  The existence of prolonged radiation domination prior to the Big Bang
Nucleosynthesis (BBN), starting just after the inflationary epoch, is not yet
established unanimously. If instead, the universe undergoes a non-standard
cosmological phase, it will alter the Hubble expansion rate significantly and
may also generate substantial entropy through non-adiabatic evolution. This
leads to a thumping impact on the properties of relic species decoupled from
the thermal bath before the revival of the standard radiation domination in the
vicinity of the BBN. In this work, considering the Dirac nature of neutrinos,
we have studied decoupling of ultra-relativistic right-handed neutrinos
($\nu_R$s) in presence of two possible non-standard cosmological phases. While
in both cases we have modified Hubble parameters causing faster expansions in
the early universe, one of the situations predicts a non-adiabatic evolution
and thereby a slower redshift of the photon temperature due to the expansion.
Considering the most general form of the collision term with Fermi-Dirac
distribution and Pauli blocking factors, we have solved the Boltzmann equation
numerically to obtain $\Delta{\rm N}_{\rm eff}$ for the three right-handed
neutrinos. We have found that for a large portion of parameter space, the
combined effect of early decoupling of $\nu_R$ as well as the slower redshift
of photon bath can easily hide the signature of right-handed neutrinos, in
spite of precise measurement of $\Delta{\rm N}_{\rm eff}$, at the next
generation CMB experiments like CMB-S4, SPT-3G etc. This however will not be
applicable for the scenarios with only fast expansion.
