Primordial Black Holes Evaporating before Big Bang Nucleosynthesis

Primordial black holes (PBHs) formed from the collapse of density fluctuations provide a unique window into the physics of the early Universe. Their evaporation through Hawking radiation around the epoch of Big Bang nucleosynthesis (BBN) can leave measurable imprints on the primordial light-element abundances. In this work, we analyze in detail the effects of PBHs evaporating before BBN, with various intermediate steps understood analytically, and obtain the BBN constraint on PBHs within a transparent and reproducible framework. We find that, to produce observable effects on BBN, the PBH mass must exceed $10^{9}$ g, a threshold higher than that reported in some earlier studies. Slightly above $10^{9}$ g, the BBN sensitivity rapidly increases with the mass and then decreases, with the turning point occurring at $2\times10^{9}$ g. For PBHs in the mass range $[10^{9},\ 10^{10}]$ g, current measurements of BBN observables set an upper bound on the initial mass fraction parameter $\beta$ ranging from $10^{-17}$ to $10^{-19}$. To facilitate future improvements, we make our code publicly available, enabling straightforward incorporation of updated nuclear reaction rates, particle-physics inputs, and cosmological data.