Context. The reionisation of the Universe is a process that is thought to have ended around z ~ 6, as inferred from spectroscopy
of distant bright background sources, such as quasars (QSO) and gamma-ray burst (GRB) afterglows. Furthermore, spectroscopy
of a GRB afterglow provides insight in its host galaxy, which is often too dim and distant to study otherwise.
For the Swift GRB 130606A at z = 5.913 we have obtained a high S/N spectrum covering the full optical and near-IR wavelength
region at intermediate spectral resolution with VLT/X-Shooter. We aim to measure the degree of ionisation of the intergalactic
medium (IGM) between z = 5.02−5.84 and to study the chemical abundance pattern and dust content of its host galaxy.
We estimated the UV continuum of the GRB afterglow using a power-law extrapolation, then measured the flux decrement due to
absorption at Lyα,β, and γ wavelength regions. Furthermore, we fitted the shape of the red damping wing of Lyα. The hydrogen
and metal absorption lines formed in the host galaxy were fitted with Voigt profiles to obtain column densities. We investigated
whether ionisation corrections needed to be applied.
Results. Our measurements of the Lyα-forest optical depth are
consistent with previous measurements of QSOs, but have a much smaller uncertainty. The analysis of the red damping wing yields
a neutral fraction xH i< 0.05 (3σ). We obtain column density measurements of H, Al, Si, and Fe; for C, O, S and Ni we obtain
limits. The ionisation due to the GRB is estimated to be negligible (corrections <0.03 dex), but larger corrections may
apply due to the pre-existing radiation field (up to 0.4 dex based on sub-DLA studies). Assuming that [ Si/Fe ] = +0.79 ±
0.13 is due to dust depletion, the dust-to-metal ratio is similar to the Galactic value.
Conclusions. Our measurements
confirm that the Universe is already predominantly ionised over the redshift range probed in this work, but was slightly more
neutral at z> 5.6. GRBs are useful probes of the ionisation state of the IGM in the early Universe, but because of internal
scatter we need a larger statistical sample to draw robust conclusions. The high [Si/Fe] in the host can be due to dust depletion,
α-element enhancement, or a combination of both. The very high value of [ Al/Fe ] = 2.40 ± 0.78 might be due to a proton capture
process and is probably connected to the stellar population history. We estimate the host metallicity to be −1.7 < [ M/H
] < −0.9 (2%−13% of solar).