Predicting the Dark Matter -- Baryon Abundance Ratio

This paper proposes that QCD axion dark matter models utilizing relaxation mechanisms can predict the observed dark matter-to-baryon abundance ratio as a discrete function of beta functions, specifically reproducing the experimental value of 5.36 within percent-level error bars when the gauge group size is $N=8$.

The Gist

The Big Mystery: Why is there so much Dark Matter?

Imagine the universe is a giant soup. In this soup, there are two main ingredients:

1. Baryons: These are the "normal" stuff we can see and touch (stars, planets, you, me).
2. Dark Matter: This is the invisible stuff that holds galaxies together but doesn't interact with light.

For a long time, scientists have been puzzled by a specific coincidence. When we measure how much of each ingredient is in the universe, we find that there is exactly 5.36 times more Dark Matter than normal matter.

This is strange because the two ingredients are made by completely different processes. It's like baking a cake and finding that you accidentally added exactly 5.36 times more chocolate chips than flour, even though you measured them separately. Usually, you'd expect the ratio to be random, like 100:1 or 1:10. The fact that it is so close to a simple number (5.36) suggests there might be a hidden rule connecting them.

The Solution: The "Relaxation" Mechanism

The authors propose a solution called a relaxation mechanism. Think of this not as a static rule, but as a dynamic process that happened in the early universe.

Imagine a thermostat (a device that adjusts temperature) that is trying to find the perfect setting.

• In this model, there is a special "scanner" field (let's call it $\phi$, or "Phi").
• This scanner acts like a dial that changes the "weight" (mass) of both the normal matter and the dark matter simultaneously as the universe evolves.
• As the universe expands, the scanner keeps turning the dial, changing the masses, until it hits a "sweet spot" where the energy of the normal matter and the dark matter balance out perfectly.

Once the scanner finds this balance, it stops moving. The universe gets "locked" into this specific ratio.

The Twist: The QCD Axion

The paper focuses on a specific type of dark matter candidate called the QCD Axion.

• The Connection: The axion is deeply tied to the physics of protons (normal matter). You can't change the axion's properties without also changing the properties of protons.
• The Scanning: When the scanner dial turns to change the axion's mass, it automatically changes the proton's mass too. They are linked like two gears on the same machine.

Because they are linked, the scanner doesn't have to guess the ratio. It just has to find the point where the "gears" mesh perfectly.

The "Prediction": Why 5.36?

Here is the most exciting part of the paper. The authors show that because of the specific way these particles are linked, the scanner can only stop at certain, discrete values. It's like a radio that only has stations at specific numbers, not in between.

The final ratio depends on a single number: $N$, which represents the size of a specific mathematical group in the theory (think of it as the number of "colors" or types of particles in a hidden sector).

• If you pick $N = 8$, the math predicts the ratio will be 5.33.
• The actual measured value is 5.36.

The authors argue that $N=8$ is the "Goldilocks" choice. It predicts the observed ratio with incredible precision (within 1% error). If $N$ were 7 or 9, the prediction would be way off. This suggests the universe isn't random; it's "predicting" this ratio based on the integer choice of $N=8$.

How the Universe Got Here (The Timeline)

The paper outlines a specific history of the early universe to make this work:

1. Inflation Ends: The universe expands rapidly and then stops.
2. Baryogenesis: A mechanism creates the imbalance of normal matter (making more matter than antimatter).
3. Relaxation Phase: The scanner field ($\phi$) starts rolling. It changes the masses of protons and axions. It keeps rolling until the energy densities of normal matter and dark matter match the ratio dictated by the math (the beta functions).
4. Locking In: Once the scanner finds the minimum energy point, a new "potential" (like a valley floor) forms, freezing the scanner in place. The ratio is now fixed forever.
5. Reheating: The universe heats up again, and standard cosmology begins.

How to Test This Idea

The paper suggests three main ways to prove or disprove this theory:

1. Measure the Ratio More Precisely: If we measure the Dark Matter to Baryon ratio more accurately and it turns out to be 5.360001 instead of 5.36, it might rule out the specific integer $N=8$ prediction.
2. Lattice QCD Calculations: Scientists need to calculate exactly how the mass of a proton depends on the fundamental forces of the universe. If the math doesn't match the paper's assumptions, the model fails.
3. Fifth Force Experiments: The model requires the scanner field to interact with normal matter. This interaction might create a tiny, new "fifth force" (beyond gravity, electromagnetism, and the nuclear forces) that could be detected in sensitive lab experiments.

Summary

The paper claims that the mysterious 5.36 ratio between Dark Matter and normal matter isn't a coincidence. It is the result of a cosmic "relaxation" process where a scanning field adjusted the masses of both until they balanced. Because of the specific rules of particle physics (involving a composite axion), this balance only happens at a specific integer setting ($N=8$), which perfectly matches what we observe in the universe today.

Technical Summary

Technical Summary: Predicting the Dark Matter - Baryon Abundance Ratio

Problem Statement
The paper addresses the "dark matter - baryon coincidence problem," the observation that the relic energy densities of cold dark matter ($\Omega_{\text{DM}}$) and baryons ($\Omega_B$) are of the same order of magnitude, with a measured ratio $r_{\text{obs}} \equiv \Omega_{\text{DM}}/\Omega_B = 5.36 \pm 0.05$. While technically an $\mathcal{O}(1)$ number, the proximity of this ratio to unity is non-trivial because the mechanisms setting these densities are typically unrelated. Baryon density is determined by the QCD confinement scale ($\Lambda_{\text{QCD}}$), generated via dimensional transmutation and exponentially sensitive to UV parameters. Conversely, dark matter (DM) mass and abundance usually depend on independent theory parameters and production mechanisms. The authors argue that the conspiring of unrelated parameters to produce a ratio of exactly 5.36 is an "astonishing" puzzle requiring a dynamical explanation.

