Gravitational baryogenesis in $F(R)$ gravity's rainbow

Technical Summary: Gravitational Baryogenesis in F(R) Gravity's Rainbow

Problem Statement
The observed baryon asymmetry of the universe (BAU), quantified by the ratio of baryon number density to entropy density ($Y_B \approx 0.864 \times 10^{-10}$), remains a significant open problem in cosmology. While the Standard Model of particle physics provides a framework for baryogenesis via the Sakharov conditions, it fails to generate sufficient CP violation or departure from thermal equilibrium to explain the observed BAU. Gravitational baryogenesis offers an alternative mechanism by coupling the derivative of the Ricci scalar curvature ($\partial_\mu R$) to the baryon current ($J^\mu$), inducing a dynamic CPT violation in an expanding universe. However, standard General Relativity (GR) and even standard modified gravity models (like $F(R)$) face challenges in generating the correct asymmetry, particularly in radiation-dominated eras. This paper investigates whether incorporating "gravity's rainbow"—a framework where spacetime metrics depend on particle energy due to quantum gravity effects—into $F(R)$ gravity can successfully produce a viable baryon asymmetry.

Methodology
The authors formulate a theoretical framework combining $F(R)$ gravity with the gravity's rainbow metric. The action includes the standard $F(R)$ term, a matter action, and a baryogenesis interaction term $S_B = \frac{1}{M_*^2} \int d^4x \sqrt{-g} (\partial_\mu R)J^\mu$, where $M_*$ is the effective theory cutoff scale.

1. Metric and Field Equations: The study utilizes a modified Friedmann-Lemaître-Robertson-Walker (FLRW) metric where the line element depends on energy-dependent rainbow functions $\tilde{f}(E)$ and $\tilde{g}(E)$. The authors derive the modified field equations for $F(R)$ gravity within this rainbow background.
2. Power-Law Solutions: The authors assume a power-law expansion for the scale factor, $a(t) = At^\gamma$ (with $\gamma > 1$), and a power-law form for the rainbow function based on the Hubble parameter, $\tilde{f} \approx (H/M)^\alpha$. They analyze two specific scenarios for the rainbow functions:
   • Case 1: $\tilde{f} = \tilde{g}$
   • Case 2: $\tilde{g} = 1$
3. Model Specification: The $F(R)$ function is chosen as a power law, $F(R) = BR^n$. The authors derive exact expressions for the Ricci scalar ($R$), energy density ($\rho$), and pressure ($P$) as functions of cosmic time and model parameters ($\alpha, \gamma, n, B$).
4. Baryon Asymmetry Calculation: Using the derived time-dependent Ricci scalar, the authors calculate the effective chemical potential for baryons. They determine the decoupling temperature ($T_D$) where baryon-number-violating interactions freeze out and subsequently derive the baryon-to-entropy ratio ($Y_B$) as a function of the model parameters and the decoupling temperature.
5. Numerical Analysis: The study performs a numerical analysis to constrain the nine model parameters ($A, \gamma, M, \alpha, B, n, M_B, m, M_*$) to satisfy physical constraints ($\rho > 0, R > 0$) and match observational data ($Y_B \approx 9.2 \times 10^{-11}$).

Key Contributions and Results

• Exact Solutions: The paper provides exact power-law solutions for the field equations in $F(R)$ gravity's rainbow for both $\tilde{f} = \tilde{g}$ and $\tilde{g} = 1$ scenarios.
• Parameter Constraints: The authors identify strict conditions for the model parameters to yield physical solutions (positive energy density and Ricci scalar).
  • For Case 1 ($\tilde{f} = \tilde{g}$), the condition $\chi^{(1)} > 0$ requires $\gamma > (1-\alpha)/2$. The positivity of energy density depends on a complex function $S^{(1)}(\alpha, \gamma, n)$.
  • For Case 2 ($\tilde{g} = 1$), the condition $\chi^{(2)} > 0$ requires $\gamma > (1+\alpha)/2$.
  • Crucially, the analysis suggests that for $n \geq 2$, viable solutions for universe expansion and baryogenesis are difficult to find without extremely large values of $\alpha$ or $\gamma$. Viable solutions are found for $n$ close to 1.4 or 1.5, provided $\alpha$ and $\gamma$ are appropriately tuned (e.g., if $n=1.4$ and $\alpha=3$, then $\gamma > 2$).
• Decoupling Temperature and Asymmetry: The study derives the decoupling temperature $T_D$ and the ratio $Y_B$ as functions of $\gamma, n, m$ (dimension of the B-violating operator), and $\alpha$.
  • Numerical results indicate that higher values of $\gamma$ generally decrease $Y_B$, while higher values of $n$ and $m$ increase it.
  • The ratio $Y_B$ reaches a maximum for specific combinations of parameters (e.g., around $\gamma \approx 3$ for certain $\alpha, n, m$ values).
  • The authors demonstrate that with specific parameter choices (e.g., $n=1.4, \gamma=3, \alpha=3, m=2$), the model can produce a baryon-to-entropy ratio consistent with observational data ($Y_B \approx 10^{-10}$).
• Starobinsky Model Limitation: The paper notes that the standard Starobinsky model ($F(R) = R + R^2/6M^2$) fails to account for inflation and subsequent baryogenesis within the power-law scale factor scenario in this framework, as the $R^2$ term dominates and disrupts the required dynamics.

Significance and Claims
The paper claims that $F(R)$ gravity's rainbow provides a novel mechanism for gravitational baryogenesis. By incorporating energy-dependent spacetime metrics (quantum gravity corrections) into modified gravity, the model alters the evolution of the Ricci scalar and the Friedmann equations. This modification allows for the generation of a viable baryon asymmetry in regimes where standard $F(R)$ gravity or GR might struggle, particularly by enabling non-zero baryon-to-entropy ratios during radiation-dominated eras. The authors conclude that their results suggest this framework can potentially explain the observed BAU while integrating quantum gravity effects, offering a unique alternative to standard baryogenesis scenarios. The study emphasizes that the viability of the model is highly sensitive to the rainbow parameter $\alpha$ and the power-law index $n$, requiring careful parameter selection to align with cosmological observations.