⚛️ phenomenology

Putting the Brakes on Axion Strings: Friction and Its Impact on the QCD Axion Abundance

This paper demonstrates that thermal friction in DFSZ-like models significantly delays the scaling dynamics of axion strings, thereby increasing axion energy density and allowing axion masses of approximately 0.1 eV, in addition to the standard meV range, to account for the observed dark matter abundance.

Original authors: Anson Hook, Rajrupa Mondal, Shourya Mukherjee

Published 2026-03-03

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CC BY 4.0

Original authors: Anson Hook, Rajrupa Mondal, Shourya Mukherjee

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 Big Picture: The Cosmic Mystery

Imagine the universe is a giant, invisible ocean. We know there is a lot of "stuff" in this ocean that we can't see, called Dark Matter. It holds galaxies together, but we don't know what it's made of.

Physicists have a favorite suspect: a tiny, ghostly particle called the Axion. It's like a cosmic whisper—so light and weak that it barely interacts with anything.

There are two main ways scientists think these axions were created in the early universe:

The "Misalignment" method: Like a pendulum that starts swinging randomly and gets stuck.
The "String" method: Like a net of cosmic spaghetti that gets tangled and snaps, shooting out axions.

This paper focuses on the String method.

The Old Story: The Cosmic Spaghetti Net

In the early universe, right after the Big Bang, a symmetry broke (think of a perfectly round ball of dough suddenly cracking). This created a network of Axion Strings.

Imagine these strings as incredibly long, thin, vibrating guitar strings stretched across the entire universe.

They wiggle and snap.
When they snap or wiggle, they shoot out axions (the dark matter particles).
For a long time, physicists thought these strings quickly settled into a steady rhythm called the "Scaling Regime." In this state, the strings behave predictably, and the amount of dark matter they produce depends on one specific number: the "decay constant" ( $f_a$ ).

The Problem: If you calculate the amount of dark matter produced by this steady rhythm, it only matches the amount we see in the universe if the axions have a very specific, tiny mass (about 0.0001 eV). This is the "standard" prediction.

The New Twist: The Cosmic Friction

The authors of this paper asked a simple question: "What if the strings aren't moving through empty space?"

In the early universe, the space wasn't empty; it was a hot, dense soup of particles (a "thermal bath"). As the axion strings moved through this soup, they didn't just glide; they dragged.

Think of it like this:

The Old View: A skier gliding on fresh, powdery snow. They move fast and smoothly.
The New View: The skier is trying to run through waist-deep water. The water creates friction, slowing them down and making it hard to move.

In models where axions talk to ordinary matter (like electrons and quarks), this friction is huge. It's like the strings are stuck in molasses.

How Friction Changes the Game

The paper argues that this friction drastically changes the story, especially for axions that are a bit heavier than the standard prediction (around 0.1 eV).

Here is the step-by-step logic using our analogies:

The "Stuck" Phase: Because of the friction, the strings can't wiggle or snap efficiently at first. They are "frozen" in the hot soup.
Delayed Release: In the old story, the strings would snap and release axions early and steadily. With friction, the strings stay stuck for a long time. They don't release their axions until much later, when the universe cools down enough for the "molasses" to thin out.
The Pile-Up: Because the strings are delayed, they don't release their energy gradually. Instead, they hold onto their energy and then release a massive burst of axions all at once when they finally break free.
The Result: This burst creates way more axions than the old "steady rhythm" model predicted.

The "Two-Door" Discovery

The most exciting part of the paper is what this means for the mass of the axion.

Door 1 (The Old Door): The standard model says axions must be very light (mass $\sim$ 0.0001 eV) to make the right amount of dark matter.
Door 2 (The New Door): The authors show that because of this friction, axions can also be 1,000 times heavier (mass $\sim$ 0.1 eV) and still produce the exact right amount of dark matter.

Why is this a big deal?

