Cosmology with the Planck $T-E$ correlation coefficient

Tensions in cosmological parameters measurement motivate a revisit of the effects of instrumental systematics. In this article, we focus on the Pearson’s correlation coefficient of the cosmic microwave background temperature and polarization E modes ${\mathcal{R}}_{\ell }^{\mathrm{TE}}$, which has the property of not being biased by multiplicative instrumental systematics. We build a ${\mathcal{R}}_{\ell }^{\mathrm{TE}}$-based likelihood for the Planck data and present the first constraints on $\mathrm{\Lambda }\mathrm{CDM}$ (Lambda cold dark matter) parameters from the correlation coefficient. Our results are compatible with parameters derived from a power-spectra-based likelihood. In particular, the value of the Hubble parameter $H_0$ characterizing the expansion of the Universe today, $67.5\pm 1.3\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$, is consistent with the ones inferred from standard cosmic microwave background analysis. We also discuss the consistency of the Planck correlation coefficient with the one computed from the most recent ACTPol power spectra.

Figure 1

One-dimensional posterior distributions of the six $\mathrm{\Lambda }\mathrm{CDM}$ parameters using a simulated dataset ($TT$, $TE$, $EE$) biased with a constant polarization efficiency (1), with a polarization transfer function (1), and with a temperature transfer function (1). The transfer function model is described in Eq. (2). We vary the ${\ell }_{\max }$ parameter from 200 to 1000. The dashed black lines correspond to the fiducial model. Gray bands correspond to $\pm 1\sigma {\mathcal{R}}_{\ell }^{\mathrm{TE}}$ constraints.

Figure 2

$\mathrm{\Lambda }\mathrm{CDM}$ parameters constraints for a set of $N_{\mathrm{sim}}=100$ simulations (gray). The average posterior distribution is shown in red. The distance from the average to the input value (in sigma units) is displayed in the top right corner of each panel. Mean posterior distributions are consistent with input values of the simulations.

Figure 3

Two-dimensional posterior distributions for $H_0$, ${\mathrm{\Omega }}_b{h}^2$, and ${\mathrm{\Omega }}_c{h}^2$ constrained using $C_{\ell }$-based likelihood (blue) or the ${\mathcal{R}}_{\ell }^{TE}$-based likelihood (red). The expected $1\sigma$ scatter between parameters estimated from the two different likelihoods is displayed with gray dotted lines.

Figure 4

One-dimensional posterior distribution for $\log ({10}^{10}{A}_s)$ using the $C_{\ell }$-based likelihood (blue) or the ${\mathcal{R}}_{\ell }^{\mathrm{TE}}$-based likelihood (red). The correlation coefficient constraint is sensitive to $A_s$ only through the effect of lensing on the power spectra.

Figure 5

Top: CMB-only ${\mathcal{R}}_{\ell }^{\mathrm{TE}}$ correlation coefficient for Planck PR4 data (red) and ACT data (green) after foreground marginalization. The best-fit correlation coefficient (gray dashed line) is computed with camb from the hillipop $C_{\ell }$-likelihood best fit cosmology. Bottom: residual plot with respect to the binned correlation coefficient best fit.