Ultrafast effective multi-level atom method for primordial hydrogen
  recombination
  Cosmological hydrogen recombination has recently been the subject of renewed
attention because of its importance for predicting the power spectrum of cosmic
microwave background anisotropies. It has become clear that it is necessary to
account for a large number n >~ 100 of energy shells of the hydrogen atom,
separately following the angular momentum substates in order to obtain
sufficiently accurate recombination histories. However, the multi-level atom
codes that follow the populations of all these levels are computationally
expensive, limiting recent analyses to only a few points in parameter space. In
this paper, we present a new method for solving the multi-level atom
recombination problem, which splits the problem into a computationally
expensive atomic physics component that is independent of the cosmology, and an
ultrafast cosmological evolution component. The atomic physics component
follows the network of bound-bound and bound-free transitions among excited
states and computes the resulting effective transition rates for the small set
of "interface" states radiatively connected to the ground state. The
cosmological evolution component only follows the populations of the interface
states. By pre-tabulating the effective rates, we can reduce the recurring cost
of multi-level atom calculations by more than 5 orders of magnitude. The
resulting code is fast enough for inclusion in Markov Chain Monte Carlo
parameter estimation algorithms. It does not yet include the radiative transfer
or high-n two-photon processes considered in some recent papers. Further work
on analytic treatments for these effects will be required in order to produce a
recombination code usable for Planck data analysis.
