Primordial helium recombination. III. Thomson scattering, isotope shifts, and cumulative results

Upcoming precision measurements of the temperature anisotropy of the cosmic microwave background at high multipoles will need to be complemented by a more complete understanding of recombination, which determines the damping of anisotropies on these scales. This is the third in a series of papers describing an accurate theory of He i and He ii recombination. Here we describe the effect of Thomson scattering, the He3 isotope shift, the contribution of rare decays, collisional processes, and peculiar motion. These effects are found to be negligible: Thomson and He3 scattering modify the free electron fraction x_e at the level of several×10^-4. The uncertainty in the 2³P^o-1¹S rate is significant, and for conservative estimates, gives uncertainties in x_e of order 10^-3. We describe several convergence tests for the atomic level code and its inputs, derive an overall C_ℓ error budget, and relate shifts in x_e(z) to the changes in C_ℓ, which are at the level of 0.5% at ℓ=3000. Finally, we summarize the main corrections developed thus far. The remaining uncertainty from known effects is ∼0.3% in x_e.