⚛️ phenomenology

Spontaneous baryogenesis with large misalignment

This paper investigates particle production by a pseudo-Nambu-Goldstone boson in the spontaneous baryogenesis scenario with large misalignment angles, demonstrating through numerical studies in both Minkowski and FLRW spacetimes that the baryon asymmetry's cubic dependence on initial phases breaks down for large oscillations, leading to saturation near $\pi$ and a strong sensitivity to the field's damping rate.

Original authors: Maxim Krasnov, Ufuk Aydemir, Maxim Khlopov

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Maxim Krasnov, Ufuk Aydemir, Maxim Khlopov

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

Imagine the universe right after the Big Bang as a giant, invisible playground. In this playground, there's a special kind of invisible ball called a pseudo-Nambu-Goldstone boson (let's call it a "pNGB" for short). Think of this ball as a marble sitting on a very specific, slightly tilted hill.

For decades, scientists have been trying to figure out why our universe is made mostly of matter (like us) instead of an equal mix of matter and anti-matter (which would have canceled each other out). The leading theory, called spontaneous baryogenesis, suggests that this "marble" (the pNGB) rolled down its hill, and as it rolled, it pushed the universe to create more matter than anti-matter.

The Old Story vs. The New Twist

Previously, scientists mostly studied this marble rolling from a spot very close to the bottom of the hill. They assumed the marble started with a tiny "misalignment"—a small nudge. In this small-angle world, the math is simple: the amount of matter created depends on the cube of how far the marble started from the bottom. It's like a gentle slide.

But here's the twist: What if the marble started at the very top of the hill?

In the real universe, quantum fluctuations (tiny, random jitters) during the inflationary expansion of the universe could have placed this marble almost anywhere. There's a real chance it started near the very peak of the hill, at an angle of roughly $\pi$ (about 3.14 radians). This is the "large misalignment" scenario. The authors of this paper asked: Does the story change if the marble starts at the very top?

The Problem: A Sticky, Memory-Keeping Hill

The problem with starting at the top is that the physics gets messy. As the marble rolls, it interacts with a sea of invisible particles (fermions). These particles don't just react instantly; they have "memory." They remember where the marble was a split second ago. This makes the equations incredibly hard to solve because the marble's current motion depends on its entire history, not just where it is right now. It's like trying to push a heavy sled that remembers every bump it hit for the last hour.

The Solution: The "Slow-Motion" Shortcut

The authors realized that even though the physics is technically "non-local" (dependent on the past), the marble moves so slowly compared to the speed of these invisible particles that we can use a shortcut. They argued that the "memory" of the particles acts like a simple, local friction force, similar to air resistance on a bike.

They proved that the conditions for this shortcut are met because the universe's energy scales are set up in a specific hierarchy (the symmetry-breaking scale $f$ is much larger than the potential scale $\Lambda$ ). This allows them to replace the complex, history-heavy math with a simpler equation that still keeps the "steepness" of the hill (the non-linear potential) intact.

What They Found: The Hill is Tricky

Using this simplified equation, they ran massive computer simulations to see what happens when the marble starts near the top ( $\theta_{in} \approx \pi$ ).

The "Small Angle" Rule Breaks: When the marble starts close to the bottom, the amount of matter created grows with the cube of the starting angle. But when the marble starts near the top, this neat rule falls apart.
Saturation at the Top: As the starting angle gets closer and closer to $\pi$ (the very top), the production of matter doesn't keep growing forever. Instead, it saturates. It hits a ceiling. Imagine trying to push a car up a hill; no matter how hard you push from the very top, you don't get infinite speed; you just get stuck or move very slowly. The simulations show that particle production levels off as the initial phase approaches $\pi$ .
The Role of Friction (Damping): The results depend heavily on how "sticky" the hill is (represented by a damping parameter $\Gamma$ ).
- If the friction is low, the marble rolls down, but the process is slow.
- If the friction is high, the marble takes a long time to even start moving, but once it does, it behaves differently.
- In the expanding universe (which is more realistic than the flat "Minkowski" space they also tested), the expansion of space itself adds extra friction. The simulations show that for high friction, the amount of matter created can actually decrease if the starting angle is too large, or show a nearly exponential dependence as it approaches $\pi$ .

The Numbers and The Reality Check

The paper doesn't claim to have solved the mystery of the universe's matter with a single magic number. Instead, they provide a map of possibilities.

They calculated that for a specific set of parameters (like a coupling constant $g = 0.01$ , a scale $f = 10^{13}$ GeV, and a potential scale $\Lambda \approx 3.7 \times 10^9$ GeV), the model can reproduce the observed baryon-to-entropy ratio of $8.6 \times 10^{-11}$ .
They emphasize that the initial phase $\theta_{in}$ is not a fixed number but a random variable. Because the universe is so vast, there are likely regions where the phase is near $\pi$ .
They explicitly rule out the idea that the "small-angle approximation" is always safe. They show that for large angles, the behavior is fundamentally different and cannot be predicted by just scaling up the small-angle math.

