$N_{\rm eff}$ as a new physics probe in the precision era of cosmology

We perform a global fit to the electroweak vertices and 4-fermion operators of the standard model effective field theory in this work using $N_{\rm eff}$ from cosmological probes, as well as data sets from colliders and low-energy experiments. We find $N_{\rm eff}$, both its current measurement and future projections, can only marginally improve the fit in both the flavor universal and the most general flavor scenarios. The resulting $1\sigma$ bound on $N_{\rm eff}$ is significantly improved from the global fit and becomes comparable to its current theoretical uncertainty, such that the latter will become important for this study at next generation experiments like future lepton colliders. $N_{\rm eff}$ from the global fit is also adopted to predict the primordial helium abundance $Y_P$, which significantly reduces the parameter space on the $Y_P$-$N_{\rm eff}$ plane. Through error propagation, we also conclude that reducing the experimental uncertainty of $Y_P$ from metal-poor galaxies down below 0.2% could play an important role in deepening our understanding on the free neutron lifetime anomaly.