J-resolved HeI emission predictions in the low-density limit

Determinations of the primordial helium abundance are used in precision cosmological tests. These require highly accurate He I recombination rate coefficients. Here we reconsider the formation of He I recombination lines in the low-density limit. This is the simplest case, and it forms the basis for the more complex situation in which collisions are important. The formation of a recombination line is a two-step process, beginning with the capture of a continuum electron into a bound state and followed by radiative cascade to ground. The rate coefficient for capture from the continuum is obtained from photoionization cross sections and detailed balancing, while radiative transition probabilities determine the cascades. We have made every effort to use today's best atomic data. Radiative decay rates are from Drake's variational calculations, which include QED, fine structure, and singlet-triplet mixing. Certain high-L fine-structure levels do not have a singlet-triplet distinction, and the singlets and triplets are free to mix in dipole-allowed radiative decays. We use quantum-defect or hydrogenic approximations to include levels higher than those treated in the variational calculations. Photoionization cross sections come from R-matrix calculations when possible. We use Seaton's method to extrapolate along sequences of transition probabilities to obtain threshold photoionization cross sections for some levels. For higher n we use scaled hydrogenic theory or an extension of quantum-defect theory. We create two independent numerical implementations to ensure that the complex bookkeeping is correct. The two codes use different (reasonable) approximations to span the gap between lower levels, having accurate data, and high levels, where scaled hydrogenic theory is appropriate. We also use different (reasonable) methods to account for recombinations above the highest levels individually considered. We compare these independent predictions to estimate the uncertainties introduced by the various approximations. Singlet-triplet mixing has little effect on the observed spectrum. While intensities of lines within multiplets change, the entire multiplet, the quantity normally observed, does not. The lack of high-precision photoionization cross sections at intermediate n and low L introduces similar to 0.5% uncertainties in intensities of some lines. The high-n unmodeled levels introduce similar to 1% uncertainties for "yrast" lines, defined as those having L = n - 1 upper levels. This last uncertainty will not be present in actual nebulae, since such high levels are held in statistical equilibrium by collisional processes. We identify those lines that are least affected by uncertainties in the atomic physics and so should be used in precision helium abundance determinations.