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Shape of the universe

From Wikipedia, the free encyclopedia
(Redirected from Open universe)

In physical cosmology, the shape of the universe refers to both its local geometry and cosmic topology. Local geometry is defined primarily by its curvature, general relativity explains how spatial curvature (local geometry) is constrained by gravity. The cosmological topology of the universe cannot be deduced from measurements of curvature inferred from observations within the family of homogeneous general relativistic models alone, due to the existence of locally indistinguishable spaces with varying global topological characteristics.[1] For example; a multiply connected space like a 3 torus has everywhere zero curvature but is finite in extent, whereas a flat simply connected space is infinite in extent (such as Euclidean space).[1]

Observational evidence (WMAP, BOOMERanG, and Planck, for example) indicates that the observable universe is spatially flat to within a 0.4% margin of error of the curvature density parameter with an unknown global topology.[2][3] It is unknown whether the universe is simply connected like euclidean space or multiply connected like a torus.[4]

Shape of the observable universe

[edit]

The universe's shape or the cosmic topology can be examined from two angles:[1]

  1. Local geometry: This relates to the curvature of the universe, primarily concerning what we can observe.
  2. Global topology: This pertains to the universe's overall shape and structure.

General relativity, the theory behind modern cosmology, is based on differential geometry and it can only model local geometry. There are an infinite number of global shapes that match Einstein's equations.[5] The global structure of the universe includes:

  • whether the universe is infinite or finite in extent,
  • whether the geometry of the global universe is flat, positively curved, or negatively curved, and,
  • whether the topology is simply connected (for example, like a sphere) or else multiply connected (for example, like a torus).[6]

The observable universe (of a given current observer) is a roughly spherical region extending about 46 billion light-years in every direction.[7] It appears older and more redshifted the deeper we look into space.[8] In theory, we could look all the way back to the Big Bang, but in practice, we can only see up to the cosmic microwave background (CMB) (roughly 370000 years after the Big Bang) as anything beyond that is opaque. Studies show that the observable universe is isotropic and homogeneous on the largest scales.[9]

If the observable universe encompasses the entire universe, we might determine its structure through observation. However, if the observable universe is smaller, we can only grasp a portion of it, making it impossible to deduce the global geometry through observation.[1] Different mathematical models of the universe's global geometry can be constructed, all consistent with observations and general relativity.[10] Hence, it is unclear whether the observable universe matches the entire universe or is significantly smaller, though it is generally accepted that the universe is larger than the observable universe.

The universe may be compact in some dimensions and not in others.[1] Such models can be tested by searching for topological lensing, such as multiple images of the same distant source or matched patterns in the cosmic microwave background.[1] A small closed universe would produce multiple images of the same object in the sky, though not necessarily of the same age.

Curved space

[edit]

The curvature is a quantity describing how the geometry of a space differs locally from flat space. The curvature of any locally isotropic space (and hence of a locally isotropic universe) falls into one of the three following cases:[1]

  1. Zero curvature (flat) – a drawn triangle's angles add up to 180° and the Pythagorean theorem holds; such 3-dimensional space is locally modeled by Euclidean space E3.
  2. Positive curvature – a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a 3-sphere S3.
  3. Negative curvature – a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space H3.

Curved geometries are in the domain of non-Euclidean geometry. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a saddle or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.[11]

The local geometry of the universe is determined by whether the density parameter Ω is greater than, less than, or equal to 1. From top to bottom: a spherical universe with Ω > 1, a hyperbolic universe with Ω < 1, and a flat universe with Ω = 1. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of (local) space.
Proper distance spacetime diagram of our flat ΛCDM universe. Particle horizon: green, Hubble radius: blue, Event horizon: purple, Light cone: orange.
Hyperbolic universe with the same radiation and matter density parameters as ours, but with negative curvature instead of dark energy (ΩΛ→Ωk).
Closed universe without dark energy and with overcritical matter density, which leads to a Big Crunch. Neither the hyperbolic nor the closed examples have an Event horizon (here the purple curve is the cosmic Antipode).

