Constraining the reionization history with CMB and spectroscopic
  observations
  We investigate the constraints on the reionization history of the Universe
from a joint analysis of the cosmic microwave background and neutral hydrogen
fraction data. The $\tanh$ parametrization and principal component analysis
methods are applied to the reionization history respectively. The commonly used
$\tanh$ parametrization is oversimplistic when the neutral hydrogen fraction
data are taken into account. Using the principal component analysis method, the
reconstructed reionization history is consistent with the neutral hydrogen
fraction data. With the principal component analysis method, we reconstruct the
neutral hydrogen fraction at $z=9.75$ as $x_{\text{HI}}=0.69^{+0.30}_{-0.32}$
for $6<z<20$ range reconstruction, and $x_{\text{HI}}=0.76^{+0.22}_{-0.27}$ for
$6<z<30$ range reconstruction. These results suggest that the Universe began to
reionize at redshift no later than $z=10$ at a $95\%$ confidence level.
