Gravitational baryogenesis beyond the spectator approximation

The standard gravitational-baryogenesis operator $\lambda\,\nabla_\mu R\,J^\mu$, with $\lambda\equiv \epsilon/M_\ast^{2}$, 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 $J^\mu=n_Xu^\mu$. Up to a boundary term the interaction is equivalent to $-\lambda R\nabla_\mu J^\mu$, making its $f(R,{\rm Matter})$ character manifest, but the metric equations remain open unless the metric dependence of $J^\mu$ is specified. For an arbitrary local realization $J^\mu(\Psi,g)$ we derive the universal part of the field equations and isolate the realization-dependent tensor generated by $\delta_g J^\mu$. In the vector-density realization the explicit $J^\alpha\nabla_\alpha R$ term cancels, but an algebraic term $-\lambda g_{\mu\nu}R\nabla_\alpha J^\alpha$ survives, so the theory admits only a partial effective-Planck-mass interpretation, $M_{\rm eff}^2=M_{\rm Pl}^2-2\lambda\nabla_\mu J^\mu$, and a time-dependent effective gravitational coupling during baryogenesis. Specializing to flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) with a homogeneous current $J^\mu=n_Xu^\mu$, 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.