General relativity explains that mass and energy determine the curvature of the universe. The relationship is quantified with the density parameter, represented with Omega (Ω). The density parameter is the average density of the universe divided by the critical energy density, that is, the mass energy needed for a universe to be flat:[1]

  • If Ω = 1, the universe is flat.
  • If Ω > 1, there is positive curvature.
  • If Ω < 1, there is negative curvature.

Scientists can calculate Ω to determine the curvature in two ways. One is to count all the mass–energy in the universe and take its average density, then divide that average by the critical energy density. Data from the Wilkinson Microwave Anisotropy Probe (WMAP) as well as the Planck spacecraft give values for the three constituents of all the mass–energy in the universe – normal mass (baryonic matter and dark matter), relativistic particles (predominantly photons and neutrinos), and dark energy or the cosmological constant:[12][13]

Ωmass ≈ 0.315±0.018
Ωrelativistic ≈ 9.24×10−5
ΩΛ ≈ 0.6817±0.0018
Ωtotal = Ωmass + Ωrelativistic + ΩΛ = 1.00±0.02

The actual value for critical density value is measured as ρcritical = 9.47×10−27 kg⋅m−3. From these values, within experimental error, the universe seems to be spatially flat. Final results of the Planck mission, released in 2018, show the cosmological curvature parameter, 1 − Ω = ΩK = −Kc2/a2H2, to be 0.0007±0.0019, consistent with a flat universe.[14]

Another way to measure Ω is to do so geometrically by measuring an angle across the observable universe. This can be done by using the CMB and measuring the power spectrum and temperature anisotropy. For instance, one can imagine finding a gas cloud that is not in thermal equilibrium due to being so large that light speed cannot propagate the thermal information. Knowing this propagation speed, we then know the size of the gas cloud as well as the distance to the gas cloud, we then have two sides of a triangle and can then determine the angles. Using a method similar to this, the BOOMERanG experiment has determined that the sum of the angles to 180° within experimental error, corresponding to Ωtotal ≈ 1.00±0.12.[15]

These and other astronomical measurements constrain the spatial curvature to be very close to zero, although they do not constrain its sign. This means that although the local geometries of spacetime are generated by the theory of relativity based on spacetime intervals, we can approximate 3-space by the familiar Euclidean geometry.

The Friedmann–Lemaître–Robertson–Walker (FLRW) model using Friedmann equations is commonly used to model the universe. The FLRW model provides a curvature of the universe based on the mathematics of fluid dynamics, that is, modeling the matter within the universe as a perfect fluid. Although stars and structures of mass can be introduced into an "almost FLRW" model, a strictly FLRW model is used to approximate the local geometry of the observable universe. Another way of saying this is that, if all forms of dark energy are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed (rather than the distortions caused by "dense" objects such as galaxies). This assumption is justified by the observations that, while the universe is "weakly" inhomogeneous and anisotropic (see the large-scale structure of the cosmos), it is on average homogeneous and isotropic when analyzed at a sufficiently large spatial scale.[16]

Size

[edit]

The size of the universe is not known.[17] The observable universe is a spherical region of the universe consisting of all matter that can be observed from Earth; the electromagnetic radiation from these astronomical objects has had time to reach the Solar System and Earth since the beginning of the cosmological expansion. The radius of this region is about 14.26 gigaparsecs (46.5 billion light-years or 4.40×1026 m).

One of the unanswered questions about the universe is whether it is infinite or finite in extent.[1] For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. Mathematically, the question of whether the universe is infinite or finite is referred to as boundedness. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance d, there are points that are of a distance at least d apart. A finite universe is a bounded metric space, where there is some distance d such that all points are within distance d of each other. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale".

Estimates based on Bayesian model averaging suggests that the volume of the whole universe must be more than 250 times larger than a Hubble sphere.[18]

It is plausible that the galaxies within the observable universe represent only a minuscule fraction of the galaxies in the universe. According to the theory of cosmic inflation initially introduced by Alan Guth and D. Kazanas,[19] if it is assumed that inflation began about 10−37 seconds after the Big Bang and that the pre-inflation size of the universe was approximately equal to the speed of light times its age, that would suggest that at present the entire universe's size is at least 1.5×1034 light-years — this is at least 3×1023 times the radius of the observable universe.[20]

Some disputed estimates for the total size of the universe, if finite, reach as high as megaparsecs, as implied by a suggested resolution of the No-Boundary Proposal.[21][a]

Boundary

[edit]

Whether space is finite in extent is distinct from whether it has a boundary. In cosmological models, space is usually represented by a three-dimensional manifold. Such a manifold may be compact without having a boundary; a compact manifold without boundary is called a closed manifold.[22] Familiar examples are the 3-sphere and the 3-torus. Both are compact and have no boundary, although the 3-sphere is simply connected and has positive curvature, whereas the 3-torus is multiply connected and flat.[23][22] Thus a spatially finite universe need not have an edge, and the absence of an edge does not imply that the universe is spatially infinite.

