On gravitational baryogenesis in inflationary f ( R ) cosmologies

Abstract The origin of the small asymmetry in the abundance of matter and antimatter in the very early universe, described by the Baryon Asymmetry Factor η = 8.65 × 10 -11 , is one of the big unanswered questions of contemporary physics. Gravitational baryogenesis postulates the baryon-antibaryon asymmetry is generated by a coupling between the baryon current and the gradient of the Ricci scalar R . As this coupling is zero in General Relativity for a radiation dominated flat early Universe, more generalized theories of gravity are required. The present study investigates the feasibility of gravitational baryogenesis during the inflationary phase of two f ( R ) modified theories of gravity: the widely used Starobinsky f ( R ) = R + R 2 / M 2 model, and the recently proposed power-law model f ( R ) = c 1 R 2+ k /4 + c 2 R + c 3 of Odintsov and Oikonomou (2025) that is constructed from the slow-roll inflation parameters, and fits the new high-multipole CMB observations reported by the Planck and ACT collaborations for k ∼ -0.03. Rather than use the Jordan frame in which these cosmologies have been formulated, the present investigation is undertaken in the Einstein frame, which is obtained from the Jordan frame by a conformal transformation Ω, which introduces the scalaron field ϕ = κ -1 √(6)lnΩ. The motion of the scalaron is studied for the slow roll inflationary era using analytic approximate solutions, from which expressions are obtained for R and hence η . Some of the expressions for the Starobinsky model were obtained first by Motohashi and Nishizawa (2012) but the expressions for the power-law model of Odintsov and Oikonomou and its generalization are new. Although the values of η calculated for both models are quite close to the observed value, they are based upon several assumptions that qualify these results. The asymmetries are generated during the slow roll inflation era and, for values generated N e-folds before the end of inflation, must be diluted by the factor Δ ≲ 10 e -3 N , making agreement with observation difficult to achieve unless baryogenesis occurs at the end of the inflation era. The calculations also assume rapid thermalization of the plasma of the gravitationally produced particles, and that the interaction rate of the baryon-violating processes is greater than the expansion rate of the Universe until decoupling. The present calculations strongly violate these assumptions and would require significant variation of input parameters, which is excluded by CMB observations, to obtain agreement. A complete treatment of thermalization of the gravitationally produced plasma, baryon-number violating freeze-out, and entropy dilution is required to establish whether or not gravitational baryogenesis during slow-roll inflation is a viable mechanism for producing the observed baryon asymmetry of the Universe for these inflationary f ( R ) cosmologies.