Quite a lot of fuss was being made in cosmological circles while I was away last week concerning a paper that had just been published in *Nature Astronomy* by Eleonora Di Valentino, Alessandro Melchiorri and Joe Silk that claims evidence from the Planck Cosmic Microwave background and other data that the Universe might be closed (or at least have positive spatial curvature) in contrast to the standard cosmological model in which the spatial geometry is Euclidean. *Nature Astronomy* is behind a paywall but the paper is available for free on the arXiv here. The abstract reads:

The recent Planck Legacy 2018 release has confirmed the presence of an enhanced lensing amplitude in CMB power spectra compared to that predicted in the standard ΛCDM model. A closed universe can provide a physical explanation for this effect, with the Planck CMB spectra now preferring a positive curvature at more than 99% C.L. Here we further investigate the evidence for a closed universe from Planck, showing that positive curvature naturally explains the anomalous lensing amplitude and demonstrating that it also removes a well-known tension within the Planck data set concerning the values of cosmological parameters derived at different angular scales. We show that since the Planck power spectra prefer a closed universe, discordances higher than generally estimated arise for most of the local cosmological observables, including BAO. The assumption of a flat universe could, therefore, mask a cosmological crisis where disparate observed properties of the Universe appear to be mutually inconsistent. Future measurements are needed to clarify whether the observed discordances are due to undetected systematics, or to new physics, or simply are a statistical fluctuation.

I think the important point to take from this study is that estimates of cosmological parameters obtained from Planck are relatively indirect, in that they involve the simultaneous determination of several parameters some of which are almost degenerate. For example, the `anomalous’ lensing amplitude discussed in this paper is degenerate with the curvature so that changing one could mimic the effect on observables of changing the other; see Figure 2 in the paper.

It’s worth mentioning another (and, in my opinion, better argued) paper on a similar topic by Will Handley of Cambridge which is on the arXiv here. The abstract of this one reads:

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 3σ. 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.

Figure 1 makes a rather neat point that the combination of Planck and Baryon Acoustic Oscillations does not separately give consistent values for the Hubble constant and the curvature and neither does the combination of Planck and direct Hubble constant estimates:

I don’t know what the resolution of these tensions is, but I think it is a bit dangerous to dismiss them simply as statistical flukes. They might be that, of course, but they also might not be. By shrugging one’s shoulders and ignoring such indications one might miss something very fundamental. On the other hand, in my opinion, there is nothing here that definitely points the finger at spatial curvature either: it is possible that there is something else missing from the standard model that, if included, would resolve these tensions. But what is the missing link?

Answers on a postcard, or through the comments box.