If the universe has a compact topology on a scale comparable to or smaller than the observable universe, light may reach an observer along more than one topologically distinct path. This can in principle produce multiple images of the same astronomical objects or repeated patterns in the cosmic microwave background.[23][22] One proposed test is the circles in the sky method, which searches for matching circles in the last-scattering surface produced by different lines of sight to the same physical region.[24] Searches using Planck data found no evidence for the compact topologies tested at scales smaller than the diameter of the last-scattering surface,[25] although substantial classes of possible topologies remain observationally unconstrained.[4]

Curvature

[edit]

The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite.[5] Many textbooks erroneously state that a flat or hyperbolic universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe.[5] For example, Euclidean space is flat, simply connected, and infinite, but there are tori that are flat, multiply connected, finite, and compact (see flat torus).

In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.

The latest research shows that even the most powerful future experiments (like the SKA) will not be able to distinguish between a flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10−4. If the true value of the cosmological curvature parameter is larger than 10−3 we will be able to distinguish between these three models even now.[26]

Universe with zero curvature

[edit]

In a universe with zero curvature, the local geometry is flat. The most familiar such global structure is that of Euclidean space, which is infinite in extent. Unusual topology like the 3-torus universe allow flat universes that are finite.[27]

In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching zero. With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases. The ultimate fate of the universe is the same as that of an open universe in the sense that space will continue expanding forever.[citation needed]

A flat universe can have zero total energy.[28]

Universe with positive curvature

[edit]

A positively curved universe is described by elliptic geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold (such as the Poincaré dodecahedral space), all of which are quotients of the 3-sphere.

Poincaré dodecahedral space is a positively curved space, colloquially described as "soccerball-shaped", as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by Jean-Pierre Luminet and colleagues in 2003[29][30] and an optimal orientation on the sky for the model was estimated in 2008.[31]

Universe with negative curvature

[edit]

A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There is a great variety of hyperbolic 3-manifolds, and their classification is not completely understood.[32] Those of finite volume can be understood via the Mostow rigidity theorem. For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called "horn topologies", so called because of the shape of the pseudosphere, a canonical model of hyperbolic geometry. An example is the Picard horn, a negatively curved space, colloquially described as "funnel-shaped".[33]

Curvature: open or closed

[edit]

When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive, respectively.[16] These meanings of open and closed are different from the mathematical meaning of open and closed used for sets in topological spaces and for the mathematical meaning of open and closed manifolds, which gives rise to ambiguity and confusion. In mathematics, there are definitions for a closed manifold (i.e., compact without boundary) and open manifold (i.e., one that is not compact and without boundary). A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the Friedmann–Lemaître–Robertson–Walker (FLRW) model, the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.

History

[edit]

Ancient mythologies variously described the universe as finite.[34]

By way of the account of Diogenes Laërtius, for Leucippus (c. 5th century BC) the universe is spatially infinite.[35][36] Eudoxus (c. 380 BC) in consideration of motion concluded that the stars were encompassed within a sphere.[b][39] In the concept of Aristotle (384–322 BC), there existed concentric spheres located outwardly from Earth, the furthest of which contained the stars and was sometimes known as the kosmos. Outside the boundary of the furthest sphere there was no place of physically existing anything and no time, in addition to this outside the last sphere's edge there was no void.[40][41][42][43][44]

