  Deviations from unitarity in the three-neutrino mixing canonical picture are
expected in many physics scenarios beyond the Standard Model. The mixing of new
heavy neutral leptons with the three light neutrinos would in principle modify
the strength and flavour structure of charged-current and neutral-current
interactions with matter. Non-unitarity effects would therefore have an impact
on the neutrino decoupling processes in the early Universe and on the value of
the effective number of neutrinos, $N_{\rm eff}$. We calculate the cosmological
energy density in the form of radiation with a non-unitary neutrino mixing
matrix, addressing the possible interplay between parameters. Highly accurate
measurements of $N_{\rm eff}$ from forthcoming cosmological observations can
provide independent and complementary limits on the departures from unitarity.
For completeness, we relate the scenario of small deviations from unitarity to
non-standard neutrino interactions and compare the forecasted constraints to
other existing limits in the literature.