The "light" axions (Door 1) are very hard to find; we haven't found them yet.
The "heavier" axions (Door 2) are much easier to detect. In fact, we might be on the verge of finding them with current experiments! Or, if they exist, they might be slightly in conflict with some existing rules about how stars burn.

The Method: A Semi-Analytical Framework

You might wonder: "If the physics is so complex, how did they figure this out?"

They couldn't run a full computer simulation because the universe is too big and the strings are too small to model everything at once. It's like trying to simulate every single water molecule in a hurricane.

Instead, they built a smart shortcut (a semi-analytical framework):

They used a known model (the VOS model) that describes how strings move.
They added a "friction term" to the equations, like adding a drag coefficient to a car's engine.
They calibrated this model using existing computer simulations of frictionless strings to make sure the math was right.
They ran two different "estimators" (ways of counting the axions) to make sure their answer was robust.

The Conclusion

The paper concludes that friction is a game-changer.

If the universe was filled with a "sticky" plasma that slowed down axion strings, then the "sweet spot" for dark matter isn't just one tiny mass. It could be a much heavier mass that is much easier to find in a lab.

In short: The universe might be full of "heavy" axions that we missed because we assumed the strings were moving smoothly. But if they were dragging through the cosmic soup, they would have dumped a huge pile of axions right when we needed them, solving the dark matter mystery with a particle we might actually be able to catch soon.

1. Problem Statement

The paper addresses a critical gap in the calculation of the QCD axion dark matter abundance produced via the post-inflationary Peccei-Quinn (PQ) symmetry breaking scenario.

Standard Assumption: Previous numerical simulations and analytical models typically assume that axion string networks quickly reach a "scaling regime" where the network's energy density scales with the Hubble radius. In this regime, friction from the thermal plasma is often neglected, particularly in KSVZ models where axion-gauge boson couplings are suppressed.
The Gap: In DFSZ-like models (and models with tree-level axion-fermion couplings), the interaction between axion strings and relativistic fermions in the thermal bath generates significant Aharonov-Bohm (AB) scattering. This creates a substantial frictional drag force.
The Consequence: Current numerical simulations cannot yet resolve the vast hierarchy of scales required to model this friction dynamically (from the string core to the Hubble radius). Consequently, the impact of this friction on the axion production rate for lower decay constants ( $f_a$ ) has been unquantified. The authors ask: How does plasma friction alter the axion string network dynamics and the resulting dark matter abundance in DFSZ-like models?

2. Methodology

Since full real-time lattice simulations including plasma drag are currently computationally infeasible, the authors develop a semi-analytical framework calibrated against existing frictionless numerical data.

Velocity-One-Scale (VOS) Model: They utilize the VOS model to evolve the string network. This model characterizes the network by a single correlation length ( $L$ $L$ ) and root-mean-square velocity ( $v$ $v$ ).
- They modify the standard VOS equations to include a friction length scale ( $\ell_f$ ) derived from microphysical scattering calculations (specifically AB scattering of SM fermions).
- The friction length scales as $\ell_f \propto f_a^2 T^{-3}$ , meaning friction is dominant at lower $f_a$ and higher temperatures.
Calibration: The VOS solution is rescaled to match the results of large-volume lattice simulations in the frictionless scaling regime. This ensures the normalization of the string energy density ( $\xi$ ) is consistent with established numerical results when friction is negligible.
Dual Estimators for Axion Production: To convert the string network history into axion number density, they employ two complementary methods:
1. Loop-based Estimator: Assumes a log-uniform distribution of string loops. It introduces a friction-limited critical loop size ( $\ell_{crit}$ ). Loops larger than the friction length $\ell_f$ are overdamped and cannot radiate efficiently.
2. IR-dominated Spectral Estimator: Adapts the emission spectrum prescription from Ref. [14]. It imposes a friction-induced infrared (IR) cutoff ( $k_{min}$ ) on the emission spectrum, preventing modes with wavelengths larger than $\ell_f$ from contributing to the radiation.
Nonlinear Dilution: They incorporate the nonlinear evolution of the axion field after the onset of oscillations ( $t_{osc}$ ). They track the IR gradient energy until it becomes comparable to the axion potential energy ( $t_\ell$ ), accounting for the "dilution" of the axion abundance due to the field remaining relativistic for longer in the presence of friction.

