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 Lyα 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 2s1/2 → 1s1/2 emission, rather than always degrading to Lyα as usually assumed. (2) The fine structure of the Lyα resonance alters the photon frequency distribution and leads to a suppression of the scattering rate. (3) The spinflip 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 WouthuysenField 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 Lyα spectral distortion based on the diffusion approximation. It is shown that (1) suppresses the Wouthuysen-Field coupling strength by a factor of up to ∼ 2, while (2) and (3) are important only at low kinetic temperatures. Effect (4) has a 6 3 per cent effect for kinetic temperatures Tk > 2K. In particular if the prereionization 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 Tk > 2K and Gunn-Peterson optical depth 10 5 < τGP < 10 7 so that all of these effects can be easily incorporated into 21 cm codes.