  We study the impact of the existence of an eV-mass scale sterile
neutrino---with parameters in the ballpark of what is required to fit the
laboratory anomalies---on the early time profile of the electron neutrino and
antineutrino fluxes associated to a core-collapse supernova (SN). In
particular, we focus on the universal feature of neutronization burst expected
in the first tens of ms of the signal: provided that a detector with sufficient
sensitivity is available, it is well-known that in the 3 neutrino framework the
detection of the neutronization burst in neutrino channel would signal inverted
mass hierarchy. This conclusion is dramatically altered in the presence of a
sterile neutrino: we study here both analytically and numerically the region in
parameter space where this characteristic signal disappears, mimicking normal
hierarchy expectations. Conversely, the detection of a peak consistent with
expectations for inverted mass hierarchy would exclude the existence of a
sterile state over a much wider parameter space than what required by
laboratory anomalies fits, or even probed by detectors coming on-line in the
near future. Additionally, we show the peculiar alteration in the energy-time
double differential flux, with a delayed peak appearing for kinematical
reasons, which might offer a remarkable signature in case of favorable
parameters and for a high statistics detection of a Galactic SN. We also
comment on additional potentially interesting effects in the electron
antineutrino channel, if more than one angle in the active-sterile sector is
non-vanishing. As an ancillary result that we derived in the technical
resolution of the equations, in an appendix we report the Cayley-Hamilton
formalism for the evolution of a four neutrino system in matter, generalizing
existing results in the literature.
