  During a failed core-collapse supernova, the protoneutron star eventually
collapses under its own gravitational field and forms a black hole. This
collapse happens quickly, on the dynamical time of the protoneutron star,
$\lesssim$0.5 ms. During this collapse, barring any excessive rotation, the
entire protoneutron star is accreted into the newly formed black hole. The main
source of neutrinos is now removed and the signal abruptly shuts off over this
formation timescale. However, while the source of neutrinos is turned off, the
arrival times at an Earth-based detector will depend on the neutrino path. We
show here that a modest amount of neutrinos, emitted just prior to the black
hole forming, scatter on the infalling material into our line of sight and
arrive after the formation of the black hole, up to 15 ms in our model. This
neutrino echo, which we characterize with Monte Carlo simulations and analytic
models, has a significantly higher average energy (upwards of $\sim$ 50 MeV)
compared to the main neutrino signal, and for the canonical failed supernova
explored here, is likely detectable in $\mathcal{O}$(10 kT) supernova neutrino
detectors for Galactic failed supernovae. The presence of this signal is
important to consider if using black hole formation as a time post for
triangulation or the post black hole timing profile for neutrino mass
measurements. On its own, it can also be used to characterize or constrain the
structure and nature of the accretion flow.
