Gravitational Baryogenesis Process

• Gravitational baryogenesis is a theoretical framework where the time-varying spacetime curvature couples with the baryonic current, inducing net baryon asymmetry even in thermal equilibrium.
• It employs a CPT-violating interaction via the derivative of the Ricci scalar (or analogous invariants), effective in various cosmological and modified gravity scenarios such as anisotropic models and f(T) theories.
• The mechanism successfully accounts for the observed baryon-to-entropy ratio while confronting challenges like higher-derivative instabilities, which can be mitigated by stabilization techniques including higher curvature terms.

Gravitational baryogenesis is a theoretical framework in which the dynamics of spacetime curvature—specifically, the time-dependent Ricci scalar or analogous invariants—couple to the baryon number current and induce CPT violation in the early universe. This mechanism generates a net baryon asymmetry even in thermal equilibrium, offering a gravitational alternative to traditional baryogenesis scenarios that require explicit CP violation and departure from equilibrium in the particle sector. The process is realized in a broad variety of cosmological backgrounds and gravity theories, including anisotropic geometries, teleparallel and higher-curvature modifications, scalar-tensor extensions, and models with gravitationally induced particle production. The baryon-to-entropy ratio predicted by gravitational baryogenesis can match observational constraints under appropriate dynamical conditions and parameter choices.

1. Fundamental Mechanism: Curvature Coupling and CPT Violation

The essential interaction at the core of gravitational baryogenesis is a dimension-six, CPT-violating operator coupling the derivative of the Ricci scalar $R$ (or a general gravitational invariant) to the baryonic current $J^\mu_B$: $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$ where $M_*$ is the cutoff scale of the effective field theory. In an expanding, time-dependent universe, $\partial_0 R$ is generically nonzero, inducing an effective chemical potential for baryons $\mu_B \sim \dot{R}/M_*^2$. This chemical potential tilts the energy spectrum between baryons and antibaryons, thereby generating a net baryon number density in equilibrium.

The baryon-to-entropy ratio at decoupling (temperature $T_D$) is given by

$$\frac{n_B}{s} \approx -\frac{15 g_b}{4\pi^2 g_*} \frac{\dot{R}}{M_*^2 T_D}$$

where $g_b$ is the number of internal baryon degrees of freedom, and $g_*$ the effective relativistic degrees of freedom. The structure of this formula holds across various gravity scenarios by replacing $J^\mu_B$0 with the relevant gravitational invariant and coupling accordingly (Saaidi et al., 2010, Oikonomou et al., 2016, Oikonomou et al., 2016, Maity et al., 2018, Lima et al., 2016).

2. Anisotropic and Nonstandard Cosmologies

In standard Friedmann–Robertson–Walker (FRW) cosmology with $J^\mu_B$1 (radiation-dominated), the Ricci scalar vanishes and gravitational baryogenesis is ineffective. However, in anisotropic models, such as Bianchi type I universes, the shear tensor and associated anisotropies introduce additional terms in $J^\mu_B$2: $J^\mu_B$3 where $J^\mu_B$4 is the shear invariant (Saaidi et al., 2010). Even for $J^\mu_B$5, the presence of anisotropy ($J^\mu_B$6) ensures a nonzero $J^\mu_B$7 and hence $J^\mu_B$8, enabling baryogenesis where FRW fails.

The inclusion of a time-dependent equation-of-state parameter $J^\mu_B$9, stemming from multiple interacting fluids (e.g., radiation and scalar fields), further affects $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$0 and its derivative. Time-dependence in $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$1 can amplify or suppress baryon asymmetry generation depending on the detailed fluid interchange and the evolution of the shear.

3. Modified Gravity Extensions

Gravitational baryogenesis is not restricted to standard GR; it admits extensive generalization.

• Teleparallel Gravity ($$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$2 models): In $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$3 gravity, the Lagrangian is generalized to $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$4, with $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$5 the torsion scalar ($$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$6). The CP-violating coupling is recast as $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$7 or its $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$8 extension (Oikonomou et al., 2016, Azhar et al., 2020). Models such as $$\mathcal{L}_{\text{int}} = \frac{1}{M_*^2} \partial_\mu R \, J_B^\mu$$9 can yield an $M_*$0 within observed limits for tuned $M_*$1 (cosmic evolution exponent), while others (exponential or Linder models) may be ruled out by sign or magnitude constraints.
• $M_*$2 and Nonminimal Theories: In $M_*$3 gravity (Sahoo et al., 2019), interactions with $M_*$4 or $M_*$5 can be considered, although only derivatives involving $M_*$6 (or the composite Lagrangian) generally produce observationally viable baryon asymmetry. The hybrid scale factor $M_*$7 provides the required transition from radiation-like to accelerated expansion.
• Running Vacuum and Brane World Models: In running vacuum models (RVMs), the vacuum energy is a function $M_*$8 of the Hubble parameter, yielding $M_*$9 in the radiation era and a nonzero $\partial_0 R$0 (Oikonomou et al., 2016). Similarly, in DGP brane cosmology, corrections from the extra dimension ensure a nonvanishing $\partial_0 R$1 and thus baryogenesis even for $\partial_0 R$2 (Atazadeh, 2020).
• Teleparallel Gauss–Bonnet and Boundary Terms: Extensions to $\partial_0 R$3 and $\partial_0 R$4 (where $\partial_0 R$5 is the teleparallel Gauss–Bonnet invariant and $\partial_0 R$6 the torsion–Ricci boundary term) provide additional sources for geometric time-dependence, allowing for more parameter freedom and phenomena such as generalized baryogenesis interactions of the form $\partial_0 R$7. Appropriate parameter choices yield $\partial_0 R$8 in line with observations (Azhar et al., 2020).
• Scalar–Tensor and Energy–Momentum Squared Gravity: In scalar–tensor theories, the expansion rate and time–temperature relation are modified, which impacts the decoupling rates and the efficacy of baryogenesis mechanisms. In $\partial_0 R$9 (energy–momentum squared gravity), an additional term depending on the self-contraction $\mu_B \sim \dot{R}/M_*^2$0 supplements the Ricci curvature. This ensures nonzero sources for baryogenesis even when $\mu_B \sim \dot{R}/M_*^2$1 vanishes, such as in radiation eras, via a coupling to $\mu_B \sim \dot{R}/M_*^2$2 (Pereira et al., 2024).

