Effects of pair freeze-out on photon distributions in BBN epoch

1. Problem Statement

The paper addresses the Primordial Lithium Problem, a persistent discrepancy in Standard Big Bang Nucleosynthesis (SBBN) where the theoretical prediction for the abundance of primordial Lithium-7 ($^7$Li) is approximately three times higher than observational data.

• Context: While SBBN successfully predicts Deuterium and Helium abundances, the $^7$Li discrepancy suggests missing physics or incorrect assumptions in the standard model.
• Previous Work: A prior study by some of the current authors (Ref. [6]) proposed that the primordial plasma might not be in perfect thermal equilibrium. By applying Tsallis statistics (a non-extensive generalization of Boltzmann-Gibbs statistics) to the photon distribution, they found that a specific entropic index ($q \approx 1.027$) could resolve the lithium problem by enhancing the high-energy tail of the photon distribution.
• The Gap: The previous work was phenomenological; it lacked a rigorous physical derivation explaining how or why the photon distribution would deviate from the Planck distribution (where $q=1$) during the BBN epoch. This paper aims to provide that physical mechanism.

2. Methodology

The authors construct a theoretical framework to derive the evolution of the photon distribution function ($f_\gamma$) using the Boltzmann equation with specific collision terms.

• Tsallis Statistics Framework:
  • The photon distribution is modeled using Tsallis statistics, characterized by an entropic index $q$ (or perturbation parameter $Q = 1-q$).
  • The distribution function is expanded to first order around the equilibrium Planck distribution: $f_\gamma \approx f^{(0)}_\gamma + f^{(1)}_\gamma$.
• Perturbed Boltzmann Equation:
  • The authors derive a linearized Boltzmann equation for the photon number density perturbation ($n^{(1)}_\gamma$).
  • Assumptions:
    1. Electrons, positrons, and nuclei remain in thermal equilibrium (Fermi-Dirac distribution).
    2. The deviation of photons from equilibrium is small ($\delta f_\gamma / f_\gamma \ll 1$).
    3. Feedback from the distorted photon distribution to the fermionic sector is negligible due to small momentum transfer and the dominance of equilibrium scatterings.
• Collision Terms:
  • The collision operator $C[f]$ includes electron-positron pair creation/annihilation ($2\gamma \leftrightarrow e^+e^-$) and Compton scattering ($\gamma e^- \leftrightarrow \gamma e^-$).
  • Higher-order processes (double Compton, bremsstrahlung) are neglected as electron density drops significantly at lower temperatures.
• Reduction to ODE:
  • By integrating the collision terms over momentum space, the complex partial differential equation (PDE) for the distribution function is reduced to a single Ordinary Differential Equation (ODE) governing the evolution of the non-extensivity parameter $Q(T)$:
    $$\frac{dQ}{dT} = \frac{M(T)}{ABT^6} Q$$
  • Here, $M(T)$ represents the net collision term derived from pair processes and Compton scattering, and $A, B$ are constants related to thermodynamics and expansion rates.

3. Key Contributions

1. Physical Mechanism for Non-Equilibrium: The paper identifies pair freeze-out as the physical driver for photon distribution distortion. As the Universe cools below $T \sim 10^9$ K, the energy of photons becomes insufficient for pair creation ($2\gamma \to e^+e^-$), breaking the detailed balance with pair annihilation ($e^+e^- \to 2\gamma$). This disruption prevents the photon distribution from relaxing instantly to a Planck spectrum.
2. Mathematical Formalism: The authors successfully derive the first-order perturbed Boltzmann equation specifically for Tsallis statistics, demonstrating that the evolution of the entropic index $q$ can be tracked via a computationally efficient ODE rather than a full PDE.
3. Validation of Previous Ansatz: The study provides a dynamical justification for the $q \approx 1.027$ value used in Ref. [6], showing that such a deviation is a natural consequence of pair freeze-out dynamics under specific initial conditions.

4. Results

• Evolution of $Q(T)$:
  • The collision term $M(T)$ is dominated by pair processes at high temperatures and Compton scattering at lower temperatures.
  • The ratio of $Q(T)$ to $Q(T-\delta T)$ shows that $Q$ changes rapidly at temperatures $T \gtrsim 0.022$ MeV.
  • Initial Condition: The authors set the initial temperature $T_0 \approx 0.022$ MeV. They choose a small, non-zero initial deviation $Q_0 \approx -4.78 \times 10^{-4}$ such that the maximum deviation reaches $q \approx 1.027$ (consistent with the lithium problem solution).
• Dynamics:
  • As the temperature drops from $T_0$, the pair freeze-out causes $q$ to increase (deviating from 1), enhancing the high-energy tail of the photon distribution.
  • Around $T \sim 0.018$ MeV, Compton scattering becomes the dominant collision process. This drives the system back toward equilibrium, causing $q$ to decrease and saturate back to $\approx 1.001$.
• Outcome: The model predicts a temporary state of chemical non-equilibrium for photons during the BBN epoch, followed by a restoration to the Planck distribution as the Universe enters the matter-dominated era. This restoration explains why the Cosmic Microwave Background (CMB) observations show a near-perfect Planck spectrum today.

5. Significance

• Resolution of the Lithium Problem: The study confirms that a transient non-extensive photon distribution, driven by pair freeze-out, can alter nuclear reaction rates sufficiently to reduce the predicted $^7$Li abundance to match observations without invoking exotic particles or new physics beyond the Standard Model.
• Refinement of SBBN: It challenges the strict assumption of instantaneous thermal equilibrium in the early Universe, suggesting that "frozen-out" non-equilibrium states can persist long enough to influence nucleosynthesis.
• Cosmological Implications: The findings suggest that the photon distribution in the BBN era contains "memory" of early-stage instabilities. While direct observation of BBN fluctuations is impossible, the link between these early fluctuations and the density perturbations seen in the CMB offers new avenues for understanding inflation and early Universe structure formation.
• Methodological Advance: The reduction of the Boltzmann equation to an ODE for the entropic index provides a powerful tool for future studies of non-equilibrium processes in cosmology.

In conclusion, this paper bridges the gap between phenomenological solutions to the lithium problem and fundamental kinetic theory, demonstrating that the freeze-out of electron-positron pairs naturally induces a deviation in the photon distribution that resolves the primordial lithium discrepancy.