With a simple power-law approximation of high-redshift ( \gtrsim 3.5 ) 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 Wilkinson Microwave Anisotropy Probe ( WMAP ) experiment , which reads \alpha \approx 2.18 \lg { \mathscr { N } _ { \gamma } } -3.89 , where the free parameter \mathscr { N } _ { \gamma } is the number of the escaped ionizing ultraviolet photons per baryon .
At the same time , the redshift z _ { f } at which the IGM is fully ionized 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 } \lesssim 7 , we obtain 0.3 \lesssim \alpha \lesssim 1.3 and 80 \lesssim \mathscr { N } _ { \gamma } \lesssim 230 .
For a typical number of \sim 4000 of ionizing photons released per baryon of normal stars , the fraction of these photons escaping from the stars , f _ { esc } , can be constrained to within the range of ( 2.0 - 5.8 ) \% .