tokens: of this procedure at z = 6 (red) and z = 7 (blue) in Figure 4 as obtained by changing only the outflows (left) or changing only the column density (right). While both scenarios produce stacks that have consistent FWHM values with observations at z = 6, the changing outflows scenario yields also a consistent FWHM with z = 7 measurements. This also confirms that changing the column density predicts very high FWHM >400 km s −1 at z = 7. As quoted in the legend for changing the outflows scenario (left), z = 6 lines are broader than those at z = 7, and consistent with Pentericci et al. ( 2018 ). The lines are highly asymmetric with extended red wings. The blue peaks disappear due to the poor resolution. Figure 4. Example for stacks of randomly selected 52 mock spectra at z = 6 (red) and 19 mock spectra at z = 7 (blue) with f esc ( z = 7) ≈ 0.5 f esc ( z = 6) in agreement with the measured drop of X Ly α . We assembled the stacks following the method outlined in Pentericci et al. ( 2018 ) with spectral resolution R = 1390 (corresponding to Δ = 216 km s −1 ). All these spectra are either obtained by changing only outflows while keeping the column density N H I fixed (left) or changing only the column density N H I while keeping outflows fixed. In all cases, dust is kept fixed. In either scenario, the FWHM of stacks at z = 6 is consistent with values reported in Pentericci et al. ( 2018 , FWHM( z ∼ 6) = 300 ± 30). The FWHM of stacks obtained by changing outflows at z = 7 is also consistent with Pentericci et al. ( 2018 ) measurements (FWHM( z ∼ 7) = 220 ± 25) while those produced by changing the column density N H I are not, and hence indicating that changing the outflow can naturally lead to a higher Ly α escape as well as broader lines at lower redshifts as observations indicate. To quantify the width using the above recipe, we generate randomly 10,000 combination of stacks at z = 6 and z = 7 from the total number of model combinations (i.e., 143 and 201) in each scenario and compute their widths. We show the resulting width distribution at these redshifts in Figure 5 . Results from changing outflows and changing column density scenarios are presented by dashed and solid lines. Shaded red and blue areas show Pentericci et al. ( 2018 ) measurements at z ∼ 6 and z ∼ 7, respectively. From this exercise alone, we constrain the FWHM, using changing column density scenario, to 261.7 ± 14.3 km s −1 at z = 6 and 434.9 ± 47.8 km s −1 at z = 7, and using changing outflows scenario, to 291.5 ± 7.9 km s −1 z = 6 and 224.6 ± 22.8 km s −1 at z = 7. It is evident that, over all possible combinations and the prior range assumed, the changing outflows scenario produces consistent width distribution within the 1 σ level of Pentericci et al. ( 2018 ) measurements. This confirms that changing the column density predicts inconsistent width distributions. It might be noteworthy that while we use the Pentericci et al. ( 2018 ) results in this section, other studies of
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