Radiative transfer effect on ultraviolet pumping of the 21 cm line in the high-redshift universe

During the epoch of reionization the 21 cm signal is sensitive to the scattering rate of the ultraviolet photons, redshifting across the Ly alpha resonance. Here we calculate the photon scattering rate profile for a single ultraviolet source. After taking into account previously neglected natural broadening of the resonance line, we find that photons approach the resonance frequency and experience most scatterings at a significantly smaller distance from the source than naively expected r = (Delta nu/nu(0))(c/H), where Delta nu = nu - nu(0) is the initial frequency offset, and the discrepancy increases as the initial frequency offset decreases. As a consequence, the scattering rate P-alpha(r) drops much faster with increasing distance than the previously assumed 1/r(2) profile. Near the source, (r less than or similar to 1 comoving Mpc), the scattering rate of photons that redshift into the Ly alpha resonance converges to P-alpha(r) proportional to r(-7/3). The scattering rate of Ly alpha photons produced by splitting of photons that redshift into a higher resonance (Ly gamma, Ly delta, etc.) is only weakly affected by the radiative transfer, while the sum of scattering rates of Ly alpha photons produced from all higher resonances also converges to P-alpha(r) proportional to r(-7/3) near the source. At 15 < z < 35, on scales of similar to 0.01-20 h(-1) Mpc (comoving), the total scattering rate of Ly alpha photons from all Lyman resonances is found to be higher by a factor of similar to 1 + 0.3[(1 + z)/20](2/3) than obtained without full radiative transfer. Consequently, during the early stage of reionization, the differential brightness of 21 cm signal against the cosmic microwave background is also boosted by a similar factor.