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Prior to recombination photons, electrons, and atomic nuclei rapidlyscattered and behaved, almost, like a single tightly-coupled photon-baryonplasma. We investigate here the accuracy of the tight-coupling approximationcommonly used to numerically evolve the baryon and photon perturbationequations at early times. By solving the exact perturbations equations with astiff solver starting deep in the radiation-dominated epoch we find the levelof inaccuracy introduced by resorting to the standard first-ordertight-coupling approximation. We develop a new second-order approximation inthe inverse Thomson opacity expansion and show that it closely tracks the fullsolution, at essentially no extra numerical cost. We find the bias on estimatesof cosmological parameters introduced by the first-order approximation is, formost parameters, negligible. Finally, we show that our second-orderapproximation can be used to reduce the time needed to compute cosmic microwavebackground angular spectra by as much as ~17%.

Photons and Baryons before Atoms: Improving the Tight-Coupling Approximation