Curvature tension: evidence for a closed universe

TL;DR

This paper investigates a curvature tension among Planck 2018 CMB data, CMB lensing, and BAO within the curved extension of the standard model ($K\Lambda$CDM). Using Bayesian evidence and the suspiciousness statistic, it finds that Planck data alone prefer a closed universe with $\Omega_K \approx -4.5\%$ and odds ~$50:1$ against flatness, while lensing and BAO pull constraints toward flatness and are inconsistent with Planck at $\sim$2.5–3$\sigma$. Including all datasets yields strong tensions, and curvature generally aggravates these tensions except in the Planck+lensing vs SH0ES case. The study highlights that curvature inferences from Planck alone should be treated with caution, and emphasizes the need for careful treatment of dataset combinations and potential systematics or new physics. Overall, it suggests that a definitive claim about a flat or curved universe requires resolving substantial cross-dataset tensions and further investigation into curvature’s role in cosmology.

Abstract

The curvature parameter tension between Planck 2018, cosmic microwave background lensing, and baryon acoustic oscillation data is measured using the suspiciousness statistic to be 2.5 to 3$σ$. Conclusions regarding the spatial curvature of the universe which stem from the combination of these data should therefore be viewed with suspicion. Without CMB lensing or BAO, Planck 2018 has a moderate preference for closed universes, with Bayesian betting odds of over 50:1 against a flat universe, and over 2000:1 against an open universe.

Figures (4)

• Figure 1: Three tensions from \ref{['fig:tensions']} plotted in the $(\Omega_K,H_0)$ plane using anestheticanesthetic. In the first panel, whilst the lensing posterior alone is barely distinguishable from the prior, when combined with Planck, lensing has draws the combined posterior significantly toward flatness, at a tension of 2.5$\sigma$. The second panel shows BAO's preference for a flat universe. The BAO posterior is disconnected from the Planck posterior, at a tension of $3\sigma$. Finally, in the third panel the Planck-S$H_0$ES inconsistency in the curved case is shown to be 4.5$\sigma$.
• Figure 2: Parameter tensions computed using anestheticanesthetic. The top line shows the severe tension between S$H_0$ES and Planck detailed in tensiondimensionality updated for Planck 2018 data. For $\Lambda$CDM Planck, lensing and BAO are all consistent. For $K\Lambda$CDM, Planck and BAO are strongly inconsistent, whilst Planck and lensing are moderately inconsistent. The ninth and final rows indicate the triple tension from \ref{['eqn:triple_tension']}, and show strong mutual inconsistency between Planck, lensing and S$H_0$ES, and moderate mutual inconsistency between Planck, lensing and BAO. Curvature in general enhances tension, but relaxes it for Planck+lensing vs S$H_0$ES. The large error bars for lensing vs S$H_0$ES occur due to the fact their shared constrained parameters have $\tilde{d}\ll1$.
• Figure 3: Model comparison between curved $K\Lambda$CDM and flat $\Lambda$CDM cosmologies. Positive $\Delta\log\mathcal{Z}$ indicates favouring of curved universes. The Occam regularisation penalty associated with the additional $\Omega_K$ parameter is shown in orange, estimated via the difference in KL divergence between the two models.
• Figure 4: One- and two-dimensional marginalised priors and posteriors for the seven cosmological parameters of the $K\Lambda$CDM cosmology, plotted with anestheticanesthetic. Plots on the diagonal are one-dimensional marginalisations, and off-diagonal plots are contour plots showing marginal iso-probability contours containing 68% and 95% of the marginalised posterior mass. The prior in blue and the Planck posterior in red are included in all plots, below-diagonal elements show lensing posteriors, above-diagonal elements show posteriors for BAO. The left-most column and top-most row detail curvature, where the tension between the posteriors can be seen clearly. The remaining parameters show no discernible tensions. Lensing constrains approximately two parameters, a linear combination of $\Omega_ch^2$ and $n_s$, and a linear combination of $\ln(10^{10}A_s)$ and $\tau$, which is reflected in the Bayesian model dimensionality of $\tilde{d}_\mathrm{lensing}=1.8\pm1$. BAO has a model dimensionality of $\tilde{d}_\mathrm{BAO}=2.6\pm0.1$, consistent with its constraints on $\ln(10^{10}A_s)$ and $\Omega_K$, and its partial constraint on $\Omega_ch^2$.