On the predictivity of axion dark matter in the presence of Peccei-Quinn breaking

This paper demonstrates that small explicit Peccei-Quinn symmetry breaking can render post-inflationary QCD axion dark matter unpredictable by introducing a new mass scale that alters the axion string-wall network dynamics and relic abundance before the QCD transition.

The Gist

The Big Picture: The "Perfect" Particle vs. Reality

Imagine the universe is a giant, complex machine. For a long time, physicists have been trying to fix a specific glitch in this machine called the Strong CP Problem. It's like a car engine that should run perfectly smooth, but for some unexplained reason, it has a tiny, annoying rattle that shouldn't be there.

To fix this, they invented a theoretical particle called the Axion. Think of the Axion as a "smart thermostat" for the universe. Its job is to sense that rattle and automatically adjust the engine until it runs perfectly smooth.

Because this Axion is so light and interacts so weakly, it also happens to be a perfect candidate for Dark Matter—the invisible stuff that holds galaxies together.

The Old Story (The "Predictive" Era):
For decades, physicists believed that if we found the Axion, we could predict exactly how much Dark Matter exists in the universe. It was a simple one-to-one map:

• Input: How heavy is the Axion?
• Output: How much Dark Matter do we have?
• Result: If you know one, you know the other. This made the Axion a very "predictive" candidate for experiments.

The New Twist (This Paper):
Author Michael Zantedeschi argues that this simple map might be wrong. He suggests that the "smart thermostat" (the Axion) might have a tiny, hidden flaw in its wiring. Even if this flaw is too small to break the car engine (satisfying current experiments), it could completely change how the Axion behaves in the early universe.

If this flaw exists, the simple map breaks. Knowing the Axion's weight no longer tells us how much Dark Matter there is.

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The Analogy: The String and the Wall

To understand why this happens, let's use a visual analogy involving a tangled ball of yarn and a sticky wall.

1. The Setup: The Tangled Yarn (Cosmic Strings)

In the early universe, when the Axion "turned on," it didn't turn on everywhere at the same time. Imagine a giant room where people are all trying to tie their shoes. Some tie them tight, some loose, some left, some right.
Because they are in different corners of the room, they can't agree. This disagreement creates a tangled mess of Cosmic Strings (like a giant, chaotic ball of yarn) floating in space.

2. The Old Scenario: The QCD "Freeze"

In the standard story, the universe cools down. When it gets cold enough (the "QCD transition"), a giant sticky wall (a domain wall) suddenly appears.

• This wall grabs the tangled yarn.
• The wall pulls the yarn tight, and the whole system collapses into a pile of Axions (Dark Matter).
• The Key: The strength of this sticky wall is determined entirely by the laws of physics we know today (QCD). So, the amount of Dark Matter produced depends only on the Axion's properties.

3. The New Scenario: The "Hidden Glitch" (Explicit Breaking)

Zantedeschi says: "What if the Axion has a tiny, permanent bias?"
Imagine that the Axion isn't a perfect thermostat, but one that slightly prefers to be turned to the "Left."

• This creates a pressure difference. It's like having a gentle, constant wind blowing on the tangled yarn.
• In the early, hot universe, the "sticky wall" (QCD) is weak and hasn't formed yet. But this "wind" (the glitch) is always there.
• The Result: The wind blows the tangled yarn apart and collapses the system long before the sticky wall ever forms.

Why Does This Matter?

In the old story, the collapse happened because of the sticky wall (QCD physics). The outcome was predictable.

In the new story, the collapse happens because of the wind (the glitch).

• The amount of Dark Matter now depends on how strong the wind is (the size of the glitch), not just the Axion's weight.
• Since we don't know exactly how strong the "wind" is (it depends on unknown physics from the very beginning of the universe), we can no longer predict the amount of Dark Matter just by measuring the Axion.

The "Quality" Problem

The paper highlights a philosophical shift:

• Old View: The Axion is a low-energy miracle. Its behavior is determined by the "infrared" (everyday) physics of the universe.
• New View: The Axion is a probe of "ultraviolet" (high-energy, early universe) physics. Even a tiny, invisible flaw in the Axion's symmetry (which we can't see in a lab today) can rewrite the history of the universe.

The Takeaway for Everyday Life

Think of it like baking a cake.

• The Old Recipe: "If you use 200g of flour, you get a perfect cake." (Predictable).
• The New Reality: "If you use 200g of flour, you usually get a cake, BUT if the oven had a tiny, invisible temperature glitch that happened 10 minutes before you turned it on, the cake might be a brick or a puddle."

Conclusion:
This paper warns physicists that the Axion might not be the "perfect" predictor they hoped for. To find the true amount of Dark Matter, we might need to understand not just the Axion itself, but the tiny, hidden "glitches" in the laws of physics that existed before the universe cooled down. It turns the search for Dark Matter from a simple measurement into a detective story about the very first moments of the Big Bang.

Technical Summary

1. Problem Statement

The standard cosmological prediction for QCD axion dark matter relies heavily on the post-inflationary scenario. In this framework, the Peccei–Quinn (PQ) symmetry breaks after inflation, leading to the formation of a cosmic string network. As the universe cools through the QCD phase transition, the axion potential turns on, domain walls form, and the string-wall network annihilates, producing cold axions.

Conventionally, the relic abundance of axions ($\Omega_a$) in this scenario is considered a predictive function of a single parameter: the axion decay constant, $f_a$. This implies a one-to-one mapping between the axion mass ($m_a$) and the dark matter density.

However, this predictivity assumes that the PQ symmetry is exact except for the QCD anomaly. The paper addresses the problem of explicit PQ symmetry breaking (violations beyond the QCD anomaly). While such breaking is tightly constrained by the Strong CP problem (specifically the neutron electric dipole moment), the author investigates whether small, phenomenologically viable explicit breaking can alter the early-time cosmological evolution of the axion field, thereby destroying the standard $f_a$-only predictivity.

