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We investigate how damped Lyman-$\alpha$ absorbers (DLAs) at z ~ 2-3,detected in large optical spectroscopic surveys of quasars, trace thepopulation of star-forming galaxies. Building on previous results, we constructa model based on observed and physically motivated scaling relations in orderto reproduce the bivariate distributions of metallicity, Z, and HI columndensity, N(HI). Furthermore, the observed impact parameters for galaxiesassociated to DLAs are in agreement with the model predictions. The modelstrongly favours a metallicity gradient, which scales with the luminosity ofthe host galaxy, with a value of $\gamma$* = -0.019 $\pm$ 0.008 dex kpc$^{-1}$for L* galaxies that gets steeper for fainter galaxies. We find that DLAs tracegalaxies over a wide range of galaxy luminosities, however, the bulk of the DLAcross-section arises in galaxies with L ~ 0.1 L* at z ~ 2.5 broadly consistentwith numerical simulations.

High-redshift Damped Ly-alpha Absorbing Galaxy Model Reproducing the N(HI)-Z Distribution