Consistent $N_{\rm eff}$ fitting in big bang nucleosynthesis analysis

The effective number of neutrino species, $N_{\rm eff}$, serves as a key fitting parameter extensively employed in cosmological studies. In this work, we point out a fundamental inconsistency in the conventional treatment of $N_{\rm eff}$ in big bang nucleosynthesis (BBN), particularly regarding its applicability to new physics scenarios where $\Delta N_{\rm eff}$, the deviation of $N_{\rm eff}$ from the standard BBN prediction, is negative. To ensure consistent interpretation, it is imperative to either restrict the allowed range of $N_{\rm eff}$ or systematically adjust neutrino-induced reaction rates based on physically motivated assumptions. As a concrete example, we consider a simple scenario in which a negative $\Delta N_{\rm eff}$ arises from entropy injection into the electromagnetic sector due to the decay of long-lived particles after neutrino decoupling. This process dilutes the neutrino density and suppresses the rate of neutrino-driven neutron-proton conversion. Under this assumption, we demonstrate that the resulting BBN constraints on $N_{\rm eff}$ deviate significantly from those obtained by the conventional, but unphysical, extrapolation of dark radiation scenarios into the $\Delta N_{\rm eff} < 0$ regime.