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. We consider Planck-like, the Simons Observatory, LiteBIRD, and CMB-S4 specifications for CMB, and Euclid-like, Vera C. Rubin Observatory, SPHEREx, EMU, and SKA1 for future galaxy surveys. We restrict ourselves to quasilinear scales in order to deliver results that are free as much as possible from the uncertainties in modeling nonlinearities. 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 $\mathrm{LiteBIRD}+\mathrm{CMB}\text{−}\mathrm{S}4\times \mathrm{SKA}1$ 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 $\mathrm{LiteBIRD}+\mathrm{CMB}\text{−}\mathrm{S}4\times \mathrm{SPHERE}\mathrm{x}$) detection of the minimal neutrino mass allowed in a normal hierarchy by using quasilinear scales only. Analogously, we find that the uncertainty in the local primordial non-Gaussianity could be as low as $\sigma (f_{\mathrm{NL}})\sim 1.5–2$ by using two-point statistics only with the combination of CMB and radio surveys, such as EMU and SKA1. Further, we quantify how cross-correlation will help characterizing the galaxy bias. Our results highlight the additional constraining power of the cross-correlation between CMB and galaxy clustering from future surveys, which is mainly based on quasilinear scales and therefore, sufficiently robust to nonlinear effects.

Figure 1

Left panel: signal and noise for the CMB temperature and polarization for the various CMB surveys considered. The solid lines correspond to the temperature and the dashed lines to polarization. Middle panel: signal and noise for the CMB lensing. The black dashed curve corresponds to the $\varphi \varphi$ angular power spectra and the solid colored lines to the noise for the various CMB surveys. Right panel: signal and noise for the galaxy number counts. The solid lines represent the $GG$ angular power spectra of the single bin configuration for the six surveys and the dashed lines the corresponding shot noise.

Figure 2

Normalized number density of objects as a function of the redshift for our six surveys. The black dashed line represents the global density distribution $\mathrm{d}N/\mathrm{d}z$ of the survey, and the colored lines represent the corresponding density distribution of the tomographic bins. We show also the central redshift of each bin.

Figure 3

Redshift evolution of the galaxy bias $b_G(z)$ for the six galaxy surveys considered.

Figure 4

Cross-correlation coefficients $r_{\ell }^{TG}$ (left panel), $r_{\ell }^{EG}$ (middle panel), and $r_{\ell }^{\varphi G}$ (right panel) for the six galaxy surveys. The solid lines are obtained using the angular power spectra calculated with linear perturbations only, and the dashed lines include also the nonlinear corrections from halofit.

Figure 5

Signal-to-noise ratio of the $TG$ (left panel) and $\varphi G$ (right panel) cross-correlations as a function of the number of bins $N$. The blue dots correspond to $Planck\otimes$ Euclid-ph-like, the yellow dots to $Planck+\mathrm{SO}$ $\otimes$ Euclid-ph-like, the cyan dots to $\mathrm{LiteBIRD}+\mathrm{S}4$ $\otimes \mathrm{LSST}$, the red dots to $\mathrm{LiteBIRD}+\mathrm{S}4$ $\otimes \mathrm{SKA}1$, and the purple dots to $\mathrm{LiteBIRD}+\mathrm{S}4$ $\otimes \mathrm{EMU}$.

Figure 6

Marginalized 68% and 95% 2D confidence regions for the constraints from $TG$, $\varphi G$ and the combination of both for the $w_0{w}_a\mathrm{CDM}$ model. The left panels correspond to $Planck\otimes$ Euclid-ph-like and the right panels to $\mathrm{LiteBIRD}+\mathrm{S}4\otimes \mathrm{SKA}1$. The green contours correspond to the temperature-galaxy cross-correlation constraints ($TG$), the blue contours to the lensing-galaxy cross-correlation ($\varphi G$), and the red contours to the sum of both ($TG+\varphi G$).

