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 ∼2 for LiteBIRD+CMB-S4 × 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 (≳ 3σ for LiteBIRD+CMB-S4 × 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 σ(fNL)∼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.
José R. Bermejo-Climent, Mario Ballardini, Fabio Finelli, Daniela Paoletti, Roy Maartens, José A. Rubiño-Martin, Luca Valenziano, Cosmological parameter forecasts by a joint 2D tomographic approach to CMB and galaxy clustering, Phys. Rev. D 103, 103502 (2021)