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Arguments against the flatness problem in classical cosmology: a review

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Abstract

Several authors (including myself) have made claims, none of which has been convincingly rebutted, that the flatness problem, as formulated by Dicke and Peebles, is not really a problem but rather a misunderstanding. In particular, we all agree that no fine-tuning in the early Universe is needed in order to explain the fact that there is no strong departure from flatness, neither in the early Universe nor now. Nevertheless, the flatness problem is still widely perceived to be real, since it is still routinely mentioned as an outstanding (in both senses) problem in cosmology in papers and books. Most of the arguments against the idea of a flatness problem are based on the change with time of the density parameter \(\varOmega \) and normalized cosmological constant \(\lambda \) (often assumed to be zero before there was strong evidence that it has a non-negligible positive value) and, since the Hubble constant H is not considered, are independent of time scale. In addition, taking the time scale into account, it is sometimes claimed that fine-tuning is required in order to produce a Universe which neither collapsed after a short time nor expanded so quickly that no structure formation could take place. None of those claims is correct, whether or not the cosmological constant is assumed to be zero. I briefly review the literature disputing the existence of the flatness problem, which is not as well known as it should be, compare it with some similar persistent misunderstandings, and wonder about the source of confusion.

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Notes

  1. Such models are often known as Friedmann–Robertson–Walker (FRW) models (Friedmann 1922, 1924; Robertson 1935, 1936; Walker 1935, 1937 the latter paper by Walker is very often incorrectly cited as having been published in 1936).

  2. For \(\lambda =0\) and \(k=0\), \(\rho = \rho _{\mathrm {crit}} = (3H^{2})/(8\pi G)\). That density is “critical” in the sense that, for \(\lambda =0\), a greater (lesser) density implies a positive (negative) curvature and a universe (assumed to be expanding now) which will collapse in the future (expand forever); similarly, for \(k=0\), a greater (lesser) density implies a negative (positive) cosmological constant and a universe (assumed to be expanding now) which will collapse in the future (expand forever). However, in the general case (\(k \ne 0\) and \(\lambda \ne 0\)), \(\rho _\mathrm {crit}\) doesn’t have any special meaning, though \(\varOmega \) remains a useful parameter.

  3. In general, one can describe the change in time of the scale factor in relation to a fiducial, usually the current, scale factor; a is often defined as the relative scale factor \(R/R_{0}\) and is thus dimensionless. Another common approach, which I use here, is to take the scale factor R to be the radius of curvature as given by Eq. (2) for \(k\ne 0\). In this case, this works for any time t, not just \(t_{0}\). For \(k=0\), \(R_{0}\), the scale factor at a fiducial time (usually taken to be the present), is arbitrary but is often set to \(c/H_{0}\). Note, however, that in general \(R\ne c/H\), including the flat case where \(R_0=c/H_{0}\) (\(R=c/H\) at all times in the special case of the relativistic equivalent of the Milne model with \(\lambda =0\) and \(\varOmega =0\) and hence \(k=-1\)).

  4. I have long used K as defined above, as do Goliath and Ellis (1999), though Wainwright and Ellis (2005) define K with the opposite sign, using it as others use \(\varOmega _{\mathrm {k}}\). One also sees K used as I use k, (e.g. Ellis et al. 2012).

  5. That is an important point. Since all non-empty big-bang models begin their evolution arbitrarily close to the Einstein-de Sitter model with \(\lambda =0\) and \(\varOmega =1\), large values of those parameters can be due only to a low value of the Hubble constant.

  6. They assumed that \(\varLambda =0\). If one replaces \(\varOmega \) with \(\varOmega +\lambda \), then some, but not all, of their arguments still hold. For example, if \(\varOmega +\lambda =1\) exactly, that holds for all time. On the other hand, the individual values of \(\varOmega \) and \(\lambda \) evolve with time (even though their sum is constant) in a flat universe (except in the cases of the Einstein–de Sitter universe with \(\varLambda =0\) and \(\varOmega =1\) and the de Sitter universe with \(\varLambda =1\) and \(\varOmega =0\); if \(\varLambda <0\) then both evolve to \((-)\infty \) at the time of maximum expansion (such universes always collapse in the future)).

