  If the first (PopIII) stars were very massive, their final fate is to
collapse into very massive black holes. Once a proto-black hole has formed into
the stellar core, accretion continues through a disk. It is widely accepted,
although not confirmed, that magnetic fields drive an energetic jet which
produces a burst of TeV neutrinos by photon-meson interaction, and eventually
breaks out of the stellar envelope appearing as a Gamma Ray Burst (GRB). Based
on recent numerical simulations and neutrino emission models, we predict the
expected neutrino diffuse flux from these PopIII GRBs and compare it with the
capabilities of present and planned detectors as AMANDA and IceCube. If beamed
into 1% of the sky, we find that the rate of PopIII GRBs is $\le 4 \times 10^6$
yr$^{-1}$. High energy neutrinos from PopIII GRBs could dominate the overall
flux in two energy bands [$10^4 - 10^5$] GeV and [$10^5 - 10^6$] GeV of
neutrino telescopes. The enhanced sensitivities of forthcoming detectors in the
high-energy band (AMANDA-II, IceCube) will provide a fundamental insight on the
characteristic explosion energies of PopIII GRBs and will constitute a unique
probe of the the Initial Mass Function (IMF) of the first stars and of the
redshift $z_f$ marking the metallicity-driven transition from a top-heavy to a
normal IMF. The current upper limit set by AMANDA-B10 implies that such
transition must have occurred not later than $z_f =9.2$ for the most plausible
jet energies. Based on such results, we speculate that PopIII GRBs, if not
chocked, could be associated with a new class of events detected by BeppoSax,
the Fast X-ray Transient (FXTs), which are bright X-ray sources, with peak
energies in the 2-10 keV band and durations between 10-200 s.

Astrophysics;High Energy Physics - Phenomenology