The unprecedentedly bright afterglow of Swift GRB 130606A at z = 5.91 gave us a unique opportunity to probe the reionization era by high precision analyses of the redward damping wing of Ly alpha absorption, but the reported constraints on the neutral hydrogen fraction (f_HI) in intergalactic medium (IGM) derived from spectra taken by different telescopes are in contradiction. Here we examine the origin of this discrepancy by analyzing the spectrum taken by VLT with our own analysis code previously used to fit the Subaru spectrum. Though the VLT team reported no evidence for IGM HI using the VLT spectrum, we confirmed our previous result of preferring non-zero IGM HI (the best-fit f_HI ~ 0.06, when IGM HI extends to the GRB redshift). The fit residuals of the VLT spectrum by the model without IGM HI show the same systematic trend as the Subaru spectrum. We consider that the likely origin of the discrepancy between the two teams is the difference of the wavelength ranges adopted in the fittings; our wavelength range is wider than that of the VLT team, and also we avoided the shortest wavelength range of deep Ly alpha absorption (lambda_obs < 8426 A), because this region is dominated by HI in the host galaxy and the systematic uncertainty about host HI velocity distribution is large. We also study the sensitivity of these results to the adopted Ly alpha cross section formulae, ranging from the classical Lorentzian function to the most recent one taking into account fully quantum mechanical scattering. It is found that the preference for non-zero IGM HI is robust against the choice of the cross section formulae, but it is quantitatively not negligible and hence one should be careful in future analyses.
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