HYREC-2: a highly accurate sub-millisecond recombination code
  We present the new recombination code HYREC-2, holding the same accuracy as
the state-of-the-art codes HYREC and COSMOREC and, at the same time, surpassing
the computational speed of the code RECFAST commonly used for CMB-anisotropy
data analyses. HYREC-2 is based on an effective 4-level atom model, accounting
exactly for the non-equilibrium of highly excited states of hydrogen, and very
accurately for radiative transfer effects with a correction to the
Lyman-$\alpha$ escape rate. The latter is computed with the original HYREC, and
tabulated, as a function of temperature, along with its derivatives with
respect to the relevant cosmological parameters. This enables the code to keep
the same accuracy as the original HYREC-2 over the full 99.7% confidence region
of cosmological parameters currently allowed by Planck, while running in under
one millisecond on a standard laptop. Our code leads to no noticeable bias in
any cosmological parameters even in the case of an ideal cosmic-variance
limited experiment up to $\ell$ = 5000. Beyond CMB anisotropy calculations,
HYREC-2 will be a useful tool to compute various observables that depend on the
recombination and thermal history, such as the recombination spectrum or the
21-cm signal.
