Inferring the IGM Neutral Fraction at z ~ 6-8 with Low-Luminosity Lyman
  Break Galaxies
  We present a Bayesian inference on the neutral hydrogen fraction of the
intergalactic medium (IGM), $x_{\textrm{HI}}$, at $z \sim$ 6-8 using the
properties of Lyman break galaxies during the Epoch of Reionization. We use
large samples of LBG candidates at $5.5 \leq z \leq 8.2$ with spectroscopy from
Keck/DEIMOS and Keck/MOSFIRE. For each galaxy, we incorporate either the
Lyman-$\alpha$ equivalent width (EW) for detections or the EW limit spectrum
for nondetections to parameterize the EW distribution at various ultraviolet
brightnesses for a given redshift. Using our reference sample of galaxy
candidates from the ionized universe at $z$ $\sim$ 6.0, we are able to infer
$x_{\textrm{HI}}$ at two redshifts: $z$ $\sim$ 6.7 and $z$ $\sim$ 7.6. This
work includes intrinsically faint, gravitationally lensed galaxies at $z$
$\sim$ 6.0 in order to constrain the intrinsic faint-end Ly$\alpha$ EW
distribution and provide a comparable population of galaxies to counterparts in
our sample that are at higher redshift. The inclusion of faint galaxy
candidates, in addition to a more sophisticated modelling framework, allows us
to better isolate effects of the interstellar medium and circumgalactic medium
on the observed Lyman-$\alpha$ distribution from those of the IGM. We infer an
upper limit of $x_{\textrm{HI}}$ $\leq$ 0.25 at $z$ = 6.7 $\pm$ 0.2 and a
neutral fraction of $x_{\textrm{HI}}$ = $0.83^{+0.08}_{-0.11}$ at $z$ =
7.6$\pm$ 0.6, both within 1$\sigma$ uncertainty, results which favor a
moderately late and rapid reionization.
