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 \ell \sim 2500 , necessitating a calculation of recombination with fractional accuracy of \approx 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 \gtrsim 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 multi-level-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 \sim 0.5 \sigma at Fisher-matrix level for n _ { max } = 128 , in the purely radiative case .