Cosmological hydrogen recombination: The effect of extremely high-n states

Calculations of cosmological hydrogen recombination are vital for the extraction of cosmological parameters from cosmic microwave background (CMB) observations, and for imposing constraints to inflation and reionization. The Planck mission and future experiments will make high precision measurements of CMB anisotropies at angular scales as small as l similar to 2500, necessitating a calculation of recombination with fractional accuracy of approximate to 10(-3). Recent work on recombination includes two-photon transitions from high excitation states and many radiative transfer effects. Modern recombination calculations separately follow angular momentum sublevels of the hydrogen atom to accurately treat nonequilibrium effects at late times (z < 900). The inclusion of extremely high-n (n greater than or similar to 100) states of hydrogen is then computationally challenging, preventing until now a determination of the maximum n needed to predict CMB anisotropy spectra with sufficient accuracy for Planck. Here, results from a new multilevel-atom code (RECSPARSE) are presented. For the first time, forbidden'' quadrupole transitions of hydrogen are included, but shown to be negligible. RECSPARSE is designed to quickly calculate recombination histories including extremely high-n states in hydrogen. Histories for a sequence of values as high as n(max) = 250 are computed, keeping track of all angular momentum sublevels and energy shells of the hydrogen atom separately. Use of an insufficiently high n(max) value (e.g., n(max) = 64) leads to errors (e.g., 1.8 sigma for Planck) in the predicted CMB power spectrum. Extrapolating errors, the resulting CMB anisotropy spectra are converged to similar to 0.5 sigma at Fisher-matrix level for n(max) = 128, in the purely radiative case.