominant.
Momentum Distribution: The produced axions have a mean momentum $\langle x \rangle \approx 0.33$ (Gaussian response) compared to $\langle x \rangle_B = 2.70$ for Bose equilibrium. This results in a "colder" spectrum despite the thermal origin.
Free Streaming: The calculated free-streaming length is $\lambda_{fs} \approx 2 \times 10^{-4}$ Mpc, corresponding to a wavenumber $k_{fs} \approx 3.1 \times 10^4$ Mpc $^{-1}$ . This scale is well above the CMB pivot and Lyman- $\alpha$ constraints, implying the model avoids structure suppression limits that affect warmer thermal relics.
Density Statistics: The normalized skewness at the free-streaming radius is suppressed to $\sim 10^{-28}$ due to spatial averaging over the microscopic correlation length ( $\sim 10^{24}$ Mpc $^{-1}$ ). Consequently, the model does not generate axion miniclusters, as the necessary angular variations on gravitational scales are absent.
Isocurvature Constraints: The coherent fraction is sufficiently small ( $f_{coh} \sim 10^{-5}$ ) that the inherited coherent isocurvature is negligible. The remaining isocurvature is constrained by hidden radiation entropy modes, requiring perturbations below $\sim 6.5 \times 10^{-6}$ .

Significance
The paper claims that thermal relaxation fundamentally alters the composition, momentum distribution, and transported structure of axion dark matter. By deriving a momentum-resolved relation between coherent attenuation and stochastic production, the authors show that the relic is not merely a rescaled thermal distribution but a specific phase-space state defined by the interplay of field mean and covariance. The significance lies in demonstrating that a thermal mechanism can produce the correct dark matter abundance while naturally suppressing coherent isocurvature and generating a sub-thermal momentum spectrum that evades standard free-streaming constraints. The work establishes that finite response times and correlation lengths are critical in determining the "velocity support" and "spatial power" of the relic, separating microscopic non-Gaussianity from cosmological smoothness.

The Invisible Ghosts and the Cosmic Fog

The Paper's Story: When Waves Break into a Crowd

Technical Summary: Coherent and Stochastic Axion Dark Matter from Thermal Relaxation

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