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The cross-correlation between the cosmic microwave background (CMB) fieldsand matter tracers carries important cosmological information. In this paper,we forecast by a signal-to-noise ratio analysis the information contained inthe cross-correlation of the CMB anisotropy fields with source counts forfuture cosmological observations and its impact on cosmological parametersuncertainties, using a joint tomographic analysis. We include temperature,polarization and lensing for the CMB fields and galaxy number counts for thematter tracers. By restricting ourselves to quasi-linear scales, we forecast bya Fisher matrix formalism the relative importance of the cross-correlation ofsource counts with the CMB in the constraints on the parameters for severalcosmological models. We obtain that the CMB-number counts cross-correlation canimprove the dark energy Figure of Merit (FoM) at most up to a factor $\sim 2$for LiteBIRD+CMB-S4 $\times$ SKA1 compared to the uncorrelated combination ofboth probes and will enable the Euclid-like photometric survey to reach thehighest FoM among those considered here. We also forecast how CMB-galaxyclustering cross-correlation could increase the FoM of the neutrino sector,also enabling a statistically significant ($\gtrsim$ 3$\sigma$ forLiteBIRD+CMB-S4 $\times$ SPHEREx) detection of the minimal neutrino massallowed in a normal hierarchy by using quasi-linear scales only. Analogously,we find that the uncertainty in the local primordial non-Gaussianity could beas low as $\sigma (f_{\rm NL}) \sim 1.5-2$ by using two-point statistics onlywith the combination of CMB and radio surveys such as EMU and SKA1. Our resultshighlight the additional constraining power of the cross-correlation betweenCMB and galaxy clustering from future surveys which is mainly based onquasi-linear scales and therefore sufficiently robust to non-linear effects.

Cosmological parameter forecasts by a joint 2D tomographic approach to CMB and galaxy clustering