How Problematic is the Near-Euclidean Spatial Geometry of the Large-Scale Universe?

Abstract

Modern observations based on general relativity indicate that the spatial geometry of the expanding, large-scale Universe is very nearly Euclidean. This basic empirical fact is at the core of the so-called “flatness problem”, which is widely perceived to be a major outstanding problem of modern cosmology and as such forms one of the prime motivations behind inflationary models. An inspection of the literature and some further critical reflection however quickly reveals that the typical formulation of this putative problem is fraught with questionable arguments and misconceptions and that it is moreover imperative to distinguish between different varieties of problem. It is shown that the observational fact that the large-scale Universe is so nearly flat is ultimately no more puzzling than similar “anthropic coincidences”, such as the specific (orders of magnitude of the) values of the gravitational and electromagnetic coupling constants. In particular, there is no fine-tuning problem in connection to flatness of the kind usually argued for. The arguments regarding flatness and particle horizons typically found in cosmological discourses in fact address a mere single issue underlying the standard FLRW cosmologies, namely the extreme improbability of these models with respect to any “reasonable measure” on the “space of all spacetimes”. This issue may be expressed in different ways and a phase space formulation, due to Penrose, is presented here. A horizon problem only arises when additional assumptions—which are usually kept implicit and at any rate seem rather speculative—are made.

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Fig. 1
Fig. 2

Notes

  1. 1.

    With respect to this latter aspect of Euclidean geometry, Hume’s verdict essentially remains accurate (i.e., theorems of Euclidean geometry have certainly not lost their validity in mathematics), but it is unlikely that this is all he intended to say; he certainly did not contemplate the possibility of non-Euclidean circles or triangles. Even better known than Hume’s quote in this regard are of course Kant’s notorious statements on the (synthetic) a priori truth of Euclidean geometry.

  2. 2.

    In general, the (Gaussian) curvature of a two-dimensional surface of course need not be constant, although it locally still determines the geometry according to the three basic types characterized by constant curvature. As will be seen however, there are good reasons to restrict attention to constant K globally and the entailed classification of geometries in the following.

  3. 3.

    Taking the Milky Way diameter as the unit of length, our nearest galactic neighbour, Andromeda (M31 in the Messier classification), is at a distance of more than twenty length units, whereas the most distant galaxy known to date, GN-z11 (the number representing the galaxy’s redshift z), is more than 300,000 galactic length units away. It is easy to forget that the very question as to whether other galaxies even existed was settled only in 1925, when Hubble was able to reliably determine distances to Cepheid variable stars in the Andromeda Nebula for the first time and thereby established the extra-Galactic nature of these variables.

  4. 4.

    The scale at which the transition to homogeneity occurs corresponds to \(\sim 300 \, \text{ Mly }\)—or about 3000 “galactic units” [14, 15]. Since any two points on a hypersurface of homogeneity are “equivalent”, (spatial) isotropy with respect to any point on such a surface implies isotropy with respect to all points (see also note 5). A slightly different justification for the “Copernican Principle” (i.e., spatial homogeneity) that is sometimes given is based on the observation that spherically symmetric solutions in general relativity are symmetric either with respect to at most two points or with respect to every point [16] and the former would only be consistent with the observed isotropy in an essentially anthropocentric universe. This way of formulating matters strongly suggests that, assuming general relativity to be correct, the large-scale Universe is spatially isotropic—and, by implication, homogeneous—everywhere, i.e., also beyond the Hubble radius. In addition to the more or less direct empirical evidence in favour of isotropy already mentioned, more sophisticated tests of the Copernican principle (so far confirmed) have also been proposed over the years [17, 18].

  5. 5.

    A spacetime, \((M,g_{ab})\), is said to be (spatially) isotropic if there exists a congruence of timelike curves, such that at each point \(x \in M\), there exists a subgroup of isometries of \(g_{ab}\), which leaves x and vectors tangent to the congruence at x fixed, but which acts transitively on the subspace of vectors orthogonal to the congruence at x [19]. In a similar fashion, \((M,g_{ab})\) is said to be (spatially) homogeneous if M can be foliated by a one-parameter family of spacelike hypersurfaces, \(\Sigma _{\tau }\), such that for all \(\tau \), any two points in \(\Sigma _{\tau }\) can be connected by an isometry. Hence, all points in any given hypersurface of homogeneity are “equivalent”. Isotropy is a more stringent condition than homogeneity, since the former implies the latter but not vice versa (intuitively, any inhomogeneities between large-scale regions would appear as large-scale anisotropies to observers equidistant from these regions so that an isotropic spacetime must be homogeneous [20]).

