  Although measuring the gas metallicity in galaxies at various redshifts is
crucial to constrain galaxy evolutionary scenarios, only rest-frame optical
emission lines have been generally used to measure the metallicity. This has
prevented us to accurately measure the metallicity of dust-obscured galaxies,
and accordingly to understand the chemical evolution of dusty populations, such
as ultraluminous infrared galaxies. Here we propose diagnostics of the gas
metallicity based on infrared fine structure emission lines, which are nearly
unaffected by dust extinction even the most obscured systems. Specifically, we
focus on fine-structure lines arising mostly from HII regions, not in
photo-dissociation regions, to minimize the dependence and uncertainties of the
metallicity diagnostics from various physical parameters. Based on
photoionization models, we show that the emission-line flux ratio of
([OIII]51.80+[OIII]88.33)/[NIII]57.21 is an excellent tracer of the gas
metallicity. The individual line ratios [OIII]51.80/[NIII]57.21 or
[OIII]88.33/[NIII]57.21 can also be used as diagnostics of the metallicity, but
they suffer a stronger dependence on the gas density. The line ratios
[OIII]88.33/[OIII]51.80 and [NII]121.7/[NIII]57.21 can be used to measure and,
therefore, account for the dependences on the of the gas density and ionization
parameter, respectively. We show that these diagnostic fine-structure lines are
detectable with Herschel in luminous infrared galaxies out z=0.4. Metallicity
measurements with these fine-structure lines will be feasible at relatively
high redshift (z=1 or more) with SPICA, the future infrared space observatory.
