Descending from on high: Lyman series cascades and spin-kinetic temperature coupling in the 21 cm line

We examine the effect of Lyman continuum photons on the 21 cm background in the high-redshift universe. The brightness temperature of this transition is determined by the spin temperature T_s, which describes the relative populations of the singlet and triplet hyperfine states. Once the first luminous sources appear, T_s is set by the Wouthuysen-Field effect, in which Lyman-series photons mix the hyperfine levels. Here we consider coupling through n>2 Lyman photons. We first show that coupling (and heating) from scattering of Lyman-n photons is negligible, because they rapidly cascade to lower-energy photons. These cascades can result in either a Lyman-alpha photon -- which will then affect T_s according to the usual Wouthuysen-Field mechanism -- or photons from the 2s -> 1s continuum, which escape without scattering. We show that a proper treatment of the cascades delays the onset of strong Wouthuysen-Field coupling and affects the power spectrum of brightness fluctuations when the overall coupling is still relatively weak (i.e., around the time of the first stars). Cascades damp fluctuations on small scales because only ~ 1/3 of Lyn photons cascade through Lyman-alpha, but they do not affect the large-scale power because that arises from those photons that redshift directly into the Lyman-alpha transition. We also comment on the utility of Lyman-n transitions in providing "standard rulers" with which to study the high-redshift universe.