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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 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 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 ≤3 per cent effect for kinetic temperatures Tk≥ 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 Tk≥ 2 K and Gunn–Peterson optical depth 105 < τGP < 107 so that all of these effects can be easily incorporated into 21-cm codes.