  We discuss Dirac neutrinos whose right-handed component $\nu_R$ 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
$\nu_R$-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 $\nu_R$ 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, $\Delta N_{\rm eff}^{\rm SM} \approx
7.5\times10^{-12}$, assuming the neutrino mass is 0.1 eV and generated by the
standard Higgs mechanism.
