Gamma-ray bursts as probes of high-redshift Lyman-alpha emitters and radiative transfer models

We present the updated census and statistics of Lyman-$\alpha$ emitting long gamma-ray bursts host galaxies (LAE-LGRBs). We investigate the properties of a sub-sample of LAE-LGRBs and test the shell model commonly used to fit Lyman-$\alpha$ (Ly$\alpha$) emission line spectra. Among the LAE-LGRBs detected to date, we select a golden sample of four LAE-LGRBs allowing us to retrieve information on the host galaxy properties and of its interstellar medium gas. We fit their Ly$\alpha$ spectra using the shell model, and constrain its parameters with the observed values. From the comparison of the statistics and properties of LAE-LGRBs to those of LAE samples in the literature, we find evidences of Ly$\alpha$ suppression in dusty systems, and a fraction of LAE-LGRBs among the overall LGRB hosts lower than that found for Lyman-break galaxy (LBG) samples at similar redshift range. However, we find that LAE-LGRBs are representative of Ly$\alpha$ emission from the bulk of UV-selected galaxies at z~2. We find that the golden sample of LAE-LGRBs are complex systems characterized by multiple emission blobs and by signs of possible galaxy interactions. The fitting procedure fails in recovering the HI column densities (NHI) measured from the afterglow spectra, and the other properties described by the shell-model parameters in the cases with very high NHI. The afterglows of most LGRBs and LAE-LGRBs show high NHI, implying that statistically the bulk of Ly$\alpha$ photons expected to be produced by massive stars in the star-forming region hosting the GRB will be surrounded by such opaque lines of sight. We interpret our results in the context of more sophisticated models and of different dominant Ly$\alpha$ emitting regions. We also compare LAE-LGRBs to LAE Lyman continuum (LyC) leakers in the literature in terms of properties identified as possible indirect indicators of LyC leakage. [Abridged]