We discuss Dirac neutrinos whose right-handed component $ u_R$ has new interactions that may lead to a measurable contribution to the effective number of relativistic neutrino species $N_{rm eff}$. We aim at a model-independent and comprehensive study on a variety of possibilities. Processes for $ u_R$-genesis from decay or scattering of thermal species, with spin-0, spin-1/2, or spin-1 initial or final states are all covered. We calculate numerically and analytically the contribution of $ u_R$ to $N_{rm eff}$ primarily in the freeze-in regime, since the freeze-out regime has been studied before. While our approximate analytical results apply only to freeze-in, our numerical calculations work for freeze-out as well, including the transition between the two regimes. Using current and future constraints on $N_{rm eff}$, we obtain limits and sensitivities of CMB experiments on masses and couplings of the new interactions. As a by-product, we obtain the contribution of Higgs-neutrino interactions, $Delta N_{rm eff}^{rm SM} approx 7.5times10^{-12}$, assuming the neutrino mass is 0.1 eV and generated by the standard Higgs mechanism.
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