3. Key Contributions

Identification of Friction Regimes: The paper rigorously defines three dynamical regimes for axion strings:
1. Stretching Regime: Strings are frozen in the plasma and stretched by Hubble expansion.
2. Kibble Regime (Friction-Dominated): Friction dominates curvature forces, suppressing small-scale structure and loop production. The network is "stiff."
3. Scaling Regime: Friction becomes negligible, and the network approaches the standard attractor solution.
Quantification of Friction in DFSZ Models: They demonstrate that for $f_a \lesssim 10^9$ GeV, the friction length $\ell_f$ exceeds the correlation length $L$ during the QCD epoch, keeping the network in the Kibble regime much longer than previously thought.
New Production Mechanism: They show that friction delays the onset of efficient axion emission, shifting the production closer to the onset of axion oscillations ( $t_{osc}$ ). This alters the scaling of the relic abundance with $f_a$ .

4. Key Results

The authors calculate the relic abundance ratio $\Omega_r$ (predicted abundance vs. required dark matter abundance) as a function of the axion decay constant $f_a$ .

High $f_a$ ( $f_a \gtrsim 10^9$ GeV): Friction is negligible. The network reaches scaling early. The results match standard predictions (scaling solution), yielding the standard axion mass window ( $m_a \sim \mu$ eV).
Intermediate $f_a$ ( $6 \times 10^7 \text{ GeV} \lesssim f_a \lesssim 10^9 \text{ GeV}$ ): The network spends a significant time in the Kibble regime. The correlation length $L$ is larger than in the scaling regime, and emission is morphology-limited. This leads to an enhancement of the axion yield compared to the frictionless case.
Low $f_a$ ( $f_a \lesssim 6 \times 10^7$ GeV): The network is deep in the friction-dominated regime at $t_{osc}$ $t_{osc}$ .
- The critical radiating size is limited by $\ell_f$ (which shrinks as $f_a$ decreases).
- While the string density increases, the suppression of radiating modes due to strong damping dominates, causing the abundance to drop slightly as $f_a$ decreases further.
The "Second Crossing": Crucially, the inclusion of friction creates a second viable solution for the correct dark matter abundance.
- Standard Solution: $f_a \sim 10^{10} - 10^{11}$ GeV ( $m_a \sim \mu$ eV).
- New Solution: $f_a \sim 10^8$ GeV ( $m_a \sim 0.1$ eV).
- In the friction-dominated regime, the predicted abundance is enhanced by a factor of $\sim 10^1 - 10^4$ compared to frictionless calculations, allowing the lower $f_a$ value to reproduce the observed dark matter density.

5. Significance

Experimental Implications: The discovery of a viable parameter space at $m_a \sim 0.1$ $m_{a} \sim 0.1$ eV is highly significant. This mass range is:
- Easier to detect: It is much more accessible to current and near-future axion haloscopes and helioscopes than the traditional $\mu$ eV window.
- Tension with Constraints: This region is currently in slight tension with astrophysical constraints (e.g., stellar cooling), making it a prime target for either discovery or robust exclusion.
Theoretical Robustness: The paper highlights that the "standard" prediction of the axion mass is model-dependent. Models with tree-level fermion couplings (DFSZ) fundamentally change the cosmological history of the axion string network.
Call for Simulation: The authors emphasize that their semi-analytical results are a strong motivation for dedicated numerical simulations that include plasma drag. Such simulations are necessary to verify the existence of the Kibble regime and refine the precise value of the friction coefficient.

In summary, the paper argues that friction is not a subdominant correction in DFSZ-like models but a dominant effect that reshapes the axion string network, potentially shifting the predicted QCD axion dark matter mass from the micro-eV scale to the 0.1 eV scale.