⚛️ phenomenology

Imperfect axions with no domain wall problem

This paper demonstrates that post-inflationary axion theories with a domain wall number greater than one can be phenomenologically viable if additional PQ-violating operators destroy axion strings before the QCD potential becomes relevant, thereby resolving the domain wall problem while predicting a QCD axion mass in the $10^{-3}$ – $10^{-2}$ eV range and a neutron electric dipole moment within reach of upcoming experiments.

Original authors: Marco Gorghetto, Edward Hardy, Gilad Perez, Stefan Stelzl

Published 2026-08-31

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CC BY 4.0

Original authors: Marco Gorghetto, Edward Hardy, Gilad Perez, Stefan Stelzl

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 is filled with invisible matter that holds galaxies together, yet we do not know what it is. One leading candidate for this dark matter is a hypothetical particle called the axion. This particle was originally proposed to solve a deep puzzle in the laws of physics regarding why the strong nuclear force does not violate a fundamental symmetry between matter and antimatter. If axions exist, they would be produced in the early universe and, if they are light enough, could account for the missing mass we observe today. However, for axions to be a viable explanation for dark matter, they must also solve a secondary problem: the formation of vast, stable structures known as domain walls. In many theoretical models, these walls would trap so much energy that they would overwhelm the universe, preventing it from evolving as we see it. For decades, this "domain wall problem" has been a major hurdle for theories involving multiple types of axion configurations.

A new study by Marco Gorghetto, Edward Hardy, Gilad Perez, and Stefan Stelzl offers a fresh perspective on this issue, suggesting that the universe might have avoided this catastrophe in a way previously overlooked. The researchers focused on a specific scenario where the axion symmetry breaks after the rapid expansion of the universe known as inflation. In this timeline, the breaking of symmetry creates a network of cosmic strings, which are thin, energetic defects in space. As the universe cools, these strings usually become attached to domain walls. If there is more than one possible stable state for the axion field, these walls form a stable, long-lived network that would overclose the universe. The standard solution to this problem involves introducing a small force that breaks the symmetry further, causing the walls to collapse. However, this force must be incredibly weak to avoid creating a measurable electric dipole moment in the neutron, a property that experiments have not yet detected. This extreme weakness usually means the walls collapse too late, leaving behind too much dark matter.

The authors of this paper demonstrate that there is a region of parameter space where the standard timeline does not apply. They show that if the axion decay constant—a measure of the energy scale at which the axion appears—is sufficiently small, specifically below ten to the power of ten gigaelectronvolts, the symmetry-breaking force becomes dominant much earlier than expected. In this regime, the force that breaks the symmetry acts on the cosmic strings before the QCD potential, which generates the mass of the axion, even has a chance to form the stable domain walls. Consequently, the cosmic strings are destroyed by this early force before they can ever become anchored to the problematic domain walls. The researchers confirmed this behavior through detailed computer simulations of the string network, observing that the network collapses rapidly once this early force takes effect, leaving no stable walls behind.

This finding opens up a viable path for axion dark matter that was previously thought to be excluded. The simulations indicate that for axion masses in the range of one thousandth to one hundredth of an electronvolt, the universe can naturally avoid the domain wall problem without requiring fine-tuned adjustments. The required symmetry-breaking force is strong enough to destroy the strings early but weak enough to remain consistent with current experimental limits on the neutron's electric dipole moment. The authors note that upcoming experiments designed to measure the neutron's electric dipole moment with higher precision will be able to probe this specific range of parameters. If these experiments detect a signal within the predicted range, it would provide strong evidence for this early-destruction scenario.

The study also clarifies what happens when the symmetry-breaking force is too weak to act early. In those cases, the domain walls do form and persist for a long time, eventually decaying and releasing a flood of axions that would overproduce the dark matter we observe. This confirms that the early-destruction mechanism is essential for the theory to work in the specific mass range identified. The researchers acknowledge that their simulations had to simplify certain physical scales due to computational limits, but they argue that the qualitative result—that early destruction is possible and effective—remains robust. By identifying this window where the axion mass is relatively heavy and the symmetry breaking acts early, the paper resolves a long-standing tension in axion cosmology and points toward a testable prediction for future experiments.

Technical Summary: Imperfect axions with no domain wall problem

Problem Statement
Post-inflationary axion cosmology with a domain wall number $N > 1$ faces the "domain wall (DW) problem." In standard scenarios, the spontaneous breaking of the Peccei–Quinn (PQ) symmetry after inflation generates a network of axion strings. As the QCD potential becomes relevant at a temperature $T_\star$ , these strings attach to $N$ degenerate domain walls. For $N > 1$ , this network is stable and would overclose the universe. To resolve this, an additional PQ-violating operator is typically introduced to lift the degeneracy, creating a bias that causes the walls to decay.

