  Interest in non-standard recombination scenarios has been spurred by recent
cosmic microwave background (CMB) results from BOOMERANG and MAXIMA, which show
an unexpectedly low second acoustic peak, resulting in a best-fit baryon
density that is 50% larger than the prediction of big-bang nucleosynthesis
(BBN). This apparent discrepancy can be avoided if the universe has a
non-standard ionization history in which the recombination of hydrogen is
significantly delayed relative to the standard model. While future CMB
observations may eliminate this discrepancy, it is useful to develop a general
framework for analyzing non-standard ionization histories. We develop such a
framework, examining non-standard models in which the hydrogen binding energy
E_b and the overall expression for the time rate of change of the ionized
fraction of electrons are multiplied by arbitrary factors. This set of models
includes a number of previously-proposed models as special cases. We find a
wide range of models with delayed recombination that are able to fit the CMB
data with a baryon density in accordance with BBN, but there are even allowed
models with earlier recombination than in the standard model. A generic
prediction of these models is that the third acoustic CMB peak should be very
low relative to what is found in the standard model. This is the case even for
the models with earlier recombination than in the standard model, because here
the third peak is lowered by an increased diffusion damping at recombination
relative to the standard model. Interestingly, the specific height of the third
peak depends sensitively on the model parameters, so that future CMB
measurements will be able to distinguish between different non-standard
recombination scenarios.