  6. 6.

    In addition to the contributions of Friedmann and Lemaître already mentioned, Robertson [21] and Walker [22] independently showed that the metric (1) applies to all locally isotropic spacetimes.

  7. 7.

    As a consequence of their constant curvature, any two hypersurfaces \(\Sigma _{\tau }\), \(\Sigma _{\tau '}\) with the same value of k are locally isometric. However, two foliations by hypersurfaces with the same value of k could in principle still represent topologically distinct spatial geometries (a flat, \(k=0\) geometry could for instance either be “open”, corresponding to standard infinite Euclidean three-space, or “closed”, corresponding to e.g. a three-dimensional torus). Although it is often argued, either explicitly or implicitly, that examples of this kind demonstrate that cosmology faces a serious, if not irreconcilable, problem of underdetermination, it seems that such arguments may underestimate human ingenuity. In addition, it is also not inconceivable that a future theory of quantum gravity could decide spatial topology.

  8. 8.

    Although all of the following discussion goes through unchanged by allowing negative values for w such that \(w > - 1/3\), pressure is, at the present, classical level of discussion, taken to be a non-negative quantity on physical grounds (similar remarks pertain to the classical energy density of matter, \(\rho \)). In particular, the matter distribution at the present epoch effectively behaves as “dust” (\(w=0\)), whereas in the early Universe, it effectively behaved as “radiation” (\(w=1/3\)). No negative-pressure classical matter is currently known to exist. Although the cosmological constant, \(\Lambda \), is sometimes interpreted as a negative-pressure, cosmic perfect fluid (with \(w=-1\)), for various reasons this view will not be adopted here. In accordance with Eq. (3), \(\Lambda \) is interpreted as being associated with “geometry”, rather than with “sources”, in the current work. Nevertheless, it will occasionally prove useful to temporarily switch views in order to facilitate comparison with existing literature. Finally, it is clear that the present discussion can be generalized to the case of non-interacting, multi-component perfect fluids (consisting e.g. of both radiation and dust simultaneously; see also note 18).

  9. 9.

    For a fairly comprehensive treatment of the various possible FLRW models, see for instance Ref. [20]. For specific treatments from a dynamical systems perspective, see Refs. [23,24,25]. The value of the cosmological constant turns out to play a critical role in the dynamical behaviour of solutions. In particular, all models with \(\Lambda < 0\) are “oscillatory” (i.e., reach a stage of maximum expansion and then re-contract), while all models with \(\Lambda > \Lambda _{\mathrm {E}}\), i.e., \(\Lambda \) greater than the critical value, \(\Lambda _{\mathrm {E}} := 4 \pi (1 + 3w) \rho \), corresponding to a general Einstein static universe, are forever expanding (so-called inflexional universes). For non-negative values of \(\Lambda \) not greater than \(\Lambda _{\mathrm {E}}\), there is a difference between models according to whether their spatial geometries have positive curvature or not: the “standard” models with \(k=+1\) behave in the same way as their counterparts with negative \(\Lambda \), i.e., they are oscillatory, while models with \(k=-1\) and \(k=0\) are forever expanding. The exceptions for the positive curvature models arise for \(0< \Lambda < \Lambda _{\mathrm {E}}\) in the form of “rebounding universes” (which start from a state of infinite expansion to reach a minimum radius and then re-expand) and for \(\Lambda = \Lambda _{\mathrm {E}}\) in the form of two non-static solutions, which respectively start from and asymptotically tend to the static solution (the former is sometimes regarded as a perturbed Einstein static universe and in the specific case of dust, \(w=0\), it is then referred to as the Eddington–Lemaître universe). As will be seen in Sect. 3.2, observations at present indicate that if in fact the Universe’s geometry is elliptic, \(\Lambda \) is significantly larger than \(\Lambda _{\mathrm {E}}\). Finally, the singularity at \(a=0\) discussed in the main text (conventionally taken to occur at the “initial time” \(\tau = 0\)) is characterized by the following features: (i) zero distance between all points of space, (ii) infinite density of matter and (iii) infinite curvature for \(k \ne 0\) (i.e., for small \(\tau \), the scalar curvature, R, tends to \(6k/a^2\)).

  10. 10.

