Removing the Cosmological Bound on the Axion Scale via Confinement During Inflation

This paper proposes that implementing early axion relaxation through high-scale confinement within an $SU(5)$ grand unified theory during inflation generates an early potential that dilutes the axion's energy density, thereby removing the cosmological upper bound on the axion decay constant and allowing it to serve as a viable dark matter candidate for arbitrarily large values across all known axion models.

The Gist

The Big Problem: The "Goldilocks" Axion

Imagine the universe is trying to solve a mystery called the Strong CP Problem. In the laws of physics, there is a hidden "knob" (called the $\theta$-angle) that controls how matter behaves. If this knob is turned even a tiny bit, the universe would be very different (protons would decay, atoms wouldn't hold together). But experiments show this knob is set to zero.

Physicists invented a particle called the Axion to explain why this knob is zero. The Axion is like a magical spring that automatically pushes the knob back to zero whenever it tries to move.

The Catch:
For the Axion to work as a dark matter candidate (the invisible stuff holding galaxies together), it has to be very light and very rare. This requires a specific setting on the Axion's "decay constant" (let's call it the Scale, $f_a$).

• The Old Rule: For decades, scientists thought the Scale had to be small (around $10^{12}$ GeV). If it were any bigger, there would be too much Axion dark matter, and the universe would have collapsed long ago.
• The Limitation: This is like saying a car engine can only be built if the pistons are exactly 2 inches wide. If you make them 3 inches, the engine explodes.

The New Idea: The "Early Reset" Button

This paper proposes a radical new idea: The Axion Scale can be huge (arbitrarily large), and the universe is still safe.

How? By realizing that the Axion didn't just wake up when the universe cooled down. It had a "childhood" during Inflation (the period of rapid expansion right after the Big Bang) where it was very active and got its act together early.

The Analogy: The Tangled Yarn

Imagine the Axion is a ball of yarn that got tangled.

1. Old View: The yarn was left in a dark room (the early universe) for billions of years. When the lights finally turned on (the universe cooled), the yarn was a massive, chaotic knot. If the knot was too big, it would crush the room.
2. New View: During the "dark room" phase (Inflation), the yarn was actually being actively untangled by a giant machine. By the time the lights turned on, the yarn was already neat and small. Even if the original ball of yarn was huge, the result is small because it was cleaned up early.

How Does the "Cleaning" Happen?

The paper suggests two ways the universe cleaned up the Axion during Inflation:

1. The "Direct Link" Mechanism
Imagine the Inflaton (the field driving the universe's expansion) is a master controller. Usually, it just pushes the universe to expand. But in this scenario, the Inflaton is also directly connected to the "strength" of the forces inside the universe (the Strong Force).

• The Metaphor: Think of the Strong Force as a rubber band. Usually, it's loose. But because the Inflaton is stretching the universe, it also pulls the rubber band tight. When the rubber band is super tight, the Axion gets a heavy "mass" (it becomes heavy and sluggish).
• The Result: Because it's heavy, the Axion stops wobbling around and settles down into its "zero" position very quickly. It relaxes.

2. The "Symmetry Restoration" Mechanism
In our current universe, the Grand Unified Theory (GUT) is broken (like a shattered vase). But during Inflation, the energy was so high that the vase was whole again.

• The Metaphor: Imagine a complex lock (the Strong Force) that is usually jammed because the keyhole is blocked. During Inflation, the blockage is removed, and the lock becomes a giant, powerful engine.
• The Result: This powerful engine forces the Axion to settle down immediately. It's like a heavy door slamming shut before the wind can blow it open.

The Twist: The Axion Changes Identity

One of the most fascinating parts of the paper is that the Axion isn't the same "thing" in the early universe as it is today.

• Today: The Axion is like a phase of a specific particle field (a scalar field).
• Then (During Inflation): The scalar field might have vanished! But the Axion didn't disappear. Instead, it "borrowed" its identity from a condensate of fermions (a soup of quarks).
• The Metaphor: Imagine a chameleon. Today, it looks like a green leaf. But during the early universe, the leaf died, so the chameleon turned into a rock. It's still the same animal (the Axion), just wearing a different costume.
• Why it matters: Even though the "costume" changed, the chameleon still knew where to sit (the zero position). It relaxed its position while wearing the "rock" costume, so when it switched back to the "leaf" costume later, it was already in the right spot.

