Quantifying Tensions between CMB and Distance Datasets in Models with Free Curvature or Lensing Amplitude

Recent measurements of the Cosmic Microwave Background (CMB) by the Planck Collaboration have produced arguably the most powerful observational evidence in support of the standard model of cosmology, i.e. the spatially flat $\Lambda$CDM paradigm. In this work, we perform model selection tests to examine whether the base CMB temperature and large scale polarization anisotropy data from Planck 2015 (P15) prefer any of eight commonly used one-parameter model extensions with respect to flat $\Lambda$CDM. We find a clear preference for models with free curvature, $\Omega_\mathrm{K}$, or free amplitude of the CMB lensing potential, $A_\mathrm{L}$. We also further develop statistical tools to measure tension between datasets. We use a Gaussianization scheme to compute tensions directly from the posterior samples using an entropy-based method, the surprise, as well as a calibrated evidence ratio presented here for the first time. We then proceed to investigate the consistency between the base P15~CMB data and six other CMB and distance datasets. In flat $\Lambda$CDM we find a $4.8\sigma$ tension between the base P15~CMB data and a distance ladder measurement, whereas the former are consistent with the other datasets. In the curved $\Lambda$CDM model we find significant tensions in most of the cases, arising from the well-known low power of the low-$\ell$ multipoles of the CMB data. In the flat $\Lambda$CDM $+A_\mathrm{L}$ model, however, all datasets are consistent with the base P15~CMB observations except for the CMB lensing measurement, which remains in significant tension. This tension is driven by the increased power of the CMB lensing potential derived from the base P15~CMB constraints in both models, pointing at either potentially unresolved systematic effects or the need for new physics beyond the standard flat $\Lambda$CDM model.