From the conceptual foundation of Aristotle[c] which became the model for Ptolemy (2nd century AD) postdate the time of the completion of his work, entitled in ancient Greek; Ὑποθέσεις τῶν πλανωμένων, the preferred general cosmology into the Middle Ages was that the cosmos was finite, which was known subsequently as Aristotelian cosmology.[46][37][47][48] In Paradiso, Dante Alighieri (1308–1320) made a universe according to the Ptolemaic mode, in which the Earth was placed at the centre of a number of spheres, the outer of which was were humanity could find the location of the realm of God, an idea of reality that was the shared perception of all of the prominent thinkers of the medieval era.[49][50] Thomas Bradwardine (1344) and Nicole Oresme during the 14th century contested the Aristotelian view on the basis of infinite God.[51]

With the advent of the heliocentric model scientific thought realized the possibility of an infinite universe.[52] By the use of the novel model of Nicolaus Copernicus, Thomas Digges in: A perfit description of the caelestiall orbs, published 1576, made explanation of such a model universe, at that time presenting an idealogical break in concept from the then tradition of the reality of a celestial outer realm known as Paradise.[53]

Albert Einstein in consideration of his 1916 theory of general relativity demonstrated in 1917 a finite universe.[54][51] The de Sitter infinite universe of 1917 was caused by the incompatibility of relativity with Euclidean space.[51] David Hilbert (1925) thought the universe was determined finite by elliptical geometry or infinite by Euclidean geometry[55] (i.e. flat).[56]

In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed[5] and applied to cosmological observations.[29][31]

In the 2000s and 2010s, it was shown that, since the universe is inhomogeneous as shown in the cosmic web of large-scale structure, acceleration effects measured on local scales in the patterns of the movements of galaxies should, in principle, reveal the global topology of the universe.[57][58][59]

See also

[edit]

Notes

[edit]
  1. ↑ Although listed in megaparsecs by the cited source, this number is so vast that its digits would remain virtually unchanged for all intents and purposes regardless of which conventional units it is listed in, whether it to be nanometers or gigaparsecs, as the differences would disappear into the error.
  2. ↑ ἐν τρισὶν ἐτίθετ᾽ εἶναι σφαίραις, ὧν τὴν μὲν πρώτην τὴν τῶν ἀπλανῶν ἄστρων εἶναι [37][38]
  3. ↑ Aristotle knew of the thoughts of Leucippus (and Democritus) and considered the possibility of an infinite universe. [45]

