This arXiv paper says:

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 $\lambda$CDM 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% confidence level.

If I understand it well, this question might be already obsolete - there is a little deviation from a completely flat Universe into a positive direction. How is it possible? As far I know, there were no recent Planck (or similar) measurements.

How believable is this new development? If it is believable (99% CL in an arXiv paper looks for me strong), what is the estimated radius of the Universe, if we assume a small, constant, positive curvature and spherical topology?

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    $\begingroup$ 99% CL in an arXiv paper looks for me strong - actually it was published on Nature Astronomy (nature.com/articles/s41550-019-0906-9), not just Arxiv. This might increase your belief that the result is well grounded. $\endgroup$
    – Prallax
    Aug 16, 2021 at 20:19
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    $\begingroup$ The bottom line of this paper really is only that there was an under-appreciated degeneracy in curvature with lensing amplitude. That's really all there is. Once you allowed curvature, you would mess up all kinds of other cosmological parameters, in violent disagreement with data. So, lensing amplitude it is. $\endgroup$
    – rfl
    Aug 17, 2021 at 7:10
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    $\begingroup$ Please note that peer-review is off-topic here. Instead of asking how "believable" this paper is (how would you judge the believability of physics answers here if you can't judge the believability of the paper, apparently?), it would be better to formulate this as an objective question, e.g. asking for different explanation of the same data and their respective evidence/arguments. $\endgroup$
    – ACuriousMind
    Aug 17, 2021 at 7:48
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    $\begingroup$ It would be rather bizarre for the universe to have had a positive curvature only for the last two years. $\endgroup$ Aug 18, 2021 at 2:08
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    $\begingroup$ Maybe something like "Did we really determine in 2019 that ...?" $\endgroup$
    – Barmar
    Aug 18, 2021 at 14:26

2 Answers 2


If you aren't already aware, that paper is controversial. That is why it's commonly not asserted that the universe is closed. This quote from the above link is especially relevant:

If this curvature were real, the best-fit cosmology from Planck would have $\Omega_m \sim 0.5$ and $H_0 \sim 50km/s/Mpc$. 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 ... 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.

So no: we have not proven that curvature exists. Also worth emphasizing: the paper also doesn't argue that the universe is closed. It only says that there are several internal inconsistencies in Planck data that can be resolved by assuming the Universe is closed, and suggests we investigate curvature as a solution to cosmological problems.


Figure 3 of the paper demonstrates that $\Omega_{m} \sim 0.5$ and $H_{0} \sim 50$ km/s/Mpc as the link in Allure's answer points out, which is in significant tension with other observations and studies.

The radius of curvature scale can be given by $R_{c} \sim (c/H_{0})|\Omega_{k}|^{-0.5}$ (which you can find on page 2 of the arxiv link).

For their estimate of $-\Omega_{k} \sim 0.007$ this would imply a radius of curvature of 170 Billion light years, assuming $H_{0} \sim 70$ km/s/Mpc. 200 billion for their estimate of $H_{0} \sim 50$ km/s/Mpc. For their higher estimate of $-\Omega_{k} \sim 0.09$, the radius of curvature would be 60 billion light years.

No single paper in any journal of any prestige can tell you if a result is correct. This is the role of scientific consensus and community. So that science is much greater than the sum of its individuals. Confidence intervals are estimates and very susceptible to error as well.

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    $\begingroup$ I disagree with "consensus and community" being the ultimate source of truth. $\endgroup$ Aug 18, 2021 at 21:59
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    $\begingroup$ I had accepted both of your answers, but I can give only a single pipe. $\endgroup$
    – peterh
    Aug 23, 2021 at 15:21

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