  A possible formation mechanism of hydrogen molecules on a galactic scale is
examined. We are interested especially in the role of hydrogen molecules for
formation and evolution of primordial galaxies. Then, formation process of
hydrogen molecules in a very low-metallicity galaxy (I Zw 18; the most typical
metal-deficient galaxy) is studied. Adopting a recent observational result of
absorption lines of hydrogen molecule in I Zw 18, we obtain the upper limit for
the ionization degree in the case where hydrogen molecules can form via the
H$^{-}$-process, although they are generally believed to form on the surface of
dust grains. Furthermore, we present a critical ionization degree, above which
H$^{-}$-process can be dominant over the formation process on the surface of
grains. Interestingly, this critical ionization degree is comparable to the
upper limit of the ionization degree for I Zw 18. For determination of the
formation process of hydrogen molecules, future observational facilities can be
useful. Thus, we examine the detectability in some wavelengths for the metal
deficient galaxies. According to our estimate, near-infrared line emission of
hydrogen molecule is observable in the level of 10 $\mu$Jy, the free-free radio
emission is in the level of mJy, and the far-infrared emission from the dust on
which hydrogen molecules form can also be detected with 10-mJy level with its
temperature of 16 K. The near-infrared line and the far-infrared continuum are
feasible for ASTRO-F observations.