  7. Although historically the issues of the flatness problem, the Anthropic Principle, fine-tuning, and the Multiverse have been intertwined, appearing in various combinations, in this paper about the flatness problem, I mention the Anthropic Principle here because it informs about Dicke’s reasoning, and otherwise only when referring to previous work which mentioned it or when it appears to be the only possible explanation. The discussion of fine-tuning is restricted to the sense in which it is usually used in connection with the flatness problem, and the Multiverse is only shorthand for any (real or imagined) ensemble of universes. (The term “Multiverse” is usually used for an ensemble of universes which exists in some physical sense, as opposed to merely a range of mathematical possibilities.) For some history of the relationship between the flatness problem and the Anthropic Principle, see Williams (2007); for the Multiverse in general, see Carr (2007); for the Anthropic Principle in general, see Barrow and Tipler (1988); for fine-tuning, including its relation to the Multiverse, see Lewis and Barnes (2017) and Adams (2019); for various different Multiverses, including the relation of some of them to fine-tuning, see Tegmark (2014). Of course, the whole point of the Anthropic Principle is to explain things which seem to be fine-tuned, and some sort of Multiverse is usually invoked to provide the necessary ensemble.

  8. Of course, wrong assumptions can nevertheless lead to interesting developments. An example from cosmology is Einstein’s mistaken belief that the Universe is static which led him to introduce the cosmological constant (Einstein 1917; O’Raifeartaigh et al. 2017), which turned out to be correct, though of course he could have introduced it even in a non-static Universe.

  9. The term “age problem” is usually used for a conflict between the age of the oldest objects in the Universe and the age of the Universe if computed using a too high value of \(H_{0}\).

  10. That paper is contained in a book (Einstein 1933) which is a collection of three works by Einstein, translated from German to French by Maurice Solovine.

  11. An aphorism along those lines is often attributed to Einstein, but, though it is probable that he said something similar, and certainly it is a valid characterization of his thought, there is no direct written evidence for such a phrase (O’Toole 2011).

  12. I distinctly remember a similar comment in a semi-popular article (including pictures of astronomers) but attributed directly to Dicke rather than Peebles, though I don’t remember if it was an actual reference, a summary of a remark he made, or just something which followed from his thinking. Any hints as to where I might find that would be highly appreciated.

  13. However, in general, if two quantities are unrelated, the default expectation should be that their ratio is not \(\approx 1\). There is some confusion here because sometimes it is claimed that “coincidences” where some ratio is \(\approx 1\) need an explanation, while other times it is claimed that ratios which are not \(\approx 1\), i.e., very large or very small numbers, are in need of an explanation. A little thought shows that one needs an explanation where some ratio is \(\approx 1\), the exception, usually in a particle-physics context, being very small numbers which come about via some sort of near-cancellation, that near-cancellation of course requiring a ratio very close to 1.

  14. See the acknowledgements below. All citation metrics mentioned in this article refer to those derived from ADS at the time of writing.

  15. But see the interesting discussion by Barrow et al. (2003) for caveats in a more general context.

  16. Under the assumption that the universe is now expanding, the Einstein–de Sitter model is a repulsor and the de Sitter model an attractor. The Milne universe a saddle point. Those remarks apply to the case of dust (i.e., non-relativistic matter) and radiation; in general the type of each fixed point depends on the equations of states of the various components (e.g. Uzan and Lehoucq 2001). The static Einstein model is approached asymptotically for expanding universes with a smaller scale factor while those with a larger scale factor, such as that favoured by Eddington, have been expanding away from it (forever in the completely unperturbed case). However, while all non-empty big-bang models start arbitrarily close to the Einstein–de Sitter model and all non-empty models which expand forever asymptotically approach the de Sitter model except that which asymptotically approaches the static Einstein model, only one infinitely fine-tuned trajectory in the \(\lambda \)–\(\varOmega \) plane includes the static Einstein model.

  17. Einstein himself seems to have been more convinced by the instability argument than by observations when he gave up his static model (Nussbaumer 2014).

  18. As discussed below, while Lake (2005) has presented perhaps the strongest argument against the flatness problem (especially with regard to the instability problem), and one which is particularly relevant in that it applies to universes, like our Universe, which have a positive cosmological constant and will expand forever, other arguments argue against the flatness problem even with the \(\varLambda =0\) assumption in the original formulation.

