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

The cross-correlation between the cosmic microwave background (CMB) fields and matter tracers carries important cosmological information. In this paper, we forecast by a signal-to-noise ratio analysis the information contained in the cross-correlation of the CMB anisotropy fields with source counts for future cosmological observations and its impact on cosmological parameters uncertainties, using a joint tomographic analysis. We include temperature, polarization and lensing for the CMB fields and galaxy number counts for the matter tracers. By restricting ourselves to quasi-linear scales, we forecast by a Fisher matrix formalism the relative importance of the cross-correlation of source counts with the CMB in the constraints on the parameters for several cosmological models. We obtain that the CMB-number counts cross-correlation can improve 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 of both probes and will enable the Euclid-like photometric survey to reach the highest FoM among those considered here. We also forecast how CMB-galaxy clustering cross-correlation could increase the FoM of the neutrino sector, also enabling a statistically significant ($\gtrsim$ 3$\sigma$ for LiteBIRD+CMB-S4 $\times$ SPHEREx) detection of the minimal neutrino mass allowed in a normal hierarchy by using quasi-linear scales only. Analogously, we find that the uncertainty in the local primordial non-Gaussianity could be as low as $\sigma (f_{\rm NL}) \sim 1.5-2$ by using two-point statistics only with the combination of CMB and radio surveys such as EMU and SKA1. Our results highlight the additional constraining power of the cross-correlation between CMB and galaxy clustering from future surveys which is mainly based on quasi-linear scales and therefore sufficiently robust to non-linear effects.