Reconstruction of Reionization History through Dispersion Measure of
  Fast Radio Bursts
  In this paper, we study the evolution of the ionization fraction $x_e(z)$
during the epoch of reionization by using the dispersion measurements (DMs) of
fast radio bursts (FRBs). Different from the previous studies, here we turn to
consider the large-scale clustering information of observed DMs of FRB catalog,
which only needs the rough redshift distribution, instead of the exact redshift
information of each FRB. Firstly, we consider the instantaneous
``\texttt{tanh}'' model for $x_e(z)$ and find that including the
auto-correlation information of the mock catalog, about $10^4$ FRBs with the
intrinsic DM scatter of 100 $\rm pc/cm^3$ spanning 20\% of all sky, could
significantly improve the constraint on the width $\Delta_z$ of the model, when
comparing with that from the CMB data alone. The evolution shape of the
ionization fraction will be tightly narrowed, namely the duration of the epoch
of reionization has been shrunk, $z_{\rm dur}<2.24$ (95\% C.L.). Furthermore,
we also use another redshift-asymmetric reionization model and obtain that the
FRB mock catalog could measure the ionization fraction at $z=6$ precisely with
the $1\sigma$ error $\Delta x_e(z=6)=0.012$, which means that the large-scale
clustering information of observed DMs of FRB catalog is very sensitive to the
ionization fraction of the end of reionization epoch. We conclude that the
observation of high-redshift FRBs could be a complementary probe to study the
reionization history in the future.
