  The detection of a sterile neutrino could constitute the first observation of
a particle that could have been produced before Big-Bang Nucleosynthesis (BBN),
and could provide information about the yet untested pre-BBN era. The
cosmological evolution in this era could be drastically different than
typically assumed in what constitutes the standard cosmology, as happens in a
variety of motivated particle models. In this work we assess the sensitivity to
different pre-BBN cosmologies in which entropy is conserved of 0.01 eV to 1 MeV
mass sterile neutrinos produced in the early Universe via resonant
active-sterile oscillations, which requires a large lepton asymmetry. We
identify mass ranges where it is possible to have two populations of the same
sterile neutrino, one with a colder and one with a hotter momentum spectra,
which is in principle an observable effect. Furthermore, we show the regions in
mass and mixing where fully resonant production (i.e. simultaneously coherent
and adiabatic) can occur. We find that in several of the cosmologies we
consider, including the standard one, for a lepton asymmetry larger than
$\sim10^{-4}$ fully resonantly produced sterile neutrinos in the eV-mass range
can evade all cosmological constraints.
