Forecasting the Impact of Source Galaxy Photometric Redshift Uncertainties on the LSST 3times2pt Analysis

Photometric redshifts of the source galaxies are a key source of systematic uncertainty in the Rubin Observatory Legacy Survey of Space and Time (LSST)'s galaxy clustering and weak lensing analysis, i.e., the $3\times 2$pt analysis. This paper introduces a Fisher forecast code FisherA2Z for the LSST Year 10 (Y10) $3 \times 2$pt and cosmic shear analyses, utilizing a 15-parameter redshift distribution model, with one redshift bias, variance, and outlier rate per tomographic bin. FisherA2Z employs the Core Cosmology Library CCL to compute the large-scale structure power spectrum and incorporates a four-parameter nonlinear alignment model for intrinsic alignments. We evaluate the impact of marginalizing over redshift distribution parameters on weak lensing, forecast biases in cosmological parameters due to redshift errors, and assess cosmological parameter sensitivity to redshift systematic parameters using decision trees. The sensitivity study reveals that for LSST $3\times2$pt analysis, $S_8$ is most sensitive to the mean redshift of the fourth out of the five source tomographic bins, while other cosmological parameters possess different sensitivities. Additionally, we provide cosmological analysis forecasts based on different scenarios of spectroscopic training datasets. We find that the figures-of-merit for the cosmological results increase with the number of spectroscopic training galaxies, and with the completeness of the training set above $z=1.6$, assuming the redshift information comes solely from the training set galaxies without other external constraints.