Effects of Redshift Uncertainty on Cross-Correlations of CMB Lensing and
  Galaxy Surveys
  We explore the effects of incorporating redshift uncertainty into
measurements of galaxy clustering and cross-correlations of galaxy positions
and cosmic microwave background (CMB) lensing maps. We use a simple Gaussian
model for a redshift distribution in a redshift bin with two parameters: the
mean, $z_0$, and the width, $\sigma_z$. We vary these parameters, as well as a
galaxy bias parameter, $b_{\text{g}}$, and a matter fluctuations parameter,
$\sigma_8$, for each redshift bin, as well as the parameter
$\Omega_{\text{m}}$, in a Fisher analysis across 12 redshift bins from $z=0-7$.
We find that incorporating redshift uncertainties degrades constraints on
$\sigma_8(z)$ in the Large Synoptic Survey Telescope (LSST)/CMB-S4 era by about
a factor of 10 compared to the case of perfect redshift knowledge. In our
fiducial analysis of LSST/CMB-S4 including redshift uncertainties, we project
constraints on $\sigma_8(z)$ for $z<3$ of less than $5 \%$. Galaxy imaging
surveys are expected to have priors on redshift parameters from photometric
redshift algorithms and other methods. When adding priors with the expected
precision for LSST redshift algorithms, the constraints on $\sigma_8(z)$ can be
improved by a factor of 2-3 compared to the case of no prior information. We
also find that `self-calibrated' constraints on the redshift parameters from
just the autocorrelation and cross-correlation measurements (with no prior
information) are competitive with photometric redshift techniques. In the
LSST/CMB-S4 era, we find uncertainty on the redshift parameters
($z_0,\sigma_z$) to be below 0.004(1+z) at $z<1$. For all parameters,
constraints improve significantly if smaller scales can be used. We also
project constraints for nearer term survey combinations, Dark Energy Survey
(DES)/SPT-SZ, DES/SPT-3G, and LSST/SPT-3G, and analyze how our constraints
depend on a variety of parameter and model choices.