4. Dynamical Sources: Particle Production and Phase Transitions

Gravitational baryogenesis efficiency is strongly affected by dynamical processes in the early universe.

• Gravitational Particle Production: During inflation and reheating, time variation of the spacetime background leads to gravitational particle production (GPP), which can generate heavy out-of-equilibrium states. Their subsequent baryon-number-violating decays, even if CP violation is small, can yield the observed baryon asymmetry. The dilution and efficiency depend sensitively on the inflaton potential (e.g., $\mu_B \sim \dot{R}/M_*^2$3-attractor T-models with exponent $\mu_B \sim \dot{R}/M_*^2$4), the particle coupling to gravity (conformal, tachyonic, etc.), and the redshift behavior during reheating (Flores et al., 2024).
• Non-Minimal QCD Coupling at the QCD Transition: Direct non-minimal curvature couplings to both gluon and quark fields lead to explicit conservation law violations, dynamically reduced quark masses, and novel CP violation. This supports a strong first-order QCD phase transition and a baryon asymmetry consistent with observations (Antunes et al., 2016).
• Particle Production–Induced Curvature: Macroscopic, adiabatic, gravitationally induced particle production modifies the effective equation of state and Ricci scalar evolution, enabling baryogenesis even when otherwise forbidden (e.g., in radiation-dominated eras) (Lima et al., 2016).

5. Instabilities, Limitations, and Stabilization Mechanisms

A major challenge in canonical gravitational baryogenesis driven by $\mu_B \sim \dot{R}/M_*^2$5 is the emergence of higher-derivative (fourth order) gravitational field equations. These possess exponentially unstable solutions: even small perturbations in $\mu_B \sim \dot{R}/M_*^2$6 can drive runaway growth, disrupting standard cosmological evolution (Arbuzova et al., 2017, Arbuzova, 2018, Arbuzova et al., 2023). The instability is generic, occurring both for scalar and fermionic baryon carriers.

Possible stabilization mechanisms include:

• Introduction of Higher Curvature Terms: Adding a $\mu_B \sim \dot{R}/M_*^2$7 term to the action provides additional damping, converting the instability into oscillatory (scalaron) modes. The stabilization condition is

$\mu_B \sim \dot{R}/M_*^2$8

(where $\mu_B \sim \dot{R}/M_*^2$9 encodes the strength of the higher-derivative term, particle number, and temperature). Even when stabilized, the background curvature can remain much larger than in standard cosmology, indicating nontrivial modifications to the cosmic expansion history (Arbuzova et al., 2023).

• Modified Gravity Models: More general $T_D$0 or $T_D$1 models offer additional degrees of freedom that may suppress or control the instability, at the cost of significantly altering the expansion history or introducing exotic phenomenology.

6. Observational Viability and Model Constraints

All viable gravitational baryogenesis models must reproduce the observed baryon-to-entropy ratio, $T_D$2, consistent with measurements from the cosmic microwave background and Big Bang nucleosynthesis. Achieving this ratio imposes stringent constraints on the parameter space of modified gravity couplings ($T_D$3, $T_D$4, $T_D$5, $T_D$6, etc.), cosmic evolution parameters, decoupling temperatures, cutoff scales, and interaction strengths (Oikonomou et al., 2016, Sahoo et al., 2019, Srivastava et al., 2020, Mishra et al., 2023). Some models (notably certain $T_D$7, exponential, or Linder models) are ruled out by yielding negative or unphysically large asymmetries, while others are viable only within tightly restricted ranges of their free parameters.

Atypical scenarios such as running vacuum, teleparallel Gauss–Bonnet, and brane-world cosmologies expand the phase space of successful models, especially by enabling baryogenesis in regimes where it is forbidden in standard GR.

7. Role in Cosmic and Particle Physics Paradigms

Gravitational baryogenesis provides a mechanism to connect the evolution of spacetime geometry directly to the microphysics of baryon number violation. It offers a route to baryon asymmetry without requiring explicit CP violation beyond gravitational CPT violation, and may operate efficiently even in phases where conventional mechanisms fail.

The framework is sensitive to early universe features such as anisotropy, inflationary reheating, particle production, and the nature of the background gravitational interaction. However, its viability is contingent on controlling higher-derivative instabilities and maintaining compatibility with observed large-scale cosmological and astrophysical behavior.

Gravitational baryogenesis thus sits at the intersection of cosmology, gravitational physics, and particle phenomenology, making it an active area for both theoretical development and observational constraint via next-generation cosmological and gravitational experiments.