Methodology and Model Construction
The authors propose a "relaxation" solution where a scalar "scanner" field, $\phi$, dynamically adjusts the masses of baryons and dark matter until their energy densities are comparable. The model is built within the context of QCD axion dark matter, specifically utilizing a composite axion scenario.

1. Scanner Field Dynamics: The scanner field $\phi$ couples to both the baryon sector and the dark sector. The masses are assumed to scale exponentially with $\phi$:
   $$m_B(\phi) = m_B(0) \exp(c_B \phi/f_\phi), \quad m_{\text{DM}}(\phi) = m_{\text{DM}}(0) \exp(c_D \phi/f_\phi)$$
   The finite-density potential $V(\phi) = m_B(\phi)n_B + m_{\text{DM}}(\phi)n_{\text{DM}}$ drives $\phi$ toward a minimum where $\partial V / \partial \phi = 0$. At this minimum, the ratio of energy densities is determined by the ratio of the coupling coefficients:
   $$\frac{\rho_{\text{DM}}}{\rho_B} \bigg|_{\phi_{\text{min}}} \simeq -\frac{c_B}{c_D}$$

2. UV Completion via Composite Axion: To realize this in a UV-complete setting, the authors employ a composite axion model based on an $SU(N)$ gauge group with massless vector-like fermions.

   • The $SU(N)$ confinement scale $\Lambda_N$ depends on $\phi$.
   • Through Renormalization Group (RG) running, $\Lambda_N$ influences the QCD scale $\Lambda_{\text{QCD}}$ and the axion decay constant $f_a$.
   • Crucially, because the axion mass $m_a \propto \Lambda_{\text{QCD}}^{3/2}/f_a$, scanning the axion mass inherently scans the baryon mass (via $\Lambda_{\text{QCD}}$).
   • The coefficients $c_B$ and $c_D$ are not arbitrary but are fixed by the gauge charges of the confined fermions and the beta functions of the gauge groups.

3. Predictive Power: The ratio of the relic densities is derived to be:
   $$\frac{\Omega_{\text{DM}}}{\Omega_B} = \frac{2N}{27 - 3N}$$
   where $N$ is the size of the $SU(N)$ gauge group (an integer degree of freedom). This implies the model can only accommodate discrete values of the ratio, effectively "predicting" the observed value if a specific integer $N$ is chosen.

4. Cosmological Timeline: The paper outlines a specific cosmological history to ensure the mechanism works:

   • Post-Inflation: The inflaton decays into a "reheaton" ($\Phi_E$) which dominates the universe and later reheats the Standard Model (SM) to a low temperature ($\sim 8.4$ MeV) to avoid thermal spoiling of the relaxation.
   • Baryogenesis: An Affleck-Dine (AD) mechanism generates the baryon asymmetry early, ensuring a sufficient non-relativistic baryon density exists before relaxation begins.
   • Relaxation Phase: Once baryons are non-relativistic, $\phi$ rolls down its potential. The baryon mass decreases as the universe expands, while the axion mass is scanned. The system settles into a minimum where $\rho_{\text{DM}} \approx \rho_B$.
   • Scanner Potential Lock-in: A temperature-dependent potential for $\phi$ (generated by a dark $SU(D)$ sector) turns on after relaxation, freezing the value of $\phi$ and locking in the ratio $\Omega_{\text{DM}}/\Omega_B$.
   • Reheating: The reheaton decays, reheating the SM to a temperature compatible with Big Bang Nucleosynthesis (BBN).

Key Results

• Prediction of the Ratio: By setting the gauge group size to $N=8$, the model predicts $\Omega_{\text{DM}}/\Omega_B = 5.33$. This falls within the percent-level error bars of the observed value ($5.36 \pm 0.05$).
• Parameter Space: The authors identify a viable parameter space for the axion decay constant $f_a$ and the initial baryon mass. The allowed range for the axion decay constant is found to be:
  $$10^8 \text{ GeV} \lesssim f_a \lesssim 3.2 \times 10^{11} \text{ GeV}$$
  This range satisfies constraints from supernova cooling (SN1987A), neutron star cooling, and the requirement that the axion misalignment angle remains $\theta_0 \lesssim 2\pi$.
• Constraints: The model is subject to several constraints, including:
  • The requirement that baryons are non-relativistic at the start of relaxation.
  • The requirement that pion contributions to the scanner potential do not displace $\phi$ from its minimum (imposing bounds on the duration of relaxation).
  • Constraints on the dark sector temperature to avoid overclosing the universe or violating $\Delta N_{\text{eff}}$ limits.

Significance and Claims
The paper claims that this framework offers a "qualitatively different resolution" to the coincidence problem compared to asymmetric dark matter or mirror world scenarios. Instead of postulating relations between number densities or masses, the ratio is dynamically selected by the cosmological evolution of a scanner field.

The authors emphasize the predictive nature of the model: the ratio is not a free parameter but is fixed by the discrete integer $N$ and the ratio of beta functions. The fact that $N=8$ reproduces the observed ratio is presented as a remarkable coincidence within the model's logic.

The paper concludes by noting that this framework is testable through:

1. More precise measurements of $\Omega_{\text{DM}}/\Omega_B$.
2. Lattice QCD determinations of the dependence of the proton mass on the high-energy QCD gauge coupling.
3. Traditional fifth-force and equivalence principle experiments, as the scanner field couples to baryons.

The authors remain modest regarding the complexity of the required cosmological timeline and the assumption that the vacuum potential of $\phi$ is sufficiently suppressed, noting these as areas for future improvement.