2. Methodology

The author employs a combination of Effective Field Theory (EFT) and cosmological scaling arguments:

• EFT Formulation: The paper utilizes the three-form formalism (following Dvali) to describe the axion and the topological susceptibility ($\chi$) of QCD. It introduces an explicit breaking term in the Lagrangian, parametrized by a mass scale $\mu$ ($\Delta \mathcal{L} = -\frac{1}{2}\mu^2 a^2$).
• Potential Analysis: The author analyzes the effective potential $V(a)$, which includes both the QCD-induced term (temperature-dependent) and the explicit breaking term (assumed temperature-independent).
• Cosmological Dynamics: The study focuses on the post-inflationary regime. It calculates the competition between the restoring force of the string tension ($\mu_{str}$) and the pressure difference (bias) induced by the explicit breaking across domain walls ($\sigma_{exp}$).
• Annihilation Condition: The paper derives the condition under which the string-wall network annihilates before the QCD transition. This occurs when the explicit breaking force dominates the QCD-induced force.
• Relic Abundance Estimation: Using scaling solutions for string networks, the author estimates the axion number density produced at the time of network annihilation ($t_{ann}$) as a function of the new parameters.

3. Key Contributions

• Loss of Predictivity: The primary contribution is the demonstration that the standard one-to-one mapping between $f_a$ and dark matter abundance is not robust. If explicit PQ breaking becomes dynamically relevant before the QCD transition, the relic abundance depends on the breaking scale $\mu$ and the initial CP-violating angle $\bar{\vartheta}_v$, in addition to $f_a$.
• Early Annihilation Mechanism: The paper identifies a regime where the explicit breaking creates an energy bias ($\Delta V \sim \mu^2 f_a^2$) that drives the collapse of the string-wall network at a temperature $T > T_{QCD}$. This happens because the QCD contribution to the axion mass is thermally suppressed at high temperatures, while the explicit breaking term remains active.
• Parametric Scaling Laws: The author derives new scaling relations for the relic abundance in this regime:
  $$\Omega_a \propto f_a \mu^{-1/2}$$
  This contrasts with the standard QCD-only prediction where $\Omega_a$ is determined solely by $f_a$.
• Constraints on Parameter Space: The paper quantifies the region where this effect is relevant. It shows that for axion masses $m_a \gtrsim 10^{-3} |\bar{\vartheta}_v|^{1/2}$ eV (corresponding to $f_a \lesssim 10^{10} |\bar{\vartheta}_v|^{-1/2}$ GeV), explicit breaking can dominate the dynamics, provided the initial CP-violating angle $\bar{\vartheta}_v$ is not vanishingly small.

4. Results

• Condition for Dominance: The explicit breaking dominates the cosmological evolution if the Hubble parameter at annihilation $H_{ann}$ satisfies:
  $$H_{ann} \simeq \frac{\mu}{\pi \log(m_\rho/H_{ann})}$$
  This occurs before the QCD transition if $\mu \gtrsim \kappa H_{QCD}$, where $\kappa \sim \mathcal{O}(100)$.
• Strong CP Constraint: The neutron electric dipole moment bound ($|d_n| < 2.9 \times 10^{-26}$ e cm) imposes a constraint on the ratio of the explicit breaking scale to the QCD mass scale:
  $$\frac{\mu^2}{m_a^2} = \frac{\bar{\vartheta}}{\bar{\vartheta}_v} \lesssim 10^{-10}$$
  Even with this tight constraint, if the initial angle $\bar{\vartheta}_v$ is $\mathcal{O}(1)$, the breaking scale $\mu$ can still be large enough to trigger early annihilation for $f_a \lesssim 10^{10}$ GeV.
• Impact on Predictivity:
  • If $\bar{\vartheta}_v \sim \mathcal{O}(1)$, the predictivity is lost for $f_a \lesssim 10^{10}$ GeV.
  • If $\bar{\vartheta}_v \ll 1$ (e.g., $< 10^{-4}$), the constraint on $\mu$ relaxes, and the region of lost predictivity expands to cover the entire phenomenologically relevant parameter space for QCD axion dark matter.
• Experimental Implications: This effect is relevant for the mass range currently targeted by haloscope experiments (micro-eV and above). The standard interpretation of null results or signals in these experiments must now account for the possibility that the axion abundance is not solely determined by infrared QCD dynamics.

5. Significance

• UV Sensitivity of Dark Matter: The paper fundamentally shifts the perspective on axion dark matter. It argues that the relic abundance is not purely an infrared (IR) prediction of QCD but is sensitive to ultraviolet (UV) physics (the quality of the PQ symmetry).
• Re-evaluation of Experimental Limits: Current experimental bounds and future discovery potential for axion dark matter rely on the assumption that $f_a$ uniquely determines the abundance. This work suggests that without knowledge of the UV breaking scale $\mu$ and the initial conditions ($\bar{\vartheta}_v$), the mapping between experimental sensitivity and the existence of axion dark matter is ambiguous.
• Quality Problem Revisited: The results reinforce the "axion quality problem." Even breaking terms that are small enough to satisfy the Strong CP bound can have macroscopic, cosmological consequences, rendering the standard post-inflationary axion scenario non-predictive unless the PQ symmetry is preserved to an extremely high degree over cosmological timescales.

In conclusion, Zantedeschi demonstrates that the "predictive power" of the QCD axion as a dark matter candidate is fragile. The presence of even minute explicit PQ breaking, consistent with all current laboratory constraints, can decouple the axion mass from the dark matter abundance, requiring a more complex theoretical framework to interpret cosmological and experimental data.