The spin-resolved atomic velocity distribution and 21-cm line profile of dark-age gas

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 HI 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 alpha 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 HI 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 similar to 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 similar 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 similar to 60 per cent relative to the velocity-independent calculation. We discuss the implications for 21-cm tomography of the dark ages.