This has now appeared in a top journal: Found. Phys., 2018, 48(11), 1617-1647. (The arXiv version has been updated to match.)

]]>A review piece has recently appeared on arXiv which gives due credit to the brave souls who have questioned the flatness-problem paradigm. Well worth reading!

]]>Here is an analogy which I hope is not misleading: Suppose someone actually got a rubber sheet, put weights on it, and from observations derived laws of motion. Note that “there does not exist a two-dimensional, cylindrically-symmetric surface that will yield rolling marble orbits that are equivalent to the particle orbits of Newtonian gravitation, or for particle orbits that arise in general relativity” (yes, someone actually did: http://arxiv.org/abs/1312.3893). Clearly, claiming that these observations say something about the real laws of motion would be false, a classic case of arguing too far from analogy.

Actually, I think the tightrope-walker analogy is even worse because it is not only quantitatively false (the equation of motion of a falling tightrope walker, or almost-balanced pencil, is not analogous to the Friedmann equation), but also qualitatively false.

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*Well, one of them.

Blowing my own horn here to some extent (not ideal, but perhaps better than not having it blown at all), but I just came across an interesting paper which provides an additional, easy-to-understand argument against the existence of the flatness problem, in addition to mine and to those of Coles, Ellis, Evrard, and Lake which I cited in my flatness-problem paper:

*Thus the deviation during nucleosynthesis was only about 10−17, [ten to the minus seventeen] an impressively small number. It follows merely from the present density deviation, which is not necessarily very small, plus the cosmological equations and the definition of the critical density. *

A useful analogy from elementary physics might be the following: consider a test particle of mass m with total energy E falling into the Newtonian gravitational field of a mass M. The ratio of this particle’s kinetic energy K = mv2/2 [m times v squared over two] to its potential energy |U| = GMm/r is K/|U| = (E/GMm)r + 1. Note that the difference K/|U| − 1 becomes arbitrarily small as one approaches r → 0, in exactly the same way that T − 1 does in cosmology as t → 0. Yet one would hardly be justified in concluding from this that E “must be” zero on the grounds of naturalness.

*In summary, the extremely small deviation of the density ratio from unity in the early Universe is a consequence of the definition of the critical density and the basic equations of relativistic cosmology for any value of k. We therefore do not agree with the viewpoint that k = 0 is necessarily the most natural interpretation of current observational data. If future experiments produce a much smaller limit on the flatness parameter ε (say, 10−5), then that might be a more convincing indication that the most natural value for k is zero.*

Couldn’t have said it better myself.

Let me point out two things. First, in both the cosmological case and in the example from Adler and Overduin quoted above, the “fine-tuning” exists whatever the value of Omega today, or whatever the initial velocity of the test particle. So, the fact that Omega is “still” not far from 1 today is a red herring. Second, thanks to Newtonian cosmology, the analogy between the elementary-physics example above and the cosmological case is much closer than many might think.

I came across this paper because it cited one of the papers I cited in my flatness-problem paper, namely that of Lake (well worth checking out!).

I recently read some lecture notes from someone who had posted a comment on a blog somewhere. The flatness-problem canard continues to be raised in almost all cosmology lectures. Forget about me; do people not even read what people like Overduin, Lake, and Ellis write (not to mention Coles)? I think most people don’t really think about it, but just quote it because the read it somewhere. Or because they are afraid of Rocky Kolb. 🙂 Interestingly, the original Peebles-and-Dicke claim never appeared in a refereed journal, but only in a volume of conference proceedings.

As far as I know, no-one has refuted any of the anti-flatness-problem arguments mentioned in the papers by the authors above. Usually, when a wrong claim appears on arXiv, be it by some relative unknown who claims that the universe is screwy, or be it by the venerable Penrose talking about circles on the CMB, or for that matter Kellermann in *Nature* claiming that the standard-rod test supports the Einstein-de Sitter universe, it is usually refuted quickly and by several people independently.

Finally, let me say that I don’t claim that no non-classical explanation is needed if the universe proves to be flat to one part in a million, say, rather than to within a per cent or so (current observational situation). Rather, my point is that the standard argument that classical cosmology leads to absurd conclusions which are manifested in the flatness problem (which was formulated at a time when, observationally, it was not at all clear that the universe is very close to flat, and also before inflation predicted a universe very close to flat) is due to a misinterpretation.

]]>The new stuff concerns cosmological models which will collapse in the future. While it is true that (the absolute values of) the cosmological parameters lambda and Omega evolve to infinity (and back to (0,1)—the values in the Einstein-de Sitter model, which is where all non-empty big-bang models start in the lambda-Omega parameter space), for most of the lifetime of the universe the values are not particularly large. Thus, we shouldn’t be puzzled if we don’t observe large values. In other words, the tightrope-walker argument is qualitatively but not quantitatively correct.

I recommend reading the cited works as well, not just because the paper is easier to understand if one is familiar with the cited works, but because the cited works themselves are well worth reading.

]]>I have looked at the paper but the main line of reasoning is so far not clear… can you summarize the logical structure of the argument?

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