Constraining the Metallicities of Damped Ly$\alpha$ Systems Using
  Extinction Curves
  In this paper, we present a new method to constrain the metallicities of high
redshift damped Ly$\alpha$ (DLA) absorbers using observed extinction curves.
This is the first time such an approach is employed to constrain the
metallicities of extragalactic absorbers. To demonstrate our method, we use the
spectra of 13 quasars and one GRB with DLA absorbers detected along their
sightlines. By using the Kramers-Kronig (KK) relation, which relates the
wavelength-integrated extinction to the total volume occupied by dust per
hydrogen nucleon, we set some robust lower limits on the metallicity of the
DLAs. The resulting lower limits are all consistent with the DLA metallicities
from the literature. The GRB extinction curve exhibits a very strong 2175 A
extinction bump. We try to constrain the metallicity of the GRB DLA by modeling
the GRB extinction curve using dust models with two (graphite and silicates)
and three (PAH, hydrogenated amorphous carbon, and silicates) dust components.
The two-component model resulted in a metallicity of $Z\sim$$-$0.45 while the
three-component model gives $Z\sim$$-$0.50. On the other hand, the lower limit
from the KK approach for this DLA is $Z\ge$$-$0.60. Modeling a large sample of
extinction curves with 2175 A extinction bump and measured DLA metallicities
would allow a thorough comparison between the KK and the model-dependent
approach. In cases where the precise measurement of the metallicity of a DLA is
not possible (e.g. due to the saturation of important absorption lines), the
approach presented in this paper can be used to constrain the metallicity.
