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.
Categories: astro-ph.CO gr-qc hep-ph