  In an earlier paper (Cen 2003) we pointed out the strong likelihood for the
universal reionization to occur twice, giving rise to a larger Thomson optical
depth. Here we perform a more focused analysis of the Thomson optical depth in
light of the WMAP observations. While the current uncertainties on the observed
Thomson optical depth are still relatively large, with tau_e=0.17+-0.04 (68%)
(Kogut et al 2003), important implications on Pop III star formation processes
at high redshift can already be inferred. We are able to draw four conclusions:
(1) in the absence of a top-heavy initial stellar mass function (IMF) for Pop
III metal-free stars and without a dramatic upturn in the star formation
efficiency and ionizing photon escape fraction at high redshift (z>6), we find
tau_e =< 0.09; (2) with a top-heavy IMF for the Pop III metal-free stars and
plausible star formation efficiency and ionizing photon escape fraction, it is
expected that tau_e =< 0.12; (3) it is possible to reach tau_e = 0.15, if the
metal enrichment efficiency of the intergalactic medium by Pop III stars is
very low thus Pop III era is prolonged; (4) to reach tau_e >= 0.17 requires
either of the following two conditions: the cosmological model power index n is
positively tilted to n >= 1.03, Pop III star formation in minihalos with
molecular hydrogen cooling has an efficiency c_*(H_2,III)>0.01 (with ionizing
photon escape fraction greater than 30%). If the current observed value of
Thomson optical depth withstands future data, we will have strong observational
evidence that Pop III stars are massive and their formation efficiency may be
much higher than current theoretical works suggest. Alternatively, there may be
unknown, non-stellar ionizing sources at very high redshift.