  19. After it had been suggested by Einstein and de Sitter (1932), the Einstein–de Sitter model was often used as a fiducial model, because it makes calculations easy. Such a model has an age which is 2/3 the Hubble time. Whenever the age of the Universe seemed to be larger than that, or even larger than the Hubble time, a model with a positive cosmological constant and an age larger than the Hubble time was sometimes invoked, the idea going back to Lemaître (1927, 1931a, 1931b, 1933, 1934). Also, a closed model with no cosmological constant was sometimes favoured, both because the evidence seemed to indicate it (e.g. Sandage 1968; Rees 1969) and because it was favoured by some on philosophical grounds, for example by Einstein (1931) (see also O’Raifeartaigh and McCann 2014 for discussion and translation). By the late 1960s, estimates of the Hubble constant were much lower, and hence the Hubble time much longer, and also the fact that different observers found very different values indicated that major uncertainty was involved, thus making the age problem in such a model less serious. However, before the Einstein–de Sitter model came to the fore around 1980, there was no “standard model” in the sense that the term has been used since then, both because of a lack of a clear choice and also because there was no paradigm in the modern sense including structure formation etc. Nevertheless, the Einstein–de Sitter model was often used as a concrete example, due to its mathematical simplicity.

  20. Inflation essentially causes the universe to increase in size by several orders of magnitude, thus making it appear flat for the same reason that we normally don’t notice the curvature of the Earth but do notice the curvature of a bowling ball. After inflation, however, the universe evolves as any other FRW model, hence if the universe is not exactly the Einstein–de Sitter universe, \(\varOmega \) (and, if non-zero, \(\lambda \)) will evolve away from the values of the Einstein–de Sitter model. Thus, as pointed out explicitly by Ellis and Rothman (1987), and also by Raine and Thomas (2001) in their section 8.2, the flatness problem will appear again in the future, at least if \(\varLambda =0\). In other words, even if inflation can solve the fine-tuning problem (though of course I argue here that it is not really a problem), it can’t solve the instability problem, at least not for all time.

  21. On the other hand, there was no direct evidence of spatial curvature, and still isn’t.

  22. Interestingly, Brawer (1996) suggested that neither the horizon problem (see McCoy (2015) for an overview of the horizon problem) nor the flatness problem was considered to be an important issue until inflation suggested a solution to them; thus, inflation might have been a solution in search of a problem. In this work, I restrict myself to classical cosmology. (By “classical cosmology” I mean the study of FRW models, which implies that general relativity is exactly valid and that the universe is, except for test particles, perfectly homogeneous and isotropic.) Also, I assume that a component with one equation of state cannot change into a component with another equation of state, an essential ingredient of inflation that, as in our Universe, stops at some point. Inflation is thus out of scope. Nevertheless, the analysis of the horizon and flatness problems (as a prelude to investigating whether inflation solves them) by McCoy (2015, 2016) is relevant in that he also concludes that the flatness problem is far from being as clear-cut as most still imagine; he takes a somewhat different view later (McCoy 2018a). See McCoy (2015) and references therein for some background on philosophical approaches to the horizon and flatness problem; those approaches “are either very limited in scope or are generally motivated by peculiar philosophical concerns distant from the kinds of concerns relevant to the practice of cosmology” and he attempts “to take a more physically motivated point of of view, and therefore takes the concerns voiced by cosmologists as a starting point”. Nevertheless, his paper is much more philosophical than the present paper.

  23. It is thus in good company with the fishy arguments in favour of the flatness problem which pervade the literature.

  24. In this paper, I use the term “fine-tuning” to mean a combination of two things: (1) changing the value by a small amount would have large consequences and (2) that value is unlikely. That is certainly the most interesting of the four possible combinations. Strictly speaking, fine-tuning refers to only the first usage, and it is a separate question how likely such a value is. However, the double meaning is used in essentially all literature on the flatness problem, so it makes sense to stick to it here.

