If neutrinos are Dirac particles the existence of light right-handed neutrinos $ u_{R}$ is implied. Those would contribute to the effective number of relativistic neutrino species $N_{{rm eff}}$ in the early Universe. With pure standard model interactions, the contribution is negligibly small. In the presence of new interactions, however, the contribution could be significantly enhanced. We consider the most general effective four-fermion interactions for neutrinos (scalar, pseudo-scalar, vector, axial-vector and tensor), and compute the contribution of right-handed neutrinos to $N_{{rm eff}}$. Taking the Planck 2018 measurement of $N_{{rm eff}}$, strong constraints on the effective four-fermion coupling are obtained, corresponding to interaction strengths of $10^{-5}sim10^{-3}$ in units of the Fermi constant. This translates in new physics scales of up to 43 TeV and higher. Future experiments such as CMB-S4 can probe or exclude the existence of effective 4-neutrino operators for Dirac neutrinos. Ways to avoid this conclusion are discussed.
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