The Conclusion: No More Limits

Because the Axion got a "head start" and settled down during Inflation:

1. It didn't accumulate enough energy to destroy the universe.
2. The "Goldilocks" limit is gone. The Axion Scale ($f_a$) can be as large as you want.
3. This opens the door for the Axion to be a much heavier, more elusive particle, which might actually be easier to find in experiments than the tiny, light versions we were looking for before.

Summary for the Everyday Reader

Scientists used to think the Axion particle had to be very light and specific to fit our universe. This paper argues that the universe had a "cleaning crew" during its infancy (Inflation) that organized the Axion early on. Because of this early cleanup, the Axion can be much "heavier" and more massive than we thought, and it still fits perfectly into our cosmic story. It's like realizing a messy room wasn't a disaster because someone had already tidied it up before you walked in.

Technical Summary

1. The Problem

The paper addresses the Strong CP problem in Quantum Chromodynamics (QCD) and the associated cosmological constraints on the axion, a hypothetical particle proposed to solve this problem.

• The Strong CP Problem: The QCD Lagrangian allows for a CP-violating term $\theta \text{tr}(G\tilde{G})$. Experimental limits on the neutron electric dipole moment require the effective parameter $\bar{\theta} \lesssim 10^{-9}$. The axion dynamically relaxes $\bar{\theta}$ to zero.
• The Cosmological Bound: In standard hot Big Bang cosmology, the axion field is initially misaligned from its potential minimum. As the universe cools and the axion acquires mass (at the QCD phase transition, $T \sim 100$ MeV), it begins coherent oscillations. The energy density of these oscillations scales as matter. To avoid overclosing the universe, the axion decay constant $f_a$ is constrained to be $f_a \lesssim 10^{12}$ GeV (assuming a maximal initial misalignment angle).
• The Limitation of Standard Models: This bound relies on the assumption that the axion field had "no prior knowledge" of its potential minimum before the QCD phase transition. The authors argue this assumption is invalid in inflationary cosmology, where field values and couplings can vary significantly during the inflationary epoch.

2. Methodology

The authors propose a mechanism for early relaxation of the axion field during inflation, implemented within a Grand Unified Theory (GUT) framework based on $SU(5)$.

• Inflationary Dynamics: They model the inflaton field $S$ as a master field that controls the masses and vacuum expectation values (VEVs) of other scalar fields (PQ field $\Phi$, GUT Higgs fields $\Sigma, H$).
• Mechanism for Strong Coupling: They identify two distinct mechanisms by which the QCD gauge coupling (or the unified $SU(5)$ coupling) can become strong during inflation, generating an early axion mass:
  1. Direct Coupling: The inflaton $S$ couples directly to gauge fields via higher-dimensional operators (e.g., $S/M \cdot \text{tr} G^2$), modifying the effective gauge coupling.
  2. Symmetry Restoration: During inflation, large positive mass terms induced by $S$ cause the VEVs of GUT Higgs fields to vanish. This restores the $SU(5)$ symmetry. In this unbroken phase, the renormalization group (RG) running of the $SU(5)$ gauge coupling leads to a strong coupling scale $\Lambda_C$ that is significantly higher than the standard QCD scale (potentially $\sim 10^7$ GeV).
• Axion Definition in Early Epochs: A crucial theoretical step is defining the axion when the PQ scalar VEV $\langle \Phi \rangle$ vanishes. The authors argue that even if $\langle \Phi \rangle = 0$, the $U(1)_{PQ}$ symmetry is spontaneously broken by the condensate of the fermionic 't Hooft determinant. The axion is then defined as the phase of this condensate (analogous to the $\eta'$ meson in QCD).
• Models Analyzed: The scenario is applied to three realizations of the invisible axion:
  • KSVZ: Heavy vector-like fermions carry the PQ charge.
  • DFSZ: Ordinary quarks/leptons carry the PQ charge (requiring two Higgs doublets).
  • Gauge Axion: The axion is a 2-form gauge field dual to the QCD gauge redundancy, without a global PQ symmetry.

