## [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

 Authors: Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk Abstract: 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 $99 \%$ 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. [PDF]  [PS]  [BibTex]  [Bookmark]

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### [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

This paper was commented on through 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.]

Alessandro Melchiorri
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

Dear (I presume since I could not find any name) Anonymous cosmologists,

- "(The paper) 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."

I am sorry but the anonymous cosmologists should take a more clear position. Is Nature Astronomy a " high-profile journal" or a "platform"? because if this is a high-profile journal we don't understand how we could use it as a "platform to argue a provocative position" and how we could manage to publish unoriginal information there. In reality, we simply wrote a paper, we submitted it to Nature Astronomy and after some rounds of reports from two (we presume high-level) referees, we got it accepted. This clearly does not assure that our paper is original but, at least, certifies that not everyone has such bad opinion of our work! Besides, since when having a provocative position in science is a bad thing? Therefore the comment above is just a personal opinion. Honestly, the tone of the comment seems to suggest that the anonymous cosmologists are so upset by our work at the level of being personally offended. Unfortunately I am a cosmologist and not a psycologist...so I can't give any help here.

- "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 𝐴𝐿=1 show an oscillatory pattern at high multipoles (particularly ℓ>1100
ℓ>1100), which looks approximately like the effect of 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.

There are several wrong statements in this comment. First of all, the AL parameter (introduced by Calabrese, Slosar, *Melchiorri*, Smoot 2008) is not a physical parameter. By varying Al, we just change the lensing signal by hand but it is clear that you can't do this physically if you also don't increase the cold dark matter density or ... modify general relativity! Anyway, the anonymous cosmologists are claiming that the residuals (respect to LCDM) follow ("approximately") the effect of an additional lensing smoothing...exactly like having more lensing. If the anonymous cosmologists, by looking at Figure 24, have the artistic impression of a statistical fluke I am happy for them. However, I would prefer a more quantitative argument here... if the data follow AL>1 why I should discard it ? just because it is not in agreement with LCDM? Finally, if the situation is present since 2013 and after several data releases should we not consider it a little bit more seriously?

- "These residuals mean that if 𝐴𝐿 is left free, the fit to the Planck temperature and polarisation data mildly favours 𝐴𝐿>1. The level of this discrepancy is a little over 2𝜎 using the CamSpec likelihood, or a little less than 3𝜎 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.

Indeed we have investigated it a little bit more... However, our paper is on curvature, not AL. In this case the indication for Omega_k<0 is stronger and well above the three standard deviations when using the Plik likelihood (that is the baseline likelihood used by the Planck collaboration). Secondly, we are not claiming a crisis because of Omega_k<0 but because when you leave this parameter to vary freely then other cosmological observables as BAO are in tension with Planck. We wrote this in the abstract of our paper but probably the anonymous cosmologist were so upset after just reading the title that could not read the rest...

-"It is worth emphasising that the 𝐴𝐿 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)."

Again here we are discussing Omega_k, not AL. From the Planck Tables, TT alone gives Omega_k=-0.055^{+0.044}_{-0.050} at 95% C.L. while TTTEEE gives Omega_k=-0.044^{+0.033}_{-0.034} at 95% c.l.. So it is true that TE brings Omega_k towards values more consistent with zero, but at the same time the error shrinks so actually the statistical significance increases. So the first part of the comment is wrong.
On the second part. It is true that CMB lensing brings Omega_k closer to zero but this dataset is now in tension with Planck. Moreover the CMB lensing likelihood has not been fully tested for closed models and assumes a fiducial flat universe (see pag. 34 of the Planck lensing paper).
This is a more general comment to any external dataset...how well these datasets have been tested in case of a closed universe or extra physics ?
Finally, the new Camspec likelihood result is interesting but a) is not the Planck baseline likelihood, b) it is not publically available. Once it will be publically available we will happy to consider it.

-"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 Ω𝑚∼0.5
. 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."

