The cosmic evolution of dust-corrected metallicity in the neutral gas
  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
have derived dust corrections based on the observed [Zn/Fe], [Si/Fe], or
[S/Fe], and confirmed their robustness. We present dust-corrected metallicities
in a scale of [Fe/H]$_{\rm tot}$ for 236 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 $0.5$~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 $\sim12.6$ billion
years ago ($z=5$). When including dust corrections, the average DLA
metallicities are 0.4--0.5~dex higher than without corrections. The upper
envelope of the relation between metallicity and $z$ reaches solar metallicity
at $z\lesssim0.5$, although some systems can have solar metallicity already out
to $z\sim3$.
