Cosmology from weak lensing, galaxy clustering, CMB lensing, and tSZ - I. 10 x 2pt modelling methodology

The overlap of galaxy surveys and cosmic microwave background (CMB) experiments presents an ideal opportunity for joint cosmological data set analyses. In this paper we develop a halo model-based method for the first joint analysis combining these two experiments using 10 correlated two-point functions (10 x 2pt) derived from galaxy position, galaxy shear, CMB lensing convergence, and Compton-y fields. We explore this method using the Vera Rubin Observatory Legacy Survey of Space and Time (LSST) and the Simons Observatory ( SO) as examples. We find such LSS x CMB joint analyses lead to significant improvement in Figure-of-Merit of Omega m and S-8 over the constraints from using LSS-only probes within Lambda cold dark matter (Lambda CDM) model. We identify that the shear-y and y-y correlations are the most valuable additions when thermal Sunyaev-Zel'dolvich (tSZ) is included. We further identify the dominant sources of halo model uncertainties in the small-scale modelling, and investigate the impact of halo self-calibration due to the inclusion of small-scale tSZ information.