However, the conventional picture posits that this PQ-breaking potential only becomes cosmologically relevant after the QCD potential has already formed the stable $N > 1$ network (at $H \approx m_a(T_\star)$ ). In this regime, the bias term is a small perturbation. Consequently, the network survives until the Hubble parameter drops significantly, releasing a large energy density of axions that typically overproduces dark matter, unless the PQ-breaking term is tuned to be extremely small. Such small breaking is constrained by the neutron electric dipole moment (nEDM), which requires the effective $\theta$ -angle to be $\lesssim 10^{-10}$ .

Methodology
The authors investigate a regime where the PQ-breaking potential dominates over the QCD potential before the QCD domain walls can form. They analyze the Lagrangian of a complex scalar field $\phi$ with a PQ-breaking operator of dimension $n+1$ , coupled to $N$ fermions. The analysis combines:

Analytical Estimation: Comparing the temperature-dependent QCD axion mass $m_a(T)$ with the temperature-independent PQ-breaking mass $m_{PQ}$ . The QCD mass scales as $m_a(T) \propto T^{-\alpha}$ (with $\alpha \approx 8$ ) at high temperatures, while $m_{PQ}$ remains constant.
Numerical Simulations: The authors simulate the evolution of the axion string-wall system using a "fat string" approximation on a lattice. They solve the equations of motion for the radial mode and the axion field, including both the QCD potential and the PQ-breaking potential. They track the number of strings per Hubble patch ( $\xi$ ) to determine when the network decays.
Extrapolation: Due to computational limitations, simulations are restricted to a logarithmic hierarchy $\log(m_r/m_{PQ}) \lesssim 6$ , whereas the physical regime is $\sim 60-70$ . The authors extrapolate simulation results to the physical point, accounting for the logarithmic enhancement of string tension which delays the decay of the network.

Key Contributions and Results

Early Destruction Regime: The paper identifies a viable parameter space where the PQ-breaking term becomes cosmologically relevant at a Hubble scale $H = m_{PQ}$ that is larger than the scale $H_\star = m_a(T_\star)$ where the QCD potential becomes relevant. This occurs for axion decay constants $f_a \lesssim 10^{10}$ GeV and PQ-breaking parameters near the maximal limit allowed by nEDM constraints ( $\theta_{\text{eff}} \sim 10^{-10}$ ).
Mechanism of Resolution: In this regime, the PQ-breaking potential forms an unstable domain wall network attached to the strings before the QCD potential can generate the stable $N > 1$ network. This early network collapses rapidly, destroying the string network before the QCD domain walls can form. Thus, the $N > 1$ DW problem is avoided without requiring the PQ-breaking term to be vanishingly small.
Dark Matter Abundance: The axions produced during the scaling regime (before the early collapse) contribute to the dark matter relic density. The authors derive that for this scenario to avoid overproduction, $f_a$ must satisfy $f_a \lesssim 5 \times 10^9$ GeV for $N=2$ and $f_a \lesssim 2 \times 10^9$ GeV for $N=6$ . This corresponds to an axion mass in the range $m_a \sim 10^{-3} - 10^{-2}$ eV.
Simulation Constraints: Simulations for $N=2$ indicate that the network decays if the ratio $m_{PQ}/m_a(T_\star)$ exceeds a critical value $c_{PQ}$ . While simulations suggest $c_{PQ} \approx 0.32$ , the authors conservatively estimate that logarithmic corrections at the physical scale could raise this to $c_{PQ} \approx 1.8$ .
nEDM Implications: The required PQ-breaking operators are large enough to induce a neutron electric dipole moment (nEDM) within two orders of magnitude of the current experimental limit. This places the scenario within the reach of upcoming nEDM experiments.

Significance and Claims
The authors claim to have identified a previously overlooked regime in post-inflationary axion cosmology that resolves the domain wall problem for $N > 1$ models (such as DFSZ) without fine-tuning the PQ-breaking term to be negligible.

Viability: The scenario is consistent with observed dark matter abundance and astrophysical bounds (including supernova constraints), provided $f_a \lesssim 10^{10}$ GeV.
Testability: Unlike scenarios requiring tiny PQ-breaking terms, this regime predicts a PQ-breaking scale large enough to be probed by next-generation nEDM experiments.
Limitations: The authors explicitly note that their conclusions rely on extrapolating simulation results from a modest hierarchy ( $\log_{PQ} \sim 4.5$ ) to the physical hierarchy ( $\log_{PQ} \sim 60$ ). They also acknowledge uncertainties regarding the potential obstruction of domain wall formation by the energy density of relativistic axions emitted during the scaling regime (the "axion-wave obstruction"), which could shrink the viable parameter space.

In summary, the paper argues that "imperfect" axions with significant PQ violation can naturally solve the domain wall problem for $N > 1$ if the axion decay constant is sufficiently low, leading to a testable prediction for the axion mass and nEDM.

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