From my studies, I remember that the quantum effects relative to the bosonic or fermonic nature of the particles play a role only in the conditions of degenerate gas: when the plasma is very dense and the temperature is very low, the bosons condensate and the fermions express their condition of being subject to the Pauli exclusion principle. In these conditions we should consider the distribution of fermions to be described by the Fermi-Dirac distribution rather than by the Maxwell-Boltzmann one.
Here (https://physicsworld.com/a/a-fermi-gas-of-atoms/) they say that the quantum behaviour emerges gradually as the fermion gas is cooled below the Fermi temperature $T_F = E_F/k_B = 1 \mu$K $\sim 8.62 \times 10^{-11}$ eV, for atomic gases.
If I am working in the conditions of the early universe, in a plasma at a temperature in the range (10 GeV, 0.1 MeV) and with zero chemical potential, I would say that is more than justified in those conditions to approximate the distribution of fermions in equilibrium in the plasma (for example neutrinos) with the Maxwell-Boltzmann distribution and to forget about the quantum effects related to the fermion's nature. However, I keep finding in papers that study the production of sterile neutrinos in the early universe (e.g. https://arxiv.org/abs/2005.03681 and https://arxiv.org/abs/2005.01629) in the mentioned conditions, the Fermi-Dirac distribution instead of the Maxwell-Boltzmann and also other quantum effects.
My questions are then:
why is not correct to do the calculations only with the Maxwell-Boltzmann distribution and forgetting quantum effects?
is the Fermi temperature $T_F$ larger than $1 \mu$K in this case because the primordial plasma is not an atomic gas?
is the density so high there to induce the condition of degenerate fermion gas and therefore to require the use of Fermi-Dirac distribution?
