Investigating baryon to entropy ratio phenomenon in f(R,∇R) gravity

This study explores baryogenesis within a modified gravitational framework where the action incorporates a non-local kernel dependent on the Ricci scalar R and its covariant derivative ∇R. This extension, referred to as $f(R,\nabla R)$ gravity, generalizes traditional $f\left(R\right)$ theories by introducing additional derivative terms that may influence early-universe dynamics. Within this framework, we examine four distinct forms of the cosmological scale factor: (i) an intermediate scaling solution, (ii) a conventional power-law expansion, (iii) a generalized hybrid evolution combining both power-law and exponential growth terms and (iv) logamediate scale factor. We introduce a charge-parity (CP) violating interaction term that scales linearly with the spacetime derivative of the combined Ricci scalar and its covariant derivative, expressed as ${\partial }_{\mu }(R+\nabla R)$. Through numerical computation, we determine the resulting baryon asymmetry by evaluating the dimensionless ratio $\frac{{\eta }_B}{s}$, where ${\eta }_B$ represents the baryon number density and s denotes the entropy density of the universe. We quantitatively compare our theoretical predictions for the baryon-to-entropy ratio, $\frac{{\eta }_B}{s}$, with the current observational constraint of $9.42\times {10}^{-11}$. Notably, our numerical results demonstrate remarkable agreement with this empirically established value, supporting the viability of our modified gravitational framework in explaining the observed baryon asymmetry of the universe.