Beyond optical depth: Future determination of ionization history from
  the CMB
  We explore the fundamental limits to which reionization histories can be
constrained using only large-scale cosmic microwave background (CMB) anisotropy
measurements. The redshift distribution of the fractional ionization $x_e(z)$
affects the angular distribution of CMB polarization. We project constraints on
the reionization history of the universe using low-noise full-sky temperature
and E-mode measurements of the CMB. We show that the measured TE power
spectrum, $\hat C_\ell^\mathrm{TE}$, has roughly one quarter of the
constraining power of $\hat C_\ell^\mathrm{EE}$ on the reionization optical
depth $\tau$, and its addition improves the precision on $\tau$ by 20% over
using $\hat C_\ell^\mathrm{EE}$ only. We also use a two-step reionization model
with an additional high redshift step, parametrized by an early ionization
fraction $x_e^\mathrm{min}$, and a late reionization step at $z_\mathrm{re}$.
We find that future high signal-to-noise measurements of the multipoles
$10\leqslant\ell<20$ are especially important for breaking the degeneracy
between $x_e^\mathrm{min}$ and $z_\mathrm{re}$. In addition, we show that the
uncertainties on these parameters determined from a map with sensitivity
$10\,\mathrm{\mu K\,arcmin}$ are less than 5% larger than the uncertainties in
the noiseless case, making this noise level a natural target for future large
sky area E-mode measurements.
