Relieving tensions related to the lensing of CMB temperature power spectra

The angular power spectra of the cosmic microwave background (CMB) temperature anisotropies reconstructed from Planck data seem to present too much gravitational lensing distortion. This is quantified by the control parameter $A_L$ that should be compatible with unity for a standard cosmology. With the Class Boltzmann solver and the profile-likelihood method, for this parameter we measure a 2.6$\sigma$ shift from 1 using the Planck public likelihoods. We show that, owing to strong correlations with the reionization optical depth $\tau$ and the primordial perturbation amplitude $A_s$, a $\sim2\sigma$ tension on $\tau$ also appears between the results obtained with the low ($\ell\leq 30$) and high ($30<\ell\lesssim 2500$) multipoles likelihoods. With Hillipop, another high-$\ell$ likelihood built from Planck data, this difference is lowered to $1.3\sigma$. In this case, the $A_L$ value is still in disagreement with unity by $2.2\sigma$, suggesting a non-trivial effect of the correlations between cosmological and nuisance parameters. To better constrain the nuisance foregrounds parameters, we include the very high $\ell$ measurements of the Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT) experiments and obtain $A_L = 1.03 \pm 0.08$. The Hillipop+ACT+SPT likelihood estimate of the optical depth is $\tau=0.052\pm{0.035,}$ which is now fully compatible with the low $\ell$ likelihood determination. After showing the robustness of our results with various combinations, we investigate the reasons for this improvement that results from a better determination of the whole set of foregrounds parameters. We finally provide estimates of the $\Lambda$CDM parameters with our combined CMB data likelihood.