Current data are consistent with flat spatial hypersurfaces in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>CDM</mml:mi></mml:math> cosmological model but favor more lensing than the model predicts

Abstract

We study the performance of three pairs of tilted, and a pair of untilted, $\mathrm{\ensuremath{\Lambda}}$ cold dark matter ($\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$) cosmological models, with three of these four pairs allowing for nonflat spatial hypersurfaces, against cosmic microwave background (CMB) temperature and polarization power spectrum data (P18), measurements of the Planck 2018 lensing potential power spectrum (lensing), and a large compilation of non-CMB data (non-CMB). For the eight models, we measure cosmological parameters and study whether or not pairs of the datasets (as well as subsets of them) are mutually consistent in these models. Half of these models allow the lensing consistency parameter ${A}_{L}$, which rescales the gravitational potential power spectrum, to be an additional free parameter to be determined from data, while the other three have ${A}_{L}=1$ which is the theoretically expected value. The pair of untilted nonflat $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ models are incompatible with P18 data. The tilted spatially flat models assume the usual primordial spatial inhomogeneity power spectrum that is a power law in wave number. The tilted nonflat models assume either the primordial power spectrum used in the Planck group analyses [Planck $P(q)$], which has recently been numerically shown to be a good approximation to what is quantum-mechanically generated from a particular choice of closed inflation model initial conditions, or a recently computed power spectrum [new $P(q)$] that quantum-mechanically follows from a different set of nonflat inflation model initial conditions. In the tilted nonflat 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 $3\ensuremath{\sigma}$ but not the new $P(q)$ model. No significant differences are found when cosmological parameter constraints obtained with two different datasets are compared within the standard tilted flat $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. While both P18 data and non-CMB data separately favor a closed geometry, with spatial curvature density parameter ${\mathrm{\ensuremath{\Omega}}}_{k}&lt;0$, when $\mathrm{P}18+\mathrm{non}\text{\ensuremath{-}}\mathrm{CMB}$ data are jointly analyzed the evidence in favor of nonflat hypersurfaces subsides. Differences between P18 data and non-CMB data cosmological constraints subside when ${A}_{L}$ is allowed to vary. From the most restrictive $\mathrm{P}18+\mathrm{lensing}+\mathrm{non}\text{\ensuremath{-}}\mathrm{CMB}$ data combination, we get almost model-independent constraints on the cosmological parameters and find that the ${A}_{L}&gt;1$ option is preferred over the ${\mathrm{\ensuremath{\Omega}}}_{k}&lt;0$ one, with the ${A}_{L}$ parameter, for all models, being larger than unity by $\ensuremath{\sim}2.5\ensuremath{\sigma}$. According to the deviance information criterion, in the $\mathrm{P}18+\mathrm{lensing}+\mathrm{non}\text{\ensuremath{-}}\mathrm{CMB}$ 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 $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. These data are consistent with flat spatial hypersurfaces but more and better data could improve the constraints on ${\mathrm{\ensuremath{\Omega}}}_{k}$ and might alter this conclusion. Error bars on some cosmological parameters are significantly reduced when non-CMB data are used jointly with $\mathrm{P}18+\mathrm{lensing}$ data. For example, in the tilted flat $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model for data, the Hubble constant ${H}_{0}=68.09\ifmmode\pm\else\textpm\fi{}0.38\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$, which is consistent with that from a median statistics analysis of a large compilation of ${H}_{0}$ measurements, as well as with a number of local measurements of the cosmological expansion rate. This ${H}_{0}$ error bar is 31% smaller than that from $\mathrm{P}18+\mathrm{lensing}$ data alone.

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