  Pop III stars are typically massive stars of primordial composition forming
at the centers of the first collapsed dark matter structures. Here we estimate
the optimal X-ray emission in the early universe for promoting the formation of
Pop III stars. This is important in determining the number of dwarf galaxies
formed before reionization and their fossils in the local universe, as well as
the number of intermediate-mass seed black holes. A mean X-ray emission per
source above the optimal level reduces the number of Pop III stars because of
the increased Jeans mass of the intergalactic medium (IGM), while a lower
emission suppresses the formation rate of H2 preventing or delaying star
formation in dark matter minihalos above the Jeans mass. The build up of the H2
dissociating background is slower than the X-ray background due to the
shielding effect of resonant hydrogen Lyman lines. Hence, the nearly
unavoidable X-ray emission from supernova remnants of Pop III stars is
sufficient to boost their number to few tens per comoving Mpc^3 by redshift
z~15. We find that there is a critical X-ray to UV energy ratio emitted per
source that produces a universe where the number of Pop III stars is largest:
400 per comoving- Mpc^3. This critical ratio is very close to the one provided
by 20-40 M_sun Pop III stars exploding as hypernovae. High mass X-ray binaries
in dwarf galaxies are far less effective at increasing the number of Pop III
stars than normal supernova remnants, we thus conclude that supernovae drove
the formation of Pop III stars.