The Big Picture

The authors conclude that if the universe started with these large misalignment angles, the resulting matter distribution wouldn't be smooth. It would be patchy. Some regions might have lots of matter, others might have very little, or even pockets of anti-matter.

This isn't just about numbers; it suggests that the early universe might have been a patchwork quilt of different baryon densities. This could explain why we see certain structures in the universe today and hints that the "marble" rolling down the hill might have left behind gravitational waves or other cosmic signatures we could one day detect.

In short: The paper suggests that the "large angle" scenario is not only possible but likely in some parts of the universe, and it changes the rules of the game from a simple cubic slide to a complex, saturated, and friction-dependent journey. The math is now ready to handle the messy, real-world scenario where the marble might start at the very top of the hill.

Technical Summary: Spontaneous Baryogenesis with Large Misalignment

Problem Statement
The paper investigates the mechanism of spontaneous baryogenesis driven by a pseudo-Nambu-Goldstone boson (pNGB) in the regime of large initial misalignment angles. While previous literature has largely relied on the small-angle approximation (linearizing the potential), the probability distribution of the phase field at the end of inflation, driven by quantum fluctuations, suggests that large initial angles are physically plausible. Specifically, regions where the initial phase $\theta_{in}$ is close to $\pi$ (the local maximum of the potential) are of particular interest. A direct treatment of this regime is complicated by the fact that the fermionic backreaction, which drives particle production, is intrinsically nonlocal, governed by a memory integral rather than a local differential equation.

Methodology
To address the nonlocal nature of the backreaction in the large-angle regime, the authors employ a local Markovian approximation. The core argument relies on the verification of adiabaticity conditions. The authors demonstrate that the model's built-in scale hierarchy ( $\Lambda \ll f$ , where $\Lambda$ is the symmetry-breaking scale and $f$ is the decay constant) ensures that the pNGB background varies slowly compared to the microscopic time scales of the fermionic response ( $\omega_{cut}^{-1} \sim f^{-1}$ ).

Consequently, the nonlocal memory integral can be approximated by a local dissipative term, $\Gamma_{eff} \dot{\theta}$ , while retaining the full nonlinear dependence of the potential, $V(\theta) = \Lambda^4(1 - \cos \theta)$ . The study proceeds in two stages:

Minkowski Spacetime: The evolution of the field is analyzed in flat space to isolate the effects of large initial phases and damping without cosmological expansion.
Conformal FLRW Spacetime: The analysis is extended to a radiation-dominated universe, incorporating the Hubble friction term and the scale factor's evolution.

The equations of motion are solved numerically for arbitrary initial phases $\theta_{in}$ (specifically near $\pi$ ) and various dimensionless damping parameters $\Gamma$ . The resulting baryon asymmetry is calculated by integrating the particle production rates derived from the field's time evolution.

Key Contributions and Results

Validation of Local Approximation: The paper establishes that the adiabatic conditions are parametrically satisfied in the model, justifying the replacement of the nonlocal backreaction with a local friction term $\Gamma_{eff} \dot{\theta}$ . This allows for the tractable numerical study of large-angle dynamics.
Breakdown of Cubic Scaling: In the small-angle limit, the baryon asymmetry is known to scale as $\theta_{in}^3$ . The numerical results confirm this behavior for small oscillations but demonstrate a clear breakdown for large initial phases.
Saturation in Minkowski Space: For Minkowski spacetime, the study finds that particle production saturates as the initial phase approaches $\pi$ . The growth of the asymmetry slows significantly compared to the cubic extrapolation, though the values remain of the same order of magnitude.
Strong Dependence on Damping in FLRW: In the expanding universe (FLRW), the generation of baryon excess shows a strong dependence on the damping rate $\Gamma$ $Γ$ .
- For small $\Gamma$ , the asymmetry decreases for large initial phases.
- For $\Gamma \gtrsim 1$ , the asymmetry exhibits a nearly exponential dependence on the initial phase as $\theta_{in} \to \pi$ .
Probability of Large Fluctuations: The authors calculate that while the probability of a single Hubble patch having a misalignment angle shifted by more than $\pi$ is small ( $\sim 10^{-5}$ ), the vast number of independent patches ( $\sim e^{180}$ ) implies that regions with $\theta_{in} \sim \pi$ almost certainly exist within the observable universe.

Significance and Claims
The paper claims to provide a necessary extension of spontaneous baryogenesis models beyond the small-angle approximation, addressing a regime that is statistically probable due to inflationary quantum fluctuations. The primary significance lies in demonstrating that the large-angle regime is tractable via a Markovian reduction and that the resulting baryon asymmetry behaves differently than predicted by linearized models.

The authors note that the strong dependence of baryon density on the initial phase implies significant inhomogeneities in the baryon distribution across the universe. They suggest that this could lead to macroscopic antibaryon domains if the phase crosses $\pi$ , or strong baryon density fluctuations that might influence early galaxy formation, potentially offering signatures relevant to multi-messenger cosmology. The paper provides a benchmark parameter set that successfully reproduces the observed baryon-to-entropy ratio ( $\Delta n_B/s \simeq 8.6 \times 10^{-11}$ ) within the large-angle framework.