Machine Learning LSST 3x2pt analyses -- forecasting the impact of systematics on cosmological constraints using neural networks

Validating modeling choices through simulated analyses and quantifying the impact of different systematic effects will form a major computational bottleneck in the preparation for 3$\times$2 analysis with Stage-IV surveys such as Vera Rubin Observatory's Legacy Survey of Space and Time (LSST). We can significantly reduce the computational requirements by using machine learning based emulators, which allow us to run fast inference while maintaining the full realism of the data analysis pipeline. In this paper, we use such an emulator to run simulated 3$\times$2 (cosmic shear, galaxy-galaxy lensing, and galaxy clustering) analyses for mock LSST-Y1/Y3/Y6/Y10 surveys and study the impact of various systematic effects (galaxy bias, intrinsic alignment, baryonic physics, shear calibration and photo-$z$ uncertainties). Closely following the DESC Science Requirement Document (with several updates) our main findings are: {\it a)} The largest contribution to the `systematic error budget' of LSST 3$\times$2 analysis comes from galaxy bias uncertainties, while the contribution of baryonic and shear calibration uncertainties are significantly less important. {\it b)} Tighter constraints on intrinsic alignment and photo-$z$ parameters can improve cosmological constraints noticeably, which illustrates synergies of LSST and spectroscopic surveys. {\it c)} The scale cuts adopted in the DESC SRD may be too conservative and pushing to smaller scales can increase cosmological information significantly. {\it d)} We investigate the impact of photo-$z$ outliers on 3$\times$2 pt analysis and find that we need to determine the outlier fraction to within $5-10\%$ accuracy to ensure robust cosmological analysis. We caution that these findings depend on analysis choices (parameterizations, priors, scale cuts) and can change for different settings.