Gravitational baryogenesis beyond the spectator approximation

The standard gravitational-baryogenesis operator λ ∇ μ R J μ , with λ ≡ ϵ / M * 2 , is usually treated as a spectator interaction that generates an effective chemical potential in a prescribed background. When included in the gravitational action, however, it defines a genuine curvature–matter-coupling variational problem, relevant for the baryon, lepton, and B − L currents, whether described microscopically by particle-physics operators or macroscopically by a fluid current J μ = n X u μ . Up to a boundary term the interaction is equivalent to − λ R ∇ μ J μ , making its f ( R , Matter) character manifest, but the metric equations remain open unless the metric dependence of J μ is specified. For an arbitrary local realization J μ (Ψ, g ) we derive the universal part of the field equations and isolate the realization-dependent tensor generated by δ g J μ . In the vector-density realization the explicit J α ∇ α R term cancels, but an algebraic term − λ g μ ν R ∇ α J α survives, so the theory admits only a partial effective-Planck-mass interpretation, M e f f 2 = M P l 2 − 2 λ ∇ μ J μ , and a time-dependent effective gravitational coupling during baryogenesis. Specializing to flat Friedmann-Lemaître-Robertson-Walker (FLRW) with a homogeneous current J μ = n X u μ , we obtain the modified Friedmann and Raychaudhuri equations, the associated continuity relation, and dimensionless diagnostics that quantify when the spectator approximation is controlled. We also discuss the implications for gravitational-baryogenesis studies in modified theories of gravity, providing a consistent General Relativity (GR) baseline for implementations in both standard cosmology and modified-gravity frameworks.