  A soft ultraviolet radiation field, 10.2 eV < E <13.6 eV, that permeates
neutral intergalactic gas during the Epoch of Reionization (EoR) excites the 2p
(directly) and 2s (indirectly) states of atomic hydrogen. Because the 2s state
is metastable, the lifetime of atoms in this level is relatively long, which
may cause the 2s state to be overpopulated relative to the 2p state. It has
recently been proposed that for this reason, neutral intergalactic atomic
hydrogen gas may be detected in absorption in its 3-cm fine-structure line
(2s_1/2 -> 2p_3/2) against the Cosmic Microwave Background out to very high
redshifts. In particular, the optical depth in the fine-structure line through
neutral intergalactic gas surrounding bright quasars during the EoR may reach
tau~1e-5. The resulting surface brightness temperature of tens of micro K (in
absorption) may be detectable with existing radio telescopes. Motivated by this
exciting proposal, we perform a detailed analysis of the transfer of Lyman
beta,gamma,delta,... radiation, and re-analyze the detectability of the
fine-structure line in neutral intergalactic gas surrounding high-redshift
quasars. We find that proper radiative transfer modeling causes the
fine-structure absorption signature to be reduced tremendously to tau< 1e-10.
We therefore conclude that neutral intergalactic gas during the EoR cannot
reveal its presence in the 3-cm fine-structure line to existing radio
telescopes.
