21 cm angular-power spectrum from the dark ages

At redshifts z30 neutral hydrogen gas absorbs cosmic microwave background radiation at the 21 cm spin-flip frequency. In principle this is observable and a high-precision probe of cosmology. We calculate the linear-theory angular-power spectrum of this signal and cross correlation between redshifts on scales much larger than the linewidth. In addition to the well-known redshift distortion and density perturbation sources, a full linear analysis gives additional contributions to the power spectrum. On small scales there is a percent-level linear effect due to perturbations in the 21 cm optical depth, and perturbed recombination modifies the gas temperature perturbation evolution (and hence spin temperature and 21 cm power spectrum). On large scales there are several post-Newtonian and velocity effects; although negligible on small scales, these additional terms can be significant at l100 and can be nonzero even when there is no background signal. We also discuss the linear effect of reionization rescattering, which damps the entire spectrum and gives a very small polarization signal on large scales. On small scales we also model the significant nonlinear effects of evolution and gravitational lensing. We include full results for numerical calculation and also various approximate analytic results for the power spectrum and evolution of small-scale perturbations.