Primordial deuterium abundance from calculations of $p(n,γ)$ and $d(p,γ)$ reactions within potential-model approach

This study calculates the primordial deuterium abundance using a consistent potential-model approach for $p(n,\gamma)$ and $d(p,\gamma)$ reactions based on the Malfliet-Tjon interaction, yielding a $\mathrm{D/H}$ ratio of $2.479^{+0.350}_{-0.177}\times 10^{-5}$ that aligns well with observational data from metal-poor damped Lyman-$\alpha$ systems.

The Gist

The Big Picture: The Universe's First Recipe

Imagine the very first few minutes of the Universe as a giant, cosmic kitchen. Right after the Big Bang, it was incredibly hot and dense. In this kitchen, the chefs (physicists) were trying to bake the very first ingredients: light elements like Hydrogen, Helium, and a tiny bit of Deuterium (a heavy version of hydrogen).

The paper you're asking about is essentially a cookbook check. The authors are trying to figure out exactly how the "recipes" (nuclear reactions) work to make Deuterium. If the recipe is slightly off, the amount of Deuterium in the universe today would be totally different from what we actually see.

The Two Key Reactions: The Bottleneck and the Burner

To make Deuterium, two specific nuclear reactions are the most important. Think of them as the two most critical steps in the recipe:

1. The "Door Opener" (p + n → d + γ):

   • What it is: A proton (hydrogen nucleus) and a neutron smash together to form Deuterium.
   • The Analogy: Imagine a traffic jam. For a long time, protons and neutrons are stuck in traffic, unable to combine because the heat is too high. This reaction is the green light that finally lets them merge. Without this, the universe stays stuck in traffic, and no heavier elements can form.
   • The Challenge: It's hard to measure this reaction in a lab because neutrons are tricky to control.

2. The "Deuterium Burner" (d + p → ³He + γ):

   • What it is: Once Deuterium is made, it can get smashed again by another proton to turn into Helium-3.
   • The Analogy: Think of Deuterium as a precious, fragile gem. This reaction is a blowtorch. If the blowtorch is too hot, it melts all the gems into Helium. If it's too weak, we are left with too many gems. We need to know exactly how strong the blowtorch is to know how many gems survive.

The Problem: Missing Data

The authors point out a major problem: We don't have enough data.

• For the "Blowtorch" reaction, the energy levels where the Big Bang happened are so low that it's like trying to see a firefly in a dark room. Even the best labs (like LUNA) struggle to get enough measurements.
• For the "Door Opener," we have very few low-energy measurements.

Because we can't measure everything perfectly in the lab, we have to use theoretical models (mathematical guesses) to fill in the gaps.

The Solution: The "Scaling Factor" (The Magic Dial)

This is where the authors' work shines. They used a method called the Potential Model.

• The Analogy: Imagine you are trying to tune a radio to find a clear station. You have a knob (a scaling factor, called λ or "lambda").
  • First, they tune the knob using the "Door Opener" reaction (p + n). They know the exact result of this reaction at low energies (like a known radio frequency). They turn the knob until their math matches the real-world data perfectly.
  • The clever part: They assume that the physics governing the "Door Opener" and the "Blowtorch" are related. So, once they set the knob for the first reaction, they use that same setting (or a mathematically derived version of it) for the second reaction.

They didn't just guess; they constrained the knob using real data. This makes their prediction much more reliable than just guessing.

The Results: A Perfect Match

When they ran their numbers through a supercomputer program (called PArthENoPE) that simulates the Big Bang:

1. The Prediction: They calculated that for every 100,000 Hydrogen atoms, there should be about 2.48 Deuterium atoms.
2. The Reality Check: Astronomers look at very old, metal-poor gas clouds in the universe (which haven't been messed up by stars) to count the Deuterium. Their measurements say the number is roughly 2.5.
3. The Verdict: The authors' calculation (2.48) matches the astronomers' observation (2.5) almost perfectly!

Why This Matters

The paper concludes that even small changes in how we tune that "knob" (the scaling factor) lead to big changes in the predicted amount of Deuterium.

• The Takeaway: By using a consistent mathematical framework that links these two reactions, the authors have reduced the "noise" in our understanding of the Big Bang. They've shown that our current understanding of nuclear physics is solid enough to explain the universe's composition.

Summary in One Sentence

The authors built a mathematical bridge between two difficult-to-measure nuclear reactions, used a "tuning knob" calibrated by one reaction to predict the other, and found that their prediction of the universe's Deuterium content matches what we actually see in the sky, confirming our understanding of how the Big Bang cooked the first ingredients of the universe.

Technical Summary

1. Problem Statement

Big Bang Nucleosynthesis (BBN) relies heavily on accurate nuclear reaction rates to predict the primordial abundances of light elements, particularly deuterium (D/H). Two radiative-capture reactions are critical:

• $p(n, \gamma)$: Initiates nucleosynthesis by overcoming the "deuterium bottleneck."
• $d(p, \gamma)$: Regulates the burning (destruction) of deuterium.