    As observed earlier (cf. note 7), both hyperbolic and Euclidean geometries admit topologically closed models, although these appear to be somewhat contrived. It should also be noted that there is a potential ambiguity in the open/closed universe terminology, as it might seem that reference could be made to either spatial geometry (as is intended here) or temporal duration. In fact, in the present context with \(\Lambda = 0\), the distinction is irrelevant as long as the standard practice of referring to negative and zero curvature models as “open” is followed. This, however, ceases to be the case for nonzero \(\Lambda \). In particular (cf. note 9), all models with open spatial geometry (\(k=-1, 0\)) are temporally closed (i.e., eventually recontract) if \(\Lambda <0\), while all models with closed spatial geometry (\(k=+1\)) are temporally open (i.e., expand indefinitely) if \(\Lambda > \Lambda _{\mathrm {E}}\).

  11. 11.

    Explicit arguments leading to the sort of numerical estimates given in the main text can be found in many references (see Refs. [30,31,32,33,34,35] for a fragmentary list of relevant arguments). It should be noted that slightly different estimates of the alleged fine-tuning can be found in the actual literature, depending for instance on what value for \(\tau _{\text{ eq }}\) is adopted or on the exact nature of the calculation (in some older references estimates can be found that are based on calculations also involving quantum statistical arguments and entropy bounds—leading, somewhat miraculously, to similar numerical estimates; more modern references typically follow the line of argument presented in the main text, being fundamentally based only on classical FLRW dynamics). However, it appears that the figure of 59 decimal places for a Planck order cut-off time may be taken as some sort of weighted average. A fundamental problem with all these approaches that is hardly ever pointed out however, is that the step of taking flat solutions as valid approximations to all viable solutions for the entire history of the Universe, of course completely begs the question. In fact, Eqs. (14) are somewhat misleading, since for flat solutions one obviously has \(\Omega _{\mathrm {m}} = 1\) for all times! But since there is no guarantee that a curved solution is in any reasonable sense near its flat counterpart if \(\Omega _{\mathrm {m}}\) is actually close to 1, except at very small times (as the cycloidal model discussed in Sect. 3.2 for instance perfectly well illustrates), the actual figures quoted, (15), (16) are rather deceptive as things stand.

  12. 12.

    It is recalled that all oscillatory models (i.e., all models with \(\Lambda < 0\), regardless of the value of k, and all models with \(0 \le \Lambda < \Lambda _{\mathrm {E}}\) and \(k=+1\)) are dynamically constrained to the region to the left of or below the locus of \(\ddot{a}=0\) in the \(a \Lambda \)-plane and, as a consequence, have \(\ddot{a}<0\) at all times, whereas observations indicate that the universal expansion is accelerating, i.e., that \(\ddot{a}>0\) at present. In fact, in terms of the current value of the “deceleration parameter”, \(q := -\, a \ddot{a} / \dot{a}^2\), one has (cf. Eq. (5)) \(q = \Omega _{\mathrm {m}}/2 - \Omega _{\Lambda } \simeq -\, 0.55\).

  13. 13.

    For \(k=+1\), \(\Lambda >0\) models, the physical interpretation of \(\alpha \) is that of the cosmological constant, \(\Lambda \), normalized with respect to the critical value \(\Lambda _{\mathrm {E}}\) corresponding to an Einstein static universe, i.e., \(\alpha = \Lambda / \Lambda _{\mathrm {E}}\) (alternatively, \(\alpha \sim \Lambda M^2\), with M denoting the total mass of the universe). As follows from the discussion in the main text, this justifies a remark made earlier, that for physically relevant elliptic FLRW models \(\Lambda \gg \Lambda _{\mathrm {E}}\).

  14. 14.

    Equation (26) follows from the obvious fact that for any reference epoch, \(t_0 H_0\) is given by

    $$\begin{aligned} t_0 H_0 \; = \; \int _0^{a_0} \frac{da}{aH/H_{0}} \; = \; \int _0^1 \frac{dx}{x E(x)} \qquad \qquad x \; := \; a/a_0 \; = \; e^{\eta } \end{aligned}$$
    (26)

    where \(E \; := \; H/H_0 = \sqrt{\Omega _{\mathrm {m},0} x^{-3} + \Omega _{\Lambda ,0} + \Omega _{k,0} x^{-2}}\). The reference subscript is implicit in (26). Alternatively,

    $$\begin{aligned} t \; = \; \frac{1}{H} \, \int ^{\infty }_0 \frac{dz}{(1 + z)\sqrt{(1 + z)^2(1 + z\Omega _{\mathrm {m}}) - z(z + 2) \Omega _{\Lambda }}} \end{aligned}$$
    (27)

    where z denotes the usual redshift factor, i.e., \((1+z)^{-1} = e^{\eta } = a/a_0\).

  15. 15.

    Although the curvature constraint—and hence the constraint (27) on \(\alpha \)—based on the Planck data is not model independent, there appear to be good reasons to expect that model-independent constraints at least as strong are a feasible prospect for the near future [50].

