  A laboratory detection of a sterile neutrino could provide the first
indication of the evolution of the Universe before Big-Bang Nucleosynthesis
(BBN), an epoch yet untested. Such "visible" sterile neutrinos are observable
in upcoming experiments such as KATRIN/TRISTAN and HUNTER in the keV mass range
and PTOLEMY and others in the eV mass range. A set of standard assumptions is
typically made about cosmology before the temperature of the Universe was 5
MeV. However, non-standard pre-BBN cosmologies based on alternative assumptions
could arise in motivated theoretical models and are equally in agreement with
all existing data. We revisit the production of sterile neutrinos of mass 0.01
eV to 1 MeV in two examples of such models: scalar-tensor and low reheating
temperature pre-BBN cosmologies. In both of them, the putative 3.5 keV X-ray
signal line corresponds to a sterile neutrino with a mixing large enough to be
tested in upcoming laboratory experiments. Additionally, the
cosmological/astrophysical upper limits on active-sterile neutrino mixings are
significantly weaker than in the standard pre-BBN cosmology, which shows that
these limits are not robust. For example, in the scalar-tensor case the
potential signal regions implied by the LSND and MiniBooNE short-baseline as
well as the DANSS and NEOS reactor experiments are entirely not bound by
cosmological restrictions. Our work highlights that sterile neutrinos may
constitute a sensitive probe of the pre-BBN epoch.
