  We construct a theoretical framework to study Population III (Pop III) star
formation in the post-reionization epoch ($z\lesssim 6$) by combining
cosmological simulation data with semi-analytical models. We find that due to
radiative feedback (i.e. Lyman-Werner and ionizing radiation) massive haloes
($M_{\rm halo}\gtrsim 10^{9}\ \rm M_{\odot}$) are the major ($\gtrsim 90$%)
hosts for potential Pop III star formation at $z\lesssim 6$, where dense
pockets of metal-poor gas may survive to form Pop III stars, under inefficient
mixing of metals released by supernovae. Metal mixing is the key process that
determines not only when Pop III star formation ends, but also the total mass,
$M_{\rm PopIII}$, of \textit{active} Pop III stars per host halo, which is a
crucial parameter for direct detection and identification of Pop III hosts.
Both aspects are still uncertain due to our limited knowledge of metal mixing
during structure formation. Current predictions range from early termination at
the end of reionization ($z\sim 5$) to continuous Pop III star formation
extended to $z=0$ at a non-negligible rate $\sim 10^{-7}\ \rm M_{\odot}\
yr^{-1}\ Mpc^{-3}$, with $M_{\rm PopIII}\sim 10^{3}-10^{6}\ \rm M_{\odot}$.
This leads to a broad range of redshift limits for direct detection of Pop III
hosts, $z_{\rm PopIII}\sim 0.5-12.5$, with detection rates $\lesssim 0.1-20\
\rm arcmin^{-2}$, for current and future space telescopes (e.g. HST, WFIRST and
JWST). Our model also predicts that the majority ($\gtrsim 90$%) of the cosmic
volume is occupied by metal-free gas. Measuring the volume filling fractions of
this metal-free phase can constrain metal mixing parameters and Pop III star
formation.
