Cosmic histories of star formation and reionization: An analysis with a
  power-law approximation
  With a simple power-law approximation of high-redshift ($\gtrsim3.5$) star
formation history, i.e., $\dot{\rho}_*(z)\propto [(1+z)/4.5]^{-\alpha}$, we
investigate the reionization of intergalactic medium (IGM) and the consequent
Thomson scattering optical depth for cosmic microwave background (CMB) photons.
A constraint on the evolution index $\alpha$ is derived from the CMB optical
depth measured by the {\it Wilkinson Microwave Anisotropy Probe} (WMAP)
experiment, which reads $\alpha\approx2.18\lg{\mathscr{N}_{\gamma}}-3.89$,
where the free parameter $\mathscr{N}_\gamma$ is the number of the escaped
ionizing ultraviolet photons per baryon. Moreover, the redshift for full
reionization, $z_f$, can also be expressed as a function of $\alpha$ as well as
$\mathscr{N}_{\gamma}$. By further taking into account the implication of the
Gunn-Peterson trough observations to quasars for the full reionization
redshift, i.e., $6\lesssim z_f \lesssim7$, we obtain
$0.3\lesssim\alpha\lesssim1.3$ and $80\lesssim\mathscr{N}_{\gamma}\lesssim230$.
For a typical number of $\sim4000$ of ionizing photons released per baryon of
normal stars, the fraction of these photons escaping from the stars, $f_{\rm
esc}$, can be constrained to within the range of $(2.0-5.8)%$.
