Big Bang nucleosynthesis and baryogenesis in power-law $f(R)$ gravity: Revised constraints from the semianalytical approach

Abstract: In this paper we investigate the primordial nucleosynthesis in $\mathscr{L}=\varepsilon^{2-2\beta}R^\beta+{16\pi}m_P^{-2}\mathscr{L}_m$ gravity, where $\varepsilon$ is a constant balancing the dimension of the field equation, and $1<\beta<(4+\sqrt{6})/5$ for the positivity of energy density and temperature. From the semianalytical approach, the influences of $\beta$ to the decoupling of neutrinos, the freeze-out temperature and concentration of nucleons, the opening of deuterium bottleneck, and the $^4$He abundance are all extensively analyzed; then $\beta$ is constrained to $1<\beta<1.05$ for $\varepsilon=1$ [1/s] and $1<\beta<1.001$ for $\varepsilon=m_P$ (Planck mass). Supplementarily from the empirical approach, abundances of the lightest elements (D, $^4$He, $^7$Li) are computed by the model-independent best-fit formulae for nonstandard primordial nucleosynthesis, and we find the constraint $1< \beta \leq 1.0505$ which corresponds to the extra number of neutrino species $0< \Delta N_\nu^{\text{eff}} \leq 0.6365$; also, the $^7$Li abundance problem cannot be solved by $\mathscr{L}=\varepsilon^{2-2\beta}R^\beta+{16\pi}m_P^{-2}\mathscr{L}_m$ gravity for this domains of $\beta$. Finally, the consistency with the mechanism of gravitational baryogenesis is estimated.