*Cosmo Comments*. The following comments can also be viewed as annotations on the paper via Hypothesis.

This paper, published in a high-profile journal (Nature Astronomy), sets out a controversial position that Planck evidence points to a closed Universe rather than flat, and that this constitutes a crisis for cosmology. It seems best to view this paper not as a traditional piece of scientific research, but rather as a newspaper opinion or editorial piece: it does not contain original information, but instead uses the Nature platform to argue a provocative position at odds with the conclusions of most previous work.

The situation with Planck data is well known and has been discussed at length in Planck papers from 2013 onwards. In a nutshell, the residuals in the TT power spectrum from the best-fit flat LCDM model with $A_L = 1$ show an oscillatory pattern at high multipoles (particularly $\ell > 1100$), which looks approximately like the effect of an additional lensing smoothing. This is shown in Figure 24 of the Planck 2018 cosmology results paper (1807.06209) and was commented on in other papers and in previous Planck releases as well.

These residuals mean that if $A_L$ is left free, the fit to the Planck temperature and polarisation data mildly favours $A_L > 1$. The level of this discrepancy is a little over $2\sigma$ using the CamSpec likelihood, or a little less than $3\sigma$ using the plik likelihood. Certainly this is something of interest and should be investigated further, but to characterise this as a “crisis” in cosmology is not justified.

It is worth emphasising that the $A_L$ discrepancy is mostly driven by the TT power spectrum. In particular, adding the CMB-lensing data, as measured from the four-point function of the temperature anisotropies, significantly pushes the best-fit model towards LCDM. The same phenomenon is observed in the recent extended Camspec likelihood analysis that also used a larger sky area (1910.00483).

Where does curvature come in? Allowing $\Omega_K$ to be non-zero opens up a very well-known degeneracy direction for fits to the CMB temperature and polarisation data, which allows $\Omega_m$ to increase (and $H_0$ to decrease), while still keeping the very well-measured angular scale to the first acoustic peak fixed. A larger $\Omega_m$ simulates the effect of enhanced lensing $A_L > 1$, thus the effect of the residuals that look like additional lensing smoothing means that the Planck TT, TE, EE + lowE data alone slightly favour a negative $\Omega_K$ (with the level of preference again depending a bit on which likelihood is used).

We stress again that all this is well known and already published (see e.g. Section 7.3 of the Planck 2018 cosmology results paper); we did not need a Nature paper to learn this. If this curvature were real, the best-fit cosmology from Planck would have $\Omega_m \sim 0.5$ and $H_0 \sim 50\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. Is this remotely reasonable given other cosmology data? No. Data from CMB lensing, BAO, weak lensing, direct distance ladder measurements and a host of other observations rule it out – again, we did not need a Nature paper to learn this. The vast majority of this paper merely consists of restating this fact in different ways.

Given this position and the fact that even a model with $A_L = 1$ and zero curvature still gives a reasonable $\chi^2$ for the fit to the Planck data, we think the natural conclusion to draw is that whatever the explanation for this moderate discrepancy is, it is not curvature.

*[These comments were shared with us by a member of the cosmology community. They do not necessarily reflect the opinion of the Cosmo Comments team.]*