Effective conductance method for the primordial recombination spectrum

Abstract

As atoms formed for the first time during primordial recombination, they emitted bound-bound and free-bound radiation leading to spectral distortions to the cosmic microwave background. These distortions might become observable in the future with high-sensitivity spectrometers, and provide a new window into physical conditions in the early universe. The standard multilevel atom method habitually used to compute the recombination spectrum is computationally expensive, impeding a detailed quantitative exploration of the information contained in spectral distortions thus far. In this work it is shown that the emissivity in optically thin allowed transitions can be factored into a computationally expensive but cosmology-independent part and a computationally cheap, cosmology-dependent part. The slow part of the computation consists in pre-computing temperature-dependent effective ``conductances,'' linearly relating line or continuum intensity to departures from Saha equilibrium of the lowest-order excited states ($2s$ and $2p$), that can be seen as ``voltages.'' The computation of these departures from equilibrium as a function of redshift is itself very fast, thanks to the effective multilevel atom method introduced in an earlier work. With this factorization, the recurring cost of a single computation of the recombination spectrum is only a fraction of a second on a standard laptop, more than four orders of magnitude shorter than standard computations. The spectrum from helium recombination can be efficiently computed in an identical way, and a fast code computing the full primordial recombination spectrum with this method will be made publicly available soon.