Transitioning from Stage-III to Stage-IV: Cosmology from galaxy$\times$CMB lensing and shear$\times$CMB lensing

We examine the cosmological constraining power from two cross-correlation probes between galaxy and CMB surveys: the cross-correlation of lens galaxy density with CMB lensing convergence $\langle\delta_g\kappa_{\rm CMB}\rangle$ and source galaxy weak lensing shear with CMB lensing convergence $\langle\gamma\kappa_{\rm CMB}\rangle$. These two cross-correlation probes provide an independent cross-check of other large-scale structure constraints and are insensitive to systematic effects that could be present in galaxy-only or CMB-only analyses. We study how the constraining power of $\langle\delta_g\kappa_{\rm CMB}\rangle+\langle\gamma\kappa_{\rm CMB}\rangle$ changes as the galaxy samples and CMB dataset qualitatively change from Stage-III (ongoing) to Stage-IV (future) surveys. Given the flexibility in selecting the lens galaxy sample, we also explore the impact on cosmological constraints when we vary the redshift range and magnitude limit of the lens galaxies using mock galaxy catalogs. We find that in our setup, cosmological constraints from $\langle\delta_g\kappa_{\rm CMB}\rangle$ and $\langle\gamma\kappa_{\rm CMB}\rangle$ are comparable in Stage-III; but as we move to Stage-IV, shot noise from the galaxy density becomes subdominant to cosmic variance, preventing the contribution from $\langle\delta_g\kappa_{\rm CMB}\rangle$ to further improve. This implies that to maximize the cosmological constraints from future $\langle\delta_g\kappa_{\rm CMB}\rangle+\langle \gamma\kappa_{\rm CMB}\rangle$ analyses, we should focus more on the requirements on $\langle\gamma \kappa_{\rm CMB}\rangle$ instead of $\langle\delta_g\kappa_{\rm CMB}\rangle$. In addition, the selection of the lens galaxy sample in $\langle\delta_g\kappa_{\rm CMB}\rangle$ should be optimized in terms of our ability to characterize its redshift or galaxy bias instead of its number density.