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Wouthuysen-Field coupling strength and application to high-redshift 21 cm radiation
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The first UV sources in the universe are expected to have coupled the HI spin temperature to the gas kinetic temperature via scattering in the Lyman-alpha resonance [the Wouthuysen-Field (WF) effect]. By establishing an HI spin temperature different from the temperature of the CMB, the WF effect should allow observations of HI during the reionization epoch in the redshifted 21 cm line. This paper investigates four mechanisms that can affect the strength of the WF effect that were not previously considered: (1) Photons redshifting into the HI Lyman resonances may excite an H atom and result in a radiative cascade terminating in two-photon 2s->1s emission, rather than always degrading to Lyman-alpha as usually assumed. (2) The fine structure of the Lyman-alpha resonance alters the photon frequency distribution and leads to a suppression of the scattering rate. (3) The spin-flip scatterings change the frequency of the photon and cause the photon spectrum to relax not to the kinetic temperature of the gas but to a temperature between the kinetic and spin temperatures, effectively reducing the strength of the Wouthuysen-Field coupling. (4) Near line centre, a photon can change its frequency by several times the line width in a single scattering event, thus potentially invalidating the usual calculation of the Lyman-alpha spectral distortion based on the diffusion approximation. It is shown that (1) suppresses the WF coupling strength by a factor of up to ~2, while (2) and (3) are important only at low kinetic temperatures. Effect (4) has a <=3% effect for kinetic temperatures T_k>=2K. If the pre-reionization IGM was efficiently heated by X-rays, only effect (1) is important. Fitting formulae are provided for the range of T_k>=2K and Gunn-Peterson optical depth 10^5--10^7. [ABRIDGED]
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