  16. 16.

    It is worth stressing that the foregoing conclusions are independent of how \(\Lambda \) is interpreted, i.e., as being associated with geometry or with a \(w=-1\) perfect fluid source. In fact, if the latter interpretation is adopted (as has implicitly been done in the discussion of multi-component FLRW models), the conclusions are invariant under perturbations about \(w=-1\) [39].

  17. 17.

    This particular way of referring to weak and strong anthropic reasoning, although in line with some authors (see e.g. Ref. [53]), does not appear to be standard terminology. It is however arguably a sensible terminology for rather obvious reasons.

  18. 18.

    In fact, the foregoing line of argument, based on Eq. (5) is a bit deceptive, as it might superficially appear that a past-singularity could be avoided in any expanding FLRW model, provided \(\Lambda \) is large enough, whereas the discussion in Sect. 2 should make clear that exactly the opposite is the case. Indeed, any dynamical trajectory in the upper-right quadrant of the \(a\Lambda \)-plane for which \(\dot{a}\) and \(\ddot{a}\) are both positive at some instant, in the reversed time direction (i.e., as obtained by following the appropriate constant-\(\Lambda \) line to the left) either hits a singularity at \(a=0\) or a “bounce” at some finite a-value, where \(\dot{a}\) changes sign, but the latter can occur only for \(k=+1\) and \(0< \Lambda < \Lambda _{\mathrm {E}}\). Although it may not seem obvious that the inevitability of a past-singularity persists in the case of an empirically adequate multi-component perfect fluid FLRW model, it in fact does and moreover so even without the need to include any prior assumptions about the value of \(\Lambda \) [54].

  19. 19.

    More precisely, the archetypal singularity theorem states that any general relativistic spacetime that satisfies (i) some appropriate causality condition (for instance, the absence of closed timelike curves), (ii) some appropriate energy condition (for instance, the strong energy condition) and (iii) some appropriate focussing condition (for instance, the existence of a so-called “trapped surface”), is necessarily inextendible. That is, such a spacetime necessarily contains at least one incomplete geodesic (i.e., a geodesic with only a finite parameter range in one direction, but inextendable in that direction) and is not isometric to a proper subset of another spacetime. It is important to note that, whereas it is very well conceivable that some of the above conditions may not strictly be satisfied at all spacetime points, it does not appear that—within a classical gravitational context—one could thereby avoid spacetime singularities in any realistic sense. For instance, violations of the (strong) energy condition can occur in the very early Universe because of the effects of quantum fields and/or inflationary theory, but if the condition holds in a spacetime average sense, singularities are still expected to occur, in general [56]. That the conditions typically assumed by the singularity theorems are merely sufficient conditions can also be understood from the interpretation of the cosmological constant as a negative-pressure perfect fluid. The net effect of that interpretation is to contribute towards violating the strong energy condition (for positive \(\Lambda \)), but this has in itself little bearing on the issue of the occurrence of spacetime singularities. For instance, as is easily verified from the discussion in Sect. 2, for \(\Lambda \) sufficiently large, all non-empty isotropic models on this view violate the strong energy condition most of the time, but are nevertheless past-singular (recall also note 18).

  20. 20.

    More accurately, according to the BKLM picture, apart from zero measure counter-examples, all spacelike initial singularities in general relativity are “vacuum dominated” (i.e., “matter does not matter”—that is, with the apparent exception of possible scalar matter), “local” and “oscillatory” (in the sense of being locally homogeneous, i.e., Bianchi type, and approached through an infinite sequence of alternating “Kasner type epochs”, as first described in a more specific context within Misner’s Mixmaster model [65]). It should be stressed however that this picture at the present time very much amounts to an unproven conjecture and that specific model studies have provided evidence both for and against it (for a recent discussion of the status of the conjecture, see Ref. [66]). One of the prima facie obstacles in obtaining a proof of the conjecture would appear to be making quantitative the notion of “genericity” (i.e., through defining a meaningful measure on the space of all singular cosmological models). However, as will become clear in the main text, even if the conjecture were to hold in the sense that generic spacetime singularities would essentially be of BKLM type, it is rather doubtful that this would establish anything about the actual Big Bang. Indeed, within a purely classical context, there is no reason to treat initial and final spacetime singularities within the mathematical formalism of general relativity differently, but that does not mean that there are no physical motivations to do so. In fact, the thermodynamic arrow of time very strongly amounts to precisely such a motivation.

  21. 21.

