Neutrino non-standard interactions meet precision measurements of
  $N_{\rm eff}$
  The number of relativistic species, $N_{\rm eff}$, has been precisely
calculated in the standard model, and would be measured to the percent level by
CMB-S4 in future. Neutral-current non-standard interactions would affect
neutrino decoupling in the early Universe, thus modifying $N_{\rm eff}$. We
parameterize those operators up to dimension-7 in the effective field theory
framework, and then provide a complete, generic and analytical dictionary for
the collision term integrals. From precision measurements of $N_{\rm eff}$, the
most stringent constraint is obtained for the dimension-6 vector-type
neutrino-electron operator, whose scale is constrained to be above about 195
(331) GeV from Planck (CMB-S4). We find our results complementary to other
experiments like neutrino coherent scattering, neutrino oscillation, collider,
and neutrino deep inelastic scattering experiments.
