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The 21-cm hyperfine line of atomic hydrogen (H I) is a promising probe of the cosmic dark ages. In past treatments of 21-cm radiation it was assumed the hyperfine level populations of H I could be characterized by a velocity-independent 'spin temperature' T s determined by a competition between 21-cm radiative transitions, spin-changing collisions, and (at lower redshifts) Ly scattering. However we show here that, if the collisional time is comparable to the radiative time, the spin temperature will depend on atomic velocity, T s = T s (v), and one must replace the usual hyperfine level rate equations with a Boltzmann equation describing the spin and velocity dependence of the H I distribution function. We construct here the Boltzmann equation relevant to the cosmic dark ages and solve it using a basis-function method. Accounting for the actual spin-resolved atomic velocity distribution results in up to a 2 per cent suppression of the 21-cm emissivity, and a redshift and angular-projection-dependent suppression or enhancement of the linear power spectrum of 21-cm fluctuations of up to 5 per cent. The effect on the 21-cm line profile is more dramatic -its full width at half-maximum can be enhanced by up to 60 per cent relative to the velocity-independent calculation. We discuss the implications for 21-cm tomography of the dark ages.
Hirata et al. (Sat,) studied this question.