  25. The flatness problem consists of the instability problem (why is \(\varOmega \) still close to 1 today?) and the fine-tuning problem (why was \(\varOmega \) so close to 1 in the early universe?). Dropping the assumption that \(\varLambda \) is 0, the two questions are still applicable, though the former could be amended to add “and \(\lambda \) still close to 0” and the latter to add “and \(\lambda \) so close to 0”. The basic idea in both cases is that the Einstein–de Sitter model is an unstable fixed point, so the real issue is not closeness to flatness but closeness to the Einstein–de Sitter model. The discussion is thus qualitatively the same whether or not \(\lambda \) is included. Historically, \(\varLambda =0\) was a common assumption when the flatness problem was initially formulated, so much of the historical discussion is in terms of \(\varOmega \) only. (One does have to keep in mind, though, that some authors use \(\varOmega \) for what in my notation is \(\varOmega +\lambda \).) \(\varLambda \) becomes relevant in two places, both discussed below: (1) inflation can make the universe flat by driving \(\varOmega +\lambda \) to 1, and thus says nothing about the individual values; and (2) one solution to the flatness problem (Lake 2005) points out that, for \(\varLambda >0\) and \(k=+1\), a spatially flat universe is not unlikely, but again says nothing about the individual values.

  26. The same point was made in a more general context by Coule (1995).

  27. That prior is the GHS measure (Gibbons et al. 1987), which implies that \(\varOmega \approx 1\) whether or not inflation occurred (Hawking and Page 1988).

  28. I always wonder when it is suggested that, in a universe which will expand forever, such as our Universe, it is somehow strange that we observe a value of \(\varOmega \) significantly greater than 0 and a value of \(\lambda \) significantly less than 1, since those are the asymptotic values. (A variant of that is the “coincidence problem”, ie, why does \(\varOmega \approx \lambda \) hold today?) However, in a universe which lasts forever, one is in some sense always infinitely close to the beginning. Nevertheless, after some point the values of the cosmological parameters will be close to their asymptotic values, but assuming that we should expect to observe such values implicitly assumes that life is possible in an arbitarily old universe, while rather general assumptions (lifetimes of stars, etc.) indicate that that is probably not the case

  29. All models with \(\lambda <1\) collapse. If \(\lambda =0\), models collapse for \(\varOmega >1\). For \(\lambda >1\), models will also collapse provided \(\varOmega >1\) and \(\lambda \) is not too large. For some of those, \(\varOmega \) and \(\lambda \) can be large for a significant time, but those are unlikely due to the argument of Lake (2005).

  30. In contrast to my notation, which is not unusual, Guth here uses the suffix 0 to refer to some “initial” time, not to the present time.

  31. Those are Rindler (1956), the standard paper on horizons and cosmology, and Dicke and Peebles (1979). Kragh counts a total of 79, but I count 86, which is strange since apparently both Kragh and I refer to the version reproduced by Bernstein and Feinberg (1986).

  32. Interestingly, the idea of eternal inflation, as developed by Linde (1986, 2007), is often touted as a “mechanism” for the Multiverse, though of course the Multiverse is a concept which exists independently of inflation (e.g. Trimble 2009; Tegmark 2014).

  33. There is some anecdotal evidence that that might be more the case in the astronomical community than in the general-relativity community.

  34. For example, his paper “Fluctuations at the Threshold of Classical Cosmology” (on what later came to be known as the Harrison–Zel’dovich spectrum of primordial fluctuations) (Harrison 1970) has, at the time of writing, according to ADS, 583 citations, and the term “Harrison-Zel’dovich spectrum” has certainly been mentioned much more often without citation.

  35. That definitive treatment was directly inspired (Rindler 2013) by confusion in the literature (Whitrow 1953). (See from around 01:33:00 to about 01:37:00 in Rindler (2013), though the entire video, not just Rindler’s contribution, is worth watching. Interestingly, at around 01:36:05, Rindler laments that his work on cosmological horizons led to the idea of inflation.)

  36. For example, see the comment by Helbig (2017) on the book by van den Heuvel (2016), prompted by a review of the book by Trimble (2017).

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Acknowledgements

I thank the referees for very useful comments and suggestions. There is some overlap between this paper and my paper discussing the time-scale argument in more detail (Helbig 2020), particularly in Sects. 2 and 9 and, naturally, to some extent in Sect. 11; that allows each paper to be read independently; I see no reason to change the text merely for the sake of change. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.

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Helbig, P. Arguments against the flatness problem in classical cosmology: a review. EPJ H 46, 10 (2021). https://doi.org/10.1140/epjh/s13129-021-00006-9

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