  Spectroscopic studies of high-redshift objects and increasingly precise data
on the cosmic microwave background (CMB) are beginning to independently place
strong complementary bounds on the epoch of hydrogen reionization. Parameter
estimation from current CMB data continues, however, to be subject to several
degeneracies. Here, we focus on those degeneracies in CMB parameter forecasts
related to the optical depth to reionization. We extend earlier work on the
mutual constraints that such analyses of CMB data and a reionization model may
place on each other to a more general parameter set, and to the case of data
anticipated from the $MAP$ satellite. We focus in particular on a semi-analytic
model of reionization by the first stars, although the methods here are easily
extended to other reionization scenarios. A reionization model can provide
useful complementary information for cosmological parameter (CP) extraction
from the CMB, particularly for the degeneracies between the optical depth and
either of the amplitude and index of the primordial scalar power spectrum,
which are still present in the most recent data. Alternatively, by using a
reionization model, known limits on astrophysical quantities can reduce the
forecasted errors on CPs. Forthcoming CMB data also have the potential to
constrain the sites of early star formation, as well as the fraction of baryons
that participate in it. Finally, we examine the implications of an independent,
e.g., spectroscopic, determination of the epoch of reionization for the
determination of CPs from the CMB. This has the potential to significantly
strengthen limits from the CMB on parameters such as the index of the power
spectrum, while having the considerable advantage of being free of the choice
of the reionization model (abridged).