I totally agree that we don't need a nature paper to understand this and indeed we hope to have cited the relevant paper that discussed back in the 90's the CMB degeneracies. And also you don't need a Nature paper to understand that a combination of Planck+BAO gives a flat universe. What was missing was probably an evaluation of how much Planck and BAO are discrepant in case of a closed Universe. Also, given the interest on the media and on the arxiv, it seems that the preference for a closed Universe from the Planck data was not clearly addressed in the Planck papers.
Maybe we restated things in a better way...

-"Given this position and the fact that even a model with 𝐴𝐿=1 and zero curvature still gives a reasonable 𝜒2 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."

In our paper we don't propose curvature as a solution. We clearly state that curvature introduce a crisis. The problem is why you should not explore curvature ? just because you don't like it ? please consider that there is no "new physics" in curvature but just general relativity...why we should accept a cosmological constant and not a closed universe?
In any case, you are certainly free to believe that LCDM is in optimal shape, but please don't complain if someone else thinks differently and publish this on Nature...is how science works!

Alessandro Melchiorri
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

For the sake of clarity let me explain here the content of our paper.

1- We explore models with curvature. This is not only legit (curvature is expected in GR) but mandatory since the only way to constrain it is with observations.
2- We see that Planck angular spectra data prefers a closed universe at more than 99% C.L. (We use the Plik official Plank baseline likelihood).
3- We found that once the curvature is considered tensions between Planck angular spectra and datasets as BAO, Cosmic Shear, H_0, etc are all present at more than 99% c.l.

Point 1 is present in the Planck papers. Point 2 is briefly mentioned. Point 3 is not present and is the core of our paper.
(Point 3 is also discussed by a nice paper by Will Hadley (https://arxiv.org/abs/1908.09139)).

Clearly, if LCDM was the correct model you would not expect points 2 and 3!
In a flat LCDM model, you would find Planck consistent with a flat universe and all datasets (modulo systematics) compatible with Planck.
We don't see this and this is the reason why we claim a possible crisis.

We don't believe that the Universe is closed but claiming now that the universe is flat with a precision below 0.5% it is also quite premature.
Clearly, we could miss something important by not considering the possibility of curvature at percent level as suggested by several models (see e.g https://arxiv.org/abs/1007.3086) given the current experimental situation.
Moreover, CMB angular spectra are the best single observable for measuring the curvature and you get a closed Universe. If you combine current BAO, SN-Ia, and H_0 data you also get a closed universe but with a different Hubble constant (see Figure in our paper).

The solutions could be additional physics, systematics or a statistical fluctuation. Given the significance of the tensions (above 99% c.l.) we think that a statistical fluctuation is less probable...

That's all and thank you for your attention...

Alessandro

Alessandro Melchiorri
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

I re-post here a couple of e-mails with the editors.

thank you for your reply but I am afraid you are completely wrong here.There are three BIG differences between an anonymous referee report from a journal and your method:
1- I receive an anonymous referee report because I want to publish my research on a journal. In this case, my paper has already been published, so why I should receive a further anonymous report?
2- The anonymous report that I receive from Nature is PRIVATE and not PUBLIC. I could bring Nature to a tribunal if they disclose it.
3- In the case of Nature, there is the possibility of a double-blinded review. Both the authors and the referee are anonymous. Do we have here the same situation?

If our paper is wrong, real scientists no matter if they are PhD or professors can easily write a comment on it and having it published on Nature! This is the correct way to have a scientific discussion. If they don't do this is probably because they know that this is not the case and that their comment would be easily rejected. So they prefer to use your platform to spread insults and fake statements without being accountable for it.
You defend your choice by saying that in this way the poor commenters "feel more free to be honest" against big and important professors (thank you, I feel like Robert Duvall in this case). But did you notice that one author of our paper is a postdoc and posting an aggressive comment like the one you posted could be harmful? you don't think that in this way the commenters are "feel more free to be aggressive" and to attack other competitors in a future search for jobs?
Your comment about that "both commenters have received a Ph.D. and are employed in respectable Universities"  makes no sense. I can't be accountable for my words if I have a Ph.D.? I also have a Ph.D. and I am employed to a respectable University as my co-authors.
I am now posting the entire discussion to cosmocoffe.

