## Evidence for a Spatially Flat Universe?

Yesterday I saw a paper by George Efstathiou and Steve Gratton on the arXiv with the title *The Evidence for a Spatially Flat Universe*. The abstract reads:

We revisit the observational constraints on spatial curvature following recent claims that the Planck data favour a closed Universe. We use a new and statistically powerful Planck likelihood to show that the Planck temperature and polarization spectra are consistent with a spatially flat Universe, though because of a geometrical degeneracy cosmic microwave background spectra on their own do not lead to tight constraints on the curvature density parameter Ω

_{K}. When combined with other astrophysical data, particularly geometrical measurements of baryon acoustic oscillations, the Universe is constrained to be spatially flat to extremely high precision, with Ω_{K}= 0.0004 ±0.0018 in agreement with the 2018 results of the Planck team. In the context of inflationary cosmology, the observations offer strong support for models of inflation with a large number of e-foldings and disfavour models of incomplete inflation.

You can download a PDF of the paper here. Here is the crucial figure:

This paper is in part a response to a paper I blogged about here and some other related work with the same general thrust. I thought I’d mention the paper here, however, because it contains some interesting comments about the appropriate choice of priors in the problem of inference in reference to cosmological parameters. I feel quite strongly that insufficient thought is given generally about how this should be done, often with nonsensical consequences. It’s quite frustrating to see researchers embracing the conceptual framework of Bayesian inference but then choosing an inappropriate prior. The prior is not an optional extra – it’s one of the key ingredients. This isn’t a problem limited to the inflationary scenarios discussed in the above paper, by the way, it arises in a much wider set of cosmological models. The real cosmological flatness problem is that too many cosmologists use flat priors everywhere!

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February 19, 2020 at 5:18 pm

How do you define an “inappropriate prior”? I do not believe in good vs bad priors, but rather look at the way data bends a prior reference measure into a posterior distribution.

February 19, 2020 at 5:21 pm

For me the prior has to make sense in terms of the physical properties of the system for which the parameter is defined, specifically the symmetries of that system.

February 19, 2020 at 5:26 pm

Bayes’ theorem is what tells you how the data bend the prior probability distribution into the posterior. The question is how to get the prior probability distribution in the first place. In the absence of testable information (meaning information relating directly to the distribution, eg the mean or standard deviation) then the prior probability distribution on a continuous space is equal to the measure for it; but whence that, in turn? I now of no answer other than symmetry considerations.

The correct prior is that which corresponds to the prior information. Sometimes intuition rebels against a certain prior distribution; that means either (a) your mind has prior information which isn’t encapsulated in the prior distribution, or (b) your intuition needs educating, of (c) both. I take “appropriate” as shorthand for all of that.

February 19, 2020 at 6:44 pm

If there is such a thing as “the correct prior”, and none other, then I do not think this is a Bayesian environment. If everyone (knowledgeable) agrees about physical impossibilities, this naturally restricts the range of potential priors, but not to a single choice.

February 19, 2020 at 10:39 pm

I disagree with your first sentence, although it might be that we are using different meanings (which unhappily exist) of the word “Bayesian”. I’m not sure what you are seeking to convey in your second sentence.

February 20, 2020 at 8:55 am

“The real cosmological flatness problem is that too many cosmologists use flat priors everywhere!”It depends on what one wants to do. Flat priors (over the entire parameters space) are of course appropriate if one wants to know what one particular test indicates. If one wants the best guess as to what the real parameters are, then of course priors make sense. But if they involve information from other cosmological tests (

e.g.Hubble constant, age of the Universe,etc., then it would make sense to combine the likelihoods directly, rather than compress the information into some sort of prior.Another issue is the question of priors completely independent of observations. For example, as you and George Ellis point out in your book, a flat prior on Ω in the early universe doesn’t make sense; however, many still believe that there is a flatness problem in classical cosmology. (To be clear, the classical flatness problem is the question why Ω isn’t orders of magnitude larger or smaller than it is; it is not the question why the Universe appears to beveryclose to being spatially flat. Inflation can explain this, and there is some other evidence that inflation occurred, so it all fits together. It is not completely clear, however, whether the argument of Lake can explain the observed degree of flatness, though it does solve the classical flatness problem.)February 20, 2020 at 8:57 am

Everyone interested in the flatness problem should read Marc Holman’s excellent paper on that topic.