Cosmological perturbations at second order and recombination perturbed

We derive the full set of second-order equations governing the evolution of cosmological perturbations, including the effects of the first-order electron number density perturbations, delta(e) We provide a detailed analysis of the perturbations to the recombination history of the universe and show that a perturbed version of the Peebles effective 3-level atom is sufficient for obtaining the evolution of delta(e) for comoving wavenumbers smaller than 1 Mpc(-1). We calculate rigorously the perturbations to the Ly alpha escape probability and show that to a good approximation it is governed by the local baryon velocity divergence. For modes longer than the photon diffusion scale, we find that delta(e) is enhanced during recombination by a factor of roughly 5 relative to other first-order quantities sourcing the CMB anisotropies at second order. Using these results, in a companion paper we calculate the CMB bispectrum generated during recombination.