Photons and Baryons before Atoms: Improving the Tight-Coupling
  Approximation
  Prior to recombination photons, electrons, and atomic nuclei rapidly
scattered and behaved, almost, like a single tightly-coupled photon-baryon
plasma. We investigate here the accuracy of the tight-coupling approximation
commonly used to numerically evolve the baryon and photon perturbation
equations at early times. By solving the exact perturbations equations with a
stiff solver starting deep in the radiation-dominated epoch we find the level
of inaccuracy introduced by resorting to the standard first-order
tight-coupling approximation. We develop a new second-order approximation in
the inverse Thomson opacity expansion and show that it closely tracks the full
solution, at essentially no extra numerical cost. We find the bias on estimates
of cosmological parameters introduced by the first-order approximation is, for
most parameters, negligible. Finally, we show that our second-order
approximation can be used to reduce the time needed to compute cosmic microwave
background angular spectra by as much as ~17%.
