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.
