  More than half of the atoms in the Universe recombined via forbidden
transitions, so that accurate treatment of the forbidden channels is important
in order to follow the cosmological recombination process with the level of
precision required by future microwave anisotropy experiments. We perform a
multi-level calculation of the recombination of hydrogen (H) and helium (He)
with the addition of the triplet 2P to singlet 1S spin-forbidden transition for
neutral helium (He I), plus the nS--1S and nD--1S two-photon transitions for H
(up to n=40) and among singlet states of He I (n<=10 and l<=7). The potential
importance of such transitions was first proposed by Dubrovich & Grachev (2005)
using an effective three-level atom model. Here, we relax the thermal
equilibrium assumption among the higher excited states to investigate the
effect of these extra forbidden transitions on the ionization fraction x_e and
the Cosmic Microwave Background (CMB) angular power spectrum C_l. The
spin-forbidden transition brings more than a percent change in x_e. The
two-photon transitions may also give non-negligible effects, but currently
accurate rates exist only for n<=3. We find that changes in both x_e and C_l
would be at about the percent level with the approximate rates given by
Dubrovich & Grachev (2005). However, the two-photon rates from 3S to 1S and 3D
to 1S of H appear to have been overestimated, and our best numerical
calculation puts the effect on x_e and C_l at below the percent level.
Nevertheless, we do not claim that we have the definite answer, since several
issues remain open; sub-percent level computation of the C_ls requires improved
calculations of atomic transition rates as well as increasingly complex
multi-level atom calculations.
