Gravitational Baryogenesis in $f(R)$ Cosmologies

Abstract

The generation of a baryon-antibaryon asymmetry in the Universe via gravitational baryogenesis is investigated for 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 $k \sim -0.03$. The present investigation is undertaken in the Einstein frame. The motion of the scalaron is studied for the slow roll inflationary era using analytic approximate solutions obtained from its potential and, from these solutions, analytic expressions for the Ricci scalar, its time derivative, the Hubble parameter and the scale factor of the Universe. These 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. The calculated values of the baryon asymmetry factor $η$ for the Starobinsky model vary from $(1.05 - 1.46) \times 10^{-11}$, and for the power-law model, from $(1.06 - 1.53) \times 10^{-11}$. The power-law values depend upon the unknown fitting parameters $c_{1}$ and $c_{2}$, and a future fit of the Odintsov and Oikonomou model to data could yield enhanced values. The values for $η$ depend upon a mass parameter $M_{\ast}$ which is expected to be of the order of $M_{Pl}=2.435 \times 10^{18}$ GeV. The values of $η$ for both models have been calculated for $M_{\ast}=M_{Pl}$ and are quite close to the observed value $η= 8.65 \times 10^{-11}$. Since $η\propto (M/M_{\ast})^{2}$, reducing $M_{\ast}$ slightly from $M_{Pl}$ to $0.4 M_{Pl}$ would bring the calculated values into agreement with the observed value.