Figure 7

Marginalized 68% and 95% 2D confidence regions for the constraints from CMB, GC, and cross-correlation independently and jointly for the three two-parameter $\mathrm{\Lambda }\mathrm{CDM}$ extensions: dark energy (top), neutrino physics (middle), and primordial universe (bottom). The left panels correspond to $Planck\otimes$ Euclid-ph-like and the right panels to $\mathrm{LiteBIRD}+\mathrm{S}4\otimes \mathrm{SKA}1$. The green contours correspond to the CMB-only constraints ($T,E,\varphi$), the yellow contours to the cross-correlation only ($TG,\varphi G$), the gray contours to the galaxy counts ($GG$), the blue contours to the CMB-GC combination as uncorrelated ($T,E,\varphi \oplus GG$), and the red contours to the combination including cross-correlation ($T,E,\varphi \otimes GG$). In the bottom panel, the green contours are not shown since $f_{\mathrm{NL}}$ is not constrained from the CMB information in our analysis.

Figure 8

Marginalized 68% and 95% 2D confidence regions for the joint constraints on $\mathrm{\Sigma }{m}_{\nu }$ and $w_0$, $w_a$ for a nine-parameter model ($w_0{w}_a\mathrm{CDM}+\mathrm{\Sigma }{m}_{\nu }$) using the combination of $Planck+\mathrm{SO}$ and Euclid-ph-like in the top panel and of $\mathrm{LiteBIRD}+\mathrm{CMB}\text{−}\mathrm{S}4$ and SKA1 in the bottom panel. The gray contours correspond LSS-only constraints ($GG$), the yellow contours to the cross-correlation only ($TG+\varphi \mathrm{G}$), the green contours to the CMB-only constraints ($T,E,\varphi$), the blue contours to the combination of CMB and galaxy clustering as uncorrelated, and the red ones to the combination including cross-correlation.

Figure 9

Marginalized 68% and 95% 2D confidence regions for the joint constraints on the bias nuisance parameters using the combination of $\mathrm{LiteBIRD}+\mathrm{S}4$ and SKA1. The gray contours correspond LSS-only constraints ($GG$), the green contours to the cross-correlation only ($TG+\varphi \mathrm{G}$), the blue contours to the combination of CMB and galaxy clusering as uncorrelated, and the red ones to the combination including cross-correlation.

Figure 10

68% marginalized constraints on the 12 cosmological parameters of the extCDM model as a function of ${\ell }_{\max }$ of the CMB for the tomographic combinations of $Planck+\mathrm{SO}$ with Euclid-ph-like and $\mathrm{LiteBIRD}+\mathrm{S}4$ with SKA1. The dashed lines correspond to the constraints from $\mathrm{CMB}\oplus \mathrm{GC}$ and the solid lines to the ones from $\mathrm{CMB}\otimes \mathrm{GC}$.

Figure 11

68% marginalized constraints on the 12 cosmological parameters of the extCDM model as a function of $k_{\max }$ for the tomographic combinations of Planck with Euclid-ph-like and $\mathrm{LiteBIRD}+\mathrm{S}4$ with SKA1. The dashed lines correspond to the constraints from $\mathrm{CMB}\oplus \mathrm{GC}$ and the solid lines to the ones from $\mathrm{CMB}\otimes \mathrm{GC}$.

Figure 12

Relative impact of the contributions to the total number counts with respect to the density-only angular power spectra for the $GG$ autospectra (top panel) and for the $TG$ (mid panel) and $\varphi G$ (bottom panel) cross-correlation spectra of the five SKA1 redshift bins. The colored lines correspond to the impact of each single term and the black lines to the sum of all the contributions.

Figure 13

Relative impact of the contributions to the total number counts with respect to the density-only angular power spectra for the $GG$ autospectra (top panel) and for the $TG$ (middle panel) and $\varphi G$ (bottom panel) cross-correlation spectra of the 10 Euclid-ph-like redshift bins. The colored lines correspond to the impact of each single term and the black lines to the sum of all the contributions.