Neutral fraction of hydrogen in the intergalactic medium surrounding high-redshift gamma-ray burst 210905A

The Epoch of Reionization (EoR) is a key period of cosmological history in which the intergalactic medium (IGM) underwent a major phase change from being neutral to almost completely ionized. Gamma-ray bursts (GRBs) are luminous and unique probes of their environments that can be used to study the timeline for the progression of the EoR. Here, we present a detailed analysis of the European Southern Observatory Very Large Telescope X-shooter spectrum of GRB 210905A, which resides at a redshift of $z\sim 6.3$. We focus on estimating the fraction of neutral hydrogen, $x_{\rm H \, {\small I}}$, on the line of sight to the host galaxy of GRB 210905A by fitting the shape of the Lyman-$\alpha$ damping wing of the afterglow spectrum. The X-shooter spectrum has a high signal-to-noise ratio, but the complex velocity structure of the host galaxy limits the precision of our conclusions. The statistically preferred model suggests a low neutral fraction with a 3$\sigma$ upper limit of $x_{\rm H \, {\small I}} \lesssim 0.15$ or $x_{\rm H \, {\small I}} \lesssim 0.23$, depending on the absence or presence of an ionized bubble around the GRB host galaxy, indicating that the IGM around the GRB host galaxy is mostly ionized. We discuss complications in current analyses and potential avenues for future studies of the progression of the EoR and its evolution with redshift.