The imprint of cosmological hydrogen recombination lines on the power spectrum of the CMB

We explore the imprint of the cosmological hydrogen recombination lines on the power spectrum of the cosmic microwave background (CMB). In particular, we focus on the three strongest lines for the Balmer, Paschen and Brackett series of hydrogen. We expect changes in the angular power spectrum due to these lines of about $0.3~\mu{\rm K}$ for the Hα line, being maximum at small angular scales ($\ell \approx 870$). The morphology of the signal is very rich. It leads to relatively narrow spectral features ($\Delta \nu / \nu \sim 10^{-1}$), with several regions in the power spectrum showing a characteristic change of sign of the effect as we probe different redshifts or different multipoles by measuring the power spectrum at different frequencies. It also has a very peculiar dependence on the multipole scale, connected with the details of the transfer function at the epoch of scattering. In order to compute the optical depths for these transitions, we have numerically evolved the populations of the levels of the hydrogen atom during recombination, simultaneously treating the evolution of helium. For the hydrogen atom, we follow each angular momentum state separately, up to the level $n=10$. Foregrounds and other frequency dependent contaminants such as Rayleigh scattering may be a important limitation for these measurements, although the peculiar frequency and angular dependences of the effect that we are discussing might permit us to separate it. Detection of this signal using future narrow-band spectral observations can give information about the details of how the cosmic recombination proceeds, and how Silk damping operates during recombination.