Recombinations to the Rydberg States of Hydrogen and Their Effect During
  the Cosmological Recombination Epoch
  In this paper we discuss the effect of recombinations to highly excited
states (n > 100) in hydrogen during the cosmological recombination epoch. For
this purpose, we developed a new ODE solver for the recombination problem,
based on an implicit Gear's method. This solver allows us to include up to 350
l-resolved shells or ~61 000 separate levels in the hydrogen model and to solve
the recombination problem for one cosmology in ~27 hours. This is a huge
improvement in performance over our previous recombination code, for which a
100-shell computation (5050 separate states) already required ~150 hours on a
single processor. We show that for 350 shells down to redshift z ~200 the
results for the free electron fraction have practically converged. The final
modification in the free electron fraction at z ~200 decreases from about
\DeltaNe/Ne ~2.8% for 100 shells to \DeltaNe/Ne ~1.6% for 350 shells. However,
the associated changes in the CMB power spectra at large multipoles l are
rather small, so that for accurate computations in connection with the analysis
of Planck data already ~100 shells are expected to be sufficient. Nevertheless,
the total value of \tau could still be affected at a significant level. We also
briefly investigate the effect of collisions on the recombination dynamics.
With our current estimates for the collisional rates we find a correction of
\DeltaNe/Ne ~ -0.088% at z ~ 700, which is mainly caused by l-changing
collisions with protons. Furthermore, we present results on the cosmological
recombination spectrum, showing that at low frequencies collisional processes
are important. However, the current accuracy of collisional rates is
insufficient for precise computations of templates for the recombination
spectrum at \nu<~1 GHz, and also the effect of collisions on the recombination
dynamics suffers from the uncertainty in these rates.