Regards
Alessandro

Dear Alessandro,
In your reply you were particularly concerned about the anonymous nature of Cosmo Comments. We might not have made it as clear as we could, but the editorial team behind Cosmo Comments are not anonymous. We are (in alphabetical order) Pierre Fleury, Shaun Hotchkiss, Robert Reischke, Benjamin Wallisch and Yuanyuan Zhang. We did not write the comment that we published on your paper, but we are the ones who are accountable regarding whether there is merit to publishing it. This does not necessarily mean we agree with the comment, but it does mean that we think it adds merit to the discussion. We also think your replies add merit to the discussion and we are happy you posted them.
We can reveal that this particular comment on your paper was actually merged from two separate comments that we received. We can also attest that both commenters are practising cosmologists who are employed at and have received PhDs from respectable international universities.
You might still disagree with even this level of anonymity (although, note that it is the precise level of anonymity present in ordinary peer review, including at Nature Astronomy). There are pros and cons on both sides of the anonymity debate. Clearly one con is that the commenter has less accountability. We hope that our own accountability mediates some of that. We also have a code of conduct such that we would not publish a comment if it was particularly unfair or personal. The main pro is that commenters feel more free to be honest. We note that two of the authors on your paper are well established professors at major universities, and your paper has received much public attention. It takes a certain level of confidence even to make an anonymous comment publicly in such a scenario.
You were of course annoyed by the comment and strongly disagree with it. We understand this and we were obviously aware that the comment was broadly negative. However, we do not want to censor our platform and only post positive comments. Nor do we wish to censor negative comments based on the reputation of the authors or the journal of publication. Without a platform like Cosmo Comments negative comments like this are still shared through local journal clubs, conferences and private online platforms, often without any rebuttal. One of our primary goals is to bring those comments into the public domain precisely so that the authors of the paper can see the comments and publicly respond to them. The rest of the community can then make their own judgement, or even participate in the discussion, having seen both the criticism and the rebuttal.
We hope this alleviates some of your concern regarding the nature of Cosmo Comments. If you still have any concerns or suggestions for improvement, we would be very keen to discuss them with you. We are at an early experimental phase and you are one of only two authors to have responded to a comment so far. This means that you can give us feedback that few others can.
Regarding your replies on CosmoCoffee, we are keen to repost authors’ responses on all the platforms we use. We do however have a code of conduct, which does not allow comments that make personal criticisms. Your second comment is great and we would like to repost it as is on all platforms. For your first comment, we would also like to repost it, but would ask if we could first remove all the parts of it that directly criticise the authors of the original comment. However, all the criticisms you make of the comment itself should and would certainly remain. Of course, we would send you a draft for approval before reposting it.
Thanks again for the reply. We are sincerely grateful for it and we expect that the community will benefit from reading it.
Best wishes,
The Cosmo Comments team: Pierre Fleury, Shaun Hotchkiss, Robert Reischke, Benjamin Wallisch and Yuanyuan Zhang.

Antony Lewis
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

Moreover the CMB lensing likelihood has not been fully tested for closed models and assumes a fiducial flat universe (see pag. 34 of the Planck lensing paper).
Actually the figure on page 34 of the lensing paper specifically shows that the lensing likelihood *is* valid for closed universes. It's true that the fiducial model is flat, but the likelihood accounts for changes away from the fiducial model (see Sec 3.2). Fig 25 of the lensing paper shows that this works well even for substantially non-flat cosmology.

Alessandro Melchiorri
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

Hi

thank you, Antony.
My question here is not if the results are consistent but how precise is the lensing likelihood in the case of omegak models.
If I understand correctly you have A=0.88\pm0.02 when fitting with a Omega_k=-0.04 model a single FP10 simulation in the range 2-2048 (that is the aggressive range). Now a Omega_k=-0.044 universe should match a lensing amplitude of 1.19, while 1/0.88=1.136. I am right that this means that you may get a possible systematic of \Delta A_L=0.054 ?
Does it makes any sense?

