Gravitational baryogenesis beyond the spectator approximation

The standard gravitational-baryogenesis operator λ\, _μR\, 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=nXu^μ. Up to a boundary term the interaction is equivalent to -λR_μJ^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 δg J^μ. In the vector-density realization the explicit J^α_αR term cancels, but an algebraic term -λg⏛⏜R_αJ^α survives, so the theory admits only a partial effective-Planck-mass interpretation, M ₄₅₅²=M ₏₋²-2λ_μJ^μ, and a time-dependent effective gravitational coupling during baryogenesis. Specializing to flat Friedmann-Lemaître-Robertson-Walker (FLRW) with a homogeneous current J^μ=nXu^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.