3. Key Contributions

• Time-Dependent Axion Composition: The paper demonstrates that the "identity" of the axion is not static. In the early universe (during inflation), if the PQ scalar VEV vanishes, the axion is predominantly composed of the phase of the fermion condensate ($\eta_F$), whereas today it is dominated by the PQ scalar phase ($\eta_\Phi$).
• Early Relaxation Mechanism: They show that if the axion mass $m_a$ exceeds the Hubble parameter $H$ during inflation (due to the high $\Lambda_C$), the axion field undergoes damped coherent oscillations. This rapidly relaxes the misalignment angle $\theta_{eff}$ toward zero before the end of inflation.
• Dilution of Energy Density: The amplitude of the axion field is reduced by a factor of $e^{-\frac{3}{2}N_e}$, where $N_e$ is the number of e-folds during the strong-coupling phase. Even a modest $N_e \sim 10$ is sufficient to dilute the axion energy density enough to allow for arbitrarily large $f_a$.
• Topological Defect Resolution: The authors analyze the fate of cosmic strings, domain walls, and magnetic monopoles. They argue that if the strong coupling phase occurs during inflation, any topological defects formed are inflated away, leaving a homogeneous axion field. They also discuss mechanisms (like unstable domain walls) to erase monopoles if they form post-inflation.

4. Results

• Lifting the Bound: The primary result is that the cosmological upper bound on the axion decay constant ($f_a \lesssim 10^{12}$ GeV) is removed. The axion can be a viable dark matter candidate for $f_a$ values up to the GUT scale or even the Planck scale, provided the early relaxation mechanism operates.
• Robustness Across Models: The mechanism works for KSVZ, DFSZ, and Gauge Axion formulations.
  • In KSVZ/DFSZ, even if the PQ scalar $\Phi$ has a vanishing VEV during inflation, the axion remains a well-defined degree of freedom via the 't Hooft determinant condensate.
  • In the Gauge Axion scenario, the axion is a fundamental degree of freedom at all times, and the mechanism relies solely on the generation of a large mass via strong coupling.
• Confinement Scale Estimates: Using RG running in the restored $SU(5)$ phase, the authors estimate the confinement scale $\Lambda_C$ to be $\sim 10^7$ GeV (for KSVZ) and $\sim 5 \times 10^7$ GeV (for DFSZ/Gauge Axion), which is sufficiently high to ensure $m_a > H$ during inflation.
• No New Constraints: The analysis of topological defects suggests that the scenario does not introduce new cosmological problems (e.g., monopole overproduction or domain wall stability issues) that would invalidate the model.

5. Significance

• Theoretical Flexibility: This work significantly broadens the viable parameter space for axion dark matter. It allows for "super-heavy" axions ($f_a \gg 10^{12}$ GeV) which are often theoretically preferred in string theory and GUTs but were previously ruled out by cosmology.
• Deepening Axion Physics: It highlights that the axion is a composite object whose constituents (scalar phases vs. fermion condensate phases) evolve with the cosmological epoch. This challenges the static view of the axion field in standard cosmology.
• Connection to Inflation: It provides a concrete link between the dynamics of inflation (field displacements, symmetry restoration) and the solution to the Strong CP problem, suggesting that the early universe's high-energy physics naturally resolves the axion overproduction problem.
• Observational Implications: While the mechanism removes the bound on $f_a$, it opens the door for new observational signatures, such as specific gravitational wave signals from string-wall systems formed during the inflationary strong-coupling phase, though these depend on specific model parameters.

In summary, Dvali, Fitz, and Komisel provide a robust theoretical framework demonstrating that the axion's early relaxation during inflation, driven by a high-scale confinement phase in a GUT context, eliminates the standard cosmological upper bound on the axion scale, making the axion a viable dark matter candidate across a much wider range of energy scales.