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 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 $\sigma_{\text{z}}$. We vary these parameters, as well as the galaxy bias, $b_{\text{g}}(z)$ and cosmological parameters, $\sigma_8(z)$ and $\Omega_{\text{m}}$ in a Fisher analysis across 12 redshift bins from $z=0-7$. We find that incorporating redshift uncertainties degrades constraints on cosmological parameters in the LSST/CMB-S4 era by about a factor of 10. In our fiducial analysis of LSST/CMB-S4, 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. We also find that constraints on the redshift parameters from just the galaxy clustering and cross-correlation measurements (with no priors) are competitive compared to photometric redshift techniques. We also project constraints for nearer term survey combinations: DES/SPT-SZ, DES/SPT-3G and LSST/SPT-3G.