What Shape Is the Universe? A New Study Suggests We’ve Got It All Wrong (by Wolchover)Di Valentino, Melchiorri, and Silk point out that the 2018 Planck data show a surplus of lensing, relatively to the Lambda-CDM model's prediction. And their straightforward interpretation is that the density of the Universe is some 5% higher than the critical one – say 6.0 instead of 5.7 hydrogen atoms per cubic meter.
Planck evidence for a closed Universe and a possible crisis for cosmology (Nature)
The degree of certainty that these two densities actually mismatch is said to be 99% which is not a very high certainty – it's really less than 3 sigma if directly translated. The authors agree that various similar conclusions obtained at this ambitious accuracy probably contradict each other. That is the content of the talk about the "crisis in cosmology".
If the density were above the critical density, then the spatial slices of the Universe would be positively curved – they would have to be spherical or orbifolds of a sphere. The radius of such a sphere would be "hundreds of billions if not a trillion or so of light years" – larger than the visible Universe, of course. I think it doesn't make much sense to even calculate the precise number because the relative inaccuracy is huge. I think that most good cosmologists still believe today that the radius of curvature of the slices is much greater than a trillion light years.
Since the late 1990s (when the cosmological constant was announced to be positive for the first time), cosmologists – and many of us who emulated them – often said that cosmology had entered the era of "high-precision cosmology" and became as hard a science as particle physics or others. In the light of this bold claim, this self-confident proposition may also be interpreted as mirage – one big fiction whose only justification was one consistency check in the measurement of the basic cosmological observables, and that precise consistency check could have been a coincidence.
A few years ago, we liked to say that the age of the Universe was 13.83 billion years, at least I liked that number, with the implicit hope that the accuracy would soon be even better than that. I think it was rubbish, the more comprehensive analyses of the data indicate. We must probably return to "roughly 14 billion" and we're not even quite sure that the age is closer to 14 than to 13 or 15 billion years.
Note that there are very good hardcore theoretical reasons to believe that this "spherical" claim is wrong. In particular, inflation or something equivalent to it is apparently needed to explain the huge enough size and flatness of the Universe that is already experimentally unquestionable. But when you have many e-foldings of inflation, it's rather unlikely that the Universe only expands to "a slightly greater size" than the current visible Universe. It's much more likely that you get more inflation than needed – and that the density is going to be much closer to the critical density than 5% away. But of course, academically speaking, a novel anthropic-like argument might hold and imply that the densities should differ by a few percent.
Qualitative, topological arguments in quantum gravity also exist. At the very vague level, I think that a spherical Universe is just fine – a nice topology for God to knead it in His hands. On the other hand, there have been claims rooted in fancy enough quantum gravity that the hyperbolic shape of the slices was preferred because of some detailed analyses of the tunneling in eternal inflation and other things. We still don't know the answer to any of these questions. We don't know what state-of-the-art theory should "clearly prefer".
I think that a much deeper understanding of the theory is more likely to make us certain enough about these questions than the experiments. With hindsight, the progress in string theory – especially progress in aspect of quantum gravity such as holography and the information loss plus the strengthening swampland-related results – seems like the most solid, irreversible progress in recent 2 decades or so.
Some rondo by Haydn. Klára Gibišová, the pianist, is eight.