Wouthuysen-Field coupling strength and application to high-redshift 21-cm radiation

The first ultraviolet sources in the universe are expected to have coupled the H I spin temperature to the gas kinetic temperature via scattering in the Ly alpha resonance (the 'Wouthuysen-Field effect'). By establishing an H I spin temperature different from the temperature of the cosmic microwave background, the Wouthuysen-Field effect should allow observations of H I during the reionization epoch in the redshifted 21-cm hyperfine line. This paper investigates four mechanisms that can affect the strength of the Wouthuysen-Field effect that were not previously considered. (1) Photons redshifting into the H I Lyman resonances may excite an H atom and result in a radiative cascade terminating in two-photon 2s(1/2) -> 1s(1/2) emission, rather than always degrading to Ly alpha as usually assumed. (2) The fine structure of the Ly 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 Ly alpha spectral distortion based on the diffusion approximation. It is shown that (1) suppresses the Wouthuysen-Field coupling strength by a factor of up to similar to 2, while (2) and (3) are important only at low kinetic temperatures. Effect (4) has a <= 3 per cent effect for kinetic temperatures T-k >= 2 K. In particular, if the pre-reionization intergalactic medium was efficiently heated by X-rays, only effect (1) is important. Fitting formulae for the Wouthuysen-Field coupling strength are provided for the range of T-k >= 2 K and Gunn-Peterson optical depth 10(5) < tau(GP) < 10(7) so that all of these effects can be easily incorporated into 21-cm codes.