    Although one could continue to speak of a “horizon problem” if conditions (i), (ii) are satisfied, it would seem more appropriate to refer to the underlying effective isotropy at “late” times as being problematic; cf. Sect. 5. It is also worth noting that, given that quantum (effective) fields had to be dominant in the very early Universe, it does not at all seem clear that isotropization could not have occurred sufficiently rapidly through non-ordinary dissipative processes involving the extreme spacelike entanglement of these fields [69].

  22. 22.

    Whether the expansion factor is actually a googol, the square root of a googol, or a googol squared, etc. is not important for present considerations and in fact it does not appear that inflationary models themselves are currently able to make a firm prediction in this regard. It should also be noted here that the original inflationary models [27,28,29] mentioned earlier were in fact somewhat different from each other. In Guth’s 1981 model (now usually referred to as “old inflation”), the exponential expansion was envisaged to take place with the inflaton sitting at the top of the Mexican hat potential, so to say (\(\dot{\phi }\) constant), whereas in the Linde-Albrecht-Steinhardt 1982 models (now usually referred to as “slow roll (or new) inflation”), the expansion occurred with the inflaton rolling down from the top of the sombrero potential (\(\dot{\phi }\) time-varying). Note however that, in view of the fact that these initial models were formulated in an (effectively) isotropic context to start with, it is rather doubtful (given in particular the dynamical behaviour of \(\Omega \) and the horizon issue, as discussed at length earlier), that they addressed any internal problem of cosmology that existed in the first place.

  23. 23.

    Although no explicit mention is made of the BKLM conjecture, Ref. [76] does speak of a standard view according to which the Universe before the Planck era is “in some chaotic quantum state” and at least some of the cited advocates of this view do approvingly refer to (part of) the conjecture (see e.g. Ref. [77]). However, in the actual model of Ref. [76] chaos does not in fact enter spatial geometry in any way. The model is chaotic only in the sense that arbitrary constant initial values of the \(\phi \) field (within some specified range) are allowed for different spatial regions (which are themselves isotropic and of typical dimensions much larger than the Planck length) at whatever time inflation is supposed to start. Nevertheless, since the consensus view appears to be that the BKLM conjecture holds, it seems that a genuinely chaotic inflationary model should also refer to chaotic initial conditions in spatial geometry (for work in this direction, see e.g. Refs. [78, 79]), especially within a framework that is at its core time-reversal invariant, as is usually taken to be the case [80]. It should also be stressed that the picture of a “chaotic ensemble” of Planck order sized patches sketched in the main text is intended for heuristic purposes only, since a clearly defined framework for these inflationary patches—let alone a mathematically rigorous treatment—is currently unavailable.

  24. 24.

    It has been demonstrated that a generic, initially expanding Bianchi model with positive cosmological constant inflates into an effective de Sitter spacetime within an exponentially short timeframe [81]. This “cosmic no-hair” theorem however crucially depends on the assumption that the matter stress-energy tensor satisfies the strong energy condition, which seems physically unreasonable in the very early Universe. In particular, it has been pointed out that the de Sitter solution is unstable if the condition is violated [82].

  25. 25.

    Such an initial patch could have been either already isotropic or anisotropic (as in the scenario of Ref. [68] and more in accordance with the BKLM picture).

  26. 26.

    It might be thought that cosmological precision tests should in practice be able to shed light on the question as to whether the early Universe actually went through an inflationary epoch, for instance by determining the parameter values of the inflaton potential. The problem is however that virtually any set of relevant observational data can be matched with an appropriate inflationary model (a situation that actually pertains to both the dynamical evolution of the scale factor and to the spectrum of density perturbations; see e.g. Refs. [57, 83]).

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Acknowledgements

This publication was made possible through the support of a Grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation. I thank the organizers of the Fourth International Conference on the Nature and Ontology of Spacetime, in Varna, Bulgaria, for providing an opportunity for me to present (most of) this work. Financial support from Stichting FOM (Foundation for Fundamental Research on Matter) in the Netherlands to attend the Conference is also gratefully acknowledged. Finally, I thank Andy Albrecht, Feraz Azhar, George Ellis, David Garfinkle and especially Phillip Helbig and Chris Smeenk for comments and/or discussions pertaining to the contents of this manuscript.

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Holman, M. How Problematic is the Near-Euclidean Spatial Geometry of the Large-Scale Universe?. Found Phys 48, 1617–1647 (2018). https://doi.org/10.1007/s10701-018-0218-4

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Keywords

  • Cosmological flatness problem
  • General relativity
  • FLRW solutions
  • Initial conditions
  • Fine-tuning
  • Inflation
  • Horizon problem
  • Second law of thermodynamics
  • Quantum gravity