Abstract
Interpreting abundances of Damped Ly-$\alpha$ Absorbers (DLAs) from
absorption-line spectroscopy has typically been a challenge because of the
presence of dust. Nevertheless, because DLAs trace distant gas-rich galaxies
regardless of their luminosity, they provide an attractive way of measuring the
evolution of the metallicity of the neutral gas with cosmic time. This has been
done extensively so far, but typically not taking proper dust corrections into
account. The aims of this paper are to: $i)$ provide a simplified way of
calculating dust corrections, based on a single observed $X$/Fe, $ii)$ assess
the importance of dust corrections for DLA metallicities and their evolution,
and $iii)$ investigate the cosmic evolution of iron for a large DLA sample. We
derive dust corrections based on the observed Zn/Fe, Si/Fe, or S/Fe, and
confirm their robustness. We present dust-corrected metallicities in a scale of
Fe/H$_tot$ for 266 DLAs over a broad range of $z$, and assess the
extent of dust corrections for different metals at different metallicities.
Dust corrections in DLAs are important even for Zn (typically of 0.1-0.2, and
up to 0.45 dex), which is often neglected. Finally, we study the evolution of
the dust-corrected metallicity with $z$. The DLA metallicities decrease with
redshift, by a factor of 50-100 from today to $\sim$ 12.6 billions ago ($z=$
5). When including dust corrections, the metallicity evolution is more steep
than previously thought. At low $z$, the average DLA metallicities are $\sim$
0.3 dex higher than without corrections. The upper envelope of the relation
between metallicity and $z$ reaches solar metallicity at $złesssim$ 0.5,
although some systems can have solar metallicity already out to $z\sim$ 3.
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