Current data are consistent with flat spatial hypersurfaces in the $\Lambda$CDM cosmological model but favor more lensing than the model predicts

Abstract

We study the performance of three pairs of tilted $\Lambda$CDM cosmological models, two pairs allowing for non-flat spatial hypersurfaces with CMB temperature and polarization power spectrum data (P18) from Planck, P18 lensing (lensing), and non-CMB data (non-CMB). For the six models, we measure cosmological parameters and study whether or not pairs of the data sets are mutually consistent in these models. Half of these models allow the lensing consistency parameter $A_L$ to be an additional free parameter, while the other three have $A_L = 1$. The tilted spatially-flat models assume the usual primordial spatial inhomogeneity power spectrum. The tilted non-flat models assume either the primordial power spectrum used in the Planck group analyses [Planck $P(q)$] or a recently computed power spectrum [new $P(q)$]. In the tilted non-flat models with $A_L=1$ we find differences between P18 data and non-CMB data cosmological parameter constraints, which are large enough to rule out the Planck $P(q)$ model at 3$\sigma$ but not the new $P(q)$ model. While both P18 data and non-CMB data separately favor a closed geometry when P18+non-CMB data are jointly analyzed the evidence in favor of non-flat hypersurfaces subsides. Differences between P18 data and non-CMB data cosmological constraints subside when $A_L$ is allowed to vary. From the most restrictive P18+lensing+non-CMB data combination we get almost model-independent constraints and find that the $A_L>1$ option is preferred over the $\Omega_k<0$ one, with the $A_L$ parameter, for all models, being larger than unity by $\sim 2.5\sigma$. According to the deviance information criterion, in the P18+lensing+non-CMB analysis, the varying $A_L$ option is on the verge of being strongly favored over the $A_L=1$ one, which could indicate a problem for the standard tilted flat $\Lambda$CDM model (Abridged abstract).