Challenges:

• Experimental Limitations: Experimental data for these reactions at low energies (relevant to BBN, specifically around 100 keV) are sparse. The Coulomb barrier suppresses the $d(p, \gamma)$ cross-section, and existing data (e.g., from LUNA) remain limited. For $p(n, \gamma)$, few low-energy measurements exist.
• Theoretical Uncertainties: While ab initio methods and Chiral Effective Field Theory ($\chi$EFT) offer high precision, they are computationally intensive. There is a need for robust, consistent theoretical frameworks that can constrain nuclear uncertainties and propagate them to cosmological observables.
• Gap: Previous potential model studies often focused on reproducing cross-sections for individual reactions without systematically exploring their impact on cosmological observables like the D/H ratio.

2. Methodology

The authors employ a consistent two-body potential framework based on the Malfiet-Tjon (MT) interaction to analyze both reactions simultaneously.

• Theoretical Framework:

  • Potential Model (PM): The nuclear interaction is modeled using the MT potential ($V_{MT}$), supplemented by a Coulomb potential ($V_C$) for the $d(p, \gamma)$ reaction.
  • Scaling Factor ($\lambda$): A single scaling factor $\lambda$ is introduced to relate the scattering potential to the bound-state potential ($V^s_{MT} = \lambda V^b_{MT}$).
  • Wave Functions: Bound and scattering wave functions are obtained by solving the two-body Schrödinger equation in the center-of-mass frame.
  • Transitions: Both Electric Dipole (E1) and Magnetic Dipole (M1) transitions are calculated. The M1 transition dominates at very low energies.
  • Approximations: The deuteron is treated as a point-like object (leading order) in the $d(p, \gamma)$ reaction, approximated by $V_{pd} \approx V_{np} + V_{pp} = 2V_{np}$. Three-body effects are neglected (impulse approximation).

• Constraint Strategy:

  • The parameter $\lambda$ is constrained using the experimental thermal neutron capture cross-section for $p(n, \gamma)$ ($\sigma_{th} = 332.6 \pm 0.7$ mb).
  • This constrained $\lambda$ is then propagated to the $d(p, \gamma)$ reaction. Specifically, $\lambda_{dpg} \approx 2\lambda_{png}$ based on the cluster picture ($V_{pd} = 2V_{np}$).
  • The uncertainty in $\lambda$ is derived from the experimental uncertainty of the thermal cross-section and propagated to the $d(p, \gamma)$ rate.

• Cosmological Implementation:

  • Calculated cross-sections are converted to Maxwellian-averaged reaction rates.
  • These rates are parametrized analytically and implemented in the PArthENoPE code to simulate BBN.
  • The study varies $\lambda$ within its uncertainty bounds to assess the sensitivity of the primordial D/H abundance.

3. Key Contributions

• Unified Framework: The paper provides a consistent description of both $p(n, \gamma)$ and $d(p, \gamma)$ using a single underlying interaction model, ensuring that uncertainties are treated coherently across both reactions.
• Constraint Propagation: Instead of treating reaction rates as independent inputs, the authors demonstrate how a precise constraint on the $p(n, \gamma)$ thermal cross-section directly limits the uncertainty in the $d(p, \gamma)$ rate and the resulting cosmological abundance.
• Analytical Parametrization: The authors provide new analytical fits for the reaction rates $R(T)$ valid up to 10 GK, facilitating their use in future BBN codes.
• Sensitivity Analysis: The work explicitly quantifies how modest variations in the nuclear potential scaling factor lead to significant changes in the predicted D/H ratio.

4. Results

• $p(n, \gamma)$ Reaction:
  • The calculated cross-section agrees well with experimental data (Suzuki et al., Nagai et al.) and $\chi$EFT calculations up to 1 MeV.
  • The M1 transition dominates below 100 keV.
  • The constrained scaling factor is $\lambda_{png} = 0.734 \pm 0.023$.
• $d(p, \gamma)$ Reaction:
  • Using $\lambda_{dpg} = 2\lambda_{png} \approx 1.470 \pm 0.046$, the calculated Astrophysical S-factor ($S(0)$) is $0.228 \pm 0.015$ eV b.
  • This value is slightly higher (~4%) than some recent determinations but remains consistent within uncertainties and agrees well with high-precision LUNA data.
  • The uncertainty band encompasses available experimental data.
• Primordial Deuterium Abundance (D/H):
  • Using the PArthENoPE code with the derived reaction rates, the predicted primordial abundance is:
    $$\text{D/H} = 2.479^{+0.350}_{-0.177} \times 10^{-5}$$
  • This result is in good agreement with values inferred from metal-poor damped Lyman-$\alpha$ systems.
  • The analysis shows that the D/H ratio is highly sensitive to the scaling factor $\lambda$, highlighting the importance of precise nuclear inputs.

5. Significance

• Cosmological Constraints: The study reinforces the link between nuclear physics and cosmology. By reducing nuclear uncertainties through a consistent potential model, it improves the reliability of BBN predictions used to constrain the cosmic baryon density ($\Omega_b$).
• Validation of Potential Models: Despite its simplicity compared to ab initio methods, the potential model approach, when constrained by low-energy data, yields results competitive with more complex theories for BBN-relevant energies.
• Future Directions: The work underscores that while current models are robust, the sensitivity of D/H to nuclear rates implies that further experimental precision on low-energy capture cross-sections is crucial for refining cosmological parameters. The provided analytical parametrizations serve as immediate inputs for the next generation of BBN simulations.