Future Constraints on the Reionization History and the Ionizing Sources
  from Gamma-ray Burst Afterglows
  We forecast the reionization history constraints, inferred from Lyman-alpha
damping wing absorption features, for a future sample of $\sim 20$ $z \geq 6$
gamma-ray burst (GRB) afterglows. We describe each afterglow spectrum by a
three-parameter model. First, L characterizes the size of the ionized region
(the "bubble size") around a GRB host halo. Second, $\langle{x_{\rm
HI}\rangle}$ is the volume-averaged neutral fraction outside of the ionized
bubble around the GRB, which is approximated as spatially uniform. Finally,
$N_{\mathrm{HI}}$ denotes the column-density of a local damped Lyman-alpha
absorber (DLA) associated with the GRB host galaxy. The size distribution of
ionized regions is extracted from a numerical simulation of reionization, and
evolves strongly across the Epoch of Reionization (EoR). The model DLA column
densities follow the empirical distribution determined from current GRB
afterglow spectra. We use a Fisher matrix formalism to forecast the
$\langle{x_{\rm HI}(z)\rangle}$ constraints that can be obtained from follow-up
spectroscopy of afterglows with SNR = 20 per R=3,000 resolution element at the
continuum. We find that the neutral fraction may be determined to better than
10-15\% (1-$\sigma$) accuracy from this data across multiple independent
redshift bins at $z \sim 6-10$, spanning much of the EoR, although the
precision degrades somewhat near the end of reionization. A more futuristic
survey with $80$ GRB afterglows at $z \geq 6$ can improve the precision here by
a factor of $2$ and extend measurements out to $z \sim 14$. We further discuss
how these constraints may be combined with estimates of the escape fraction of
ionizing photons, derived from the DLA column density distribution towards GRBs
extracted at slightly lower redshift. This combination will help in testing
whether we have an accurate census of the sources that reionized the universe.