Many thanks
Alessandro

Antony Lewis
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

I think the comparison is between our Eq 74 and Eq 75b - the same fixed realisation analysed assuming different fiducial models - which are consistent to < 1% in lensing amplitude (A_L)

Lloyd E Knox
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

I think papers that don't have "new results," but do have a new analysis of what to take away from existing results, can be very valuable. I read this paper with interest. But to the extent that the main benefit of extra curvature is increased lensing power, at the end of the day, it seems pretty clear curvature is not the answer, or even part of the answer. We already know, from the less model-dependent lensing reconstruction, that whatever the solution to the Alens problem is, it's NOT extra lensing power. I think I first heard this statement in a talk by Antony Lewis and I agree with him.

If curvature were a part of the explanation of "Alens> 1" then we would also need late time modifications to cosmology to explain a host of late-time measurements. These late-time modifications would have to be such as to reduce the reconstructed CMB lensing power, which would then defeat what Omega_k was bringing us in the first place. I guess this is why the authors speculate about systematic errors causing the difference between, e.g., H0=55 and 74, and for the BAO discrepancies too. But I don't find it plausible that the systematic errors have been that badly underestimated.

I do like the plot showing the increased consistency of parameter estimates across angular scale. I would like to see such plots for other alternative cosmologies that *increase* consistency with low-redshift probes, such as early dark energy models. In the Hubble Hunter's Guide Marius Millea and I speculate that these discrepancies across angular scale are what one might expect, from a LCDM analysis, if the true model is one that lowers the sound horizon. And in Smith, Poulin, and Amin they show a discrepancy across angular scale arising in LCDM analysis of data generated under assumption of an early dark energy model.

Alessandro wrote,
"We don't believe that the Universe is closed but claiming now that the universe is flat with a precision below 0.5% it is also quite premature."
I guess I would have to agree with that. I did find the paper stimulating, Alessandro.

Alessandro Melchiorri
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### Re: [1911.02087] Planck evidence for a closed Universe and a possible crisis for cosmology

Hi Lloyd,

thank you for your comment. Let me add a couple of personal thoughts...

As I wrote we don't think that curvature (alone) is the answer. However, this does not mean that we should not consider curvature!
In my opinion, we have to be very conservative here and first remove any assumption that forces concordance in order to have a proper idea of the amplitude of the tensions. For sure there is no proof that the Hubble tension and this "Omega syndrome" are connected, but you may miss one piece of the puzzle if you just concentrate on one of them.
At the same time, even if I have been a great fan of it in the past, I think that we should not focus on A_L since is not physical. Changing the lensing amplitude just by hand is like changing H_0 just on the angular diameter distance but not in the Friedmann's equations. For these reasons, I am not sure that all the tests made in the lensing likelihood could be immediately applied to omega_k.
I understand that Antony considers the lensing likelihood reliable for these models and I trust him. However if you take the cmb lensing likelihood is clearly written: "Running a full set of FFP10 simulations with a different fiducial model would be numerically very expensive, so we instead generate a set of simpler idealized isotropic-beam simulations with the FFP10 fiducial model and the test ΩK model". I am not claiming that we should discard so easily the lensing result. However, closed models have not been investigated with the same level of accuracy of flat ones.
You may don't like that Planck closed model gives H_0 around 55, but remember that this is obtained under the assumption of a LCDM+Omega_k model. Since we have very few clues about dark energy, dark matter, inflation etc it is clearly not difficult to further change the closed model to have H_0 higher.
For sure, you can still say that total chi-square from Planck for a flat model is reasonable and that therefore you can still believe in a flat universe. However, if you follow this line of reasoning, you must also give up on "precision cosmology": the same 3.5 sigma fluctuation could well be in any other parameter you are constraining. I wonder how you can constrain the neutrino mass hierarchy in this case...

All the best
Alessandro