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 $(2.0-5.8)%$. 
Yun-Wei Yu, K. S. Cheng, M. C. Chu, S. Yeung
astro-ph.CO astro-ph.HE