References

[edit]
  1. 1 2 3 4 5 6 7 8 9 Luminet, Jean-Pierre (15 January 2016). "The Status of Cosmic Topology after Planck Data". Universe. 2 (1): 1. arXiv:1601.03884. Bibcode:2016Univ....2....1L. doi:10.3390/universe2010001. ISSN 2218-1997. S2CID 7331164.
  2. ↑ "Will the Universe expand forever?". NASA. 24 January 2014. Retrieved 16 March 2015.
  3. ↑ Biron, Lauren (7 April 2015). "Our universe is Flat". Symmetry Magazine. FermiLab/SLAC.
  4. 1 2 Yashar Akrami; Stefano Anselmi; Craig J. Copi; Johannes R. Eskilt; Andrew H. Jaffe (2024). "Promise of Future Searches for Cosmic Topology". Physical Review Letters. 132 171501. arXiv:2210.11426. doi:10.1103/physrevlett.132.171501. PMID 38728711. Wikidata Q136902920. While unambiguous indicators of topology have yet to be detected, ... Much more can be done to discover, or constrain, the topology of space.
  5. 1 2 3 4 Lachièze-Rey, Marc; Luminet, Jean-Pierre (March 1995). "Cosmic topology". Physics Reports. 254 (3): 135–214. arXiv:gr-qc/9605010. Bibcode:1995PhR...254..135L. doi:10.1016/0370-1573(94)00085-H. S2CID 119500217.
  6. ↑ Davies, Paul (1977). Space and Time in the Modern Universe. Cambridge: Cambridge University Press. ISBN 978-0-521-29151-4.
  7. ↑ Crane, Leah (29 June 2024). de Lange, Catherine (ed.). "How big is the universe, really?". New Scientist. p. 31.
  8. ↑ "WMAP Overview". NASA Science. NASA. 26 September 2025. Retrieved 25 May 2026.
  9. ↑ Primack, Joel R. "Formation of Structure in the Universe". NASA/IPAC Extragalactic Database. Caltech. Retrieved 25 May 2026.
  10. ↑ Akrami, Yashar; Anselmi, Stefano; Copi, Craig J.; et al. (2024). "Promise of Future Searches for Cosmic Topology". Physical Review Letters. 132 (17) 171501. arXiv:2210.11426. Bibcode:2024PhRvL.132q1501A. doi:10.1103/PhysRevLett.132.171501. PMID 38728711.
  11. ↑ Ball, Philip (8 June 2024). "'We're trying to find the shape of space': scientists wonder if the universe is like a doughnut". The Guardian. Retrieved 25 May 2026.
  12. ↑ "Density Parameter, Omega". hyperphysics.phy-astr.gsu.edu. Retrieved 2015-06-01.
  13. ↑ Ade, P. A. R.; Aghanim, N.; Armitage-Caplan, C.; et al. (Planck Collaboration) (November 2014). "Planck 2013 results. XVI. Cosmological parameters". Astronomy & Astrophysics. 571: A16. arXiv:1303.5076. Bibcode:2014A&A...571A..16P. doi:10.1051/0004-6361/201321591. ISSN 0004-6361. S2CID 118349591.
  14. ↑ Aghanim, N.; Akrami, Y.; Ashdown, M.; et al. (Planck Collaboration) (September 2020). "Planck 2018 results: VI. Cosmological parameters". Astronomy & Astrophysics. 641: A6. arXiv:1807.06209. Bibcode:2020A&A...641A...6P. doi:10.1051/0004-6361/201833910. ISSN 0004-6361. S2CID 119335614.
  15. ↑ de Bernardis, P.; Ade, P. A. R.; Bock, J. J.; et al. (April 2000). "A flat Universe from high-resolution maps of the cosmic microwave background radiation". Nature. 404 (6781): 955–959. arXiv:astro-ph/0004404. Bibcode:2000Natur.404..955D. doi:10.1038/35010035. ISSN 0028-0836. PMID 10801117. S2CID 4412370.
  16. 1 2 Dodelson, Scott (2003). Modern Cosmology. Academic Press. ISBN 978-0-12-219141-1.
  17. ↑ Liddle, Andrew (2015). An Introduction to Modern Cosmology. John Wiley. ISBN 978-1-118-50214-3.
  18. ↑ Vardanyan, M.; Trotta, R.; Silk, J. (January 28, 2011). "Applications of Bayesian model averaging to the curvature and size of the Universe". Monthly Notices of the Royal Astronomical Society: Letters. 413 (1): L91–L95. arXiv:1101.5476. Bibcode:2011MNRAS.413L..91V. doi:10.1111/j.1745-3933.2011.01040.x. S2CID 2616287.
  19. ↑ Kazanas, D. (1980). "Dynamics of the universe and spontaneous symmetry breaking". The Astrophysical Journal. 241: L59–L63. Bibcode:1980ApJ...241L..59K. doi:10.1086/183361.
  20. ↑ Guth, Alan H. (1997). The inflationary universe: the quest for a new theory of cosmic origins. Basic Books. pp. 186–. ISBN 978-0-201-32840-0. Retrieved 1 May 2011.
  21. ↑ Don N. Page (2007). "Susskind's Challenge to the Hartle-Hawking No-Boundary Proposal and Possible Resolutions". Journal of Cosmology and Astroparticle Physics. 2007 (1): 004. arXiv:hep-th/0610199. Bibcode:2007JCAP...01..004P. doi:10.1088/1475-7516/2007/01/004. S2CID 17403084.
  22. 1 2 3 Levin, Janna (2002). "Topology and the cosmic microwave background". Physics Reports. 365 (4): 251–333. arXiv:gr-qc/0108043. doi:10.1016/S0370-1573(02)00018-2.
  23. 1 2 Lachièze-Rey, Marc; Luminet, Jean-Pierre (1995). "Cosmic topology". Physics Reports. 254 (3): 135–214. arXiv:gr-qc/9605010. doi:10.1016/0370-1573(94)00085-H.
  24. ↑ Cornish, Neil J.; Spergel, David N.; Starkman, Glenn D. (1998). "Circles in the sky: Finding topology with the microwave background radiation". Classical and Quantum Gravity. 15 (9): 2657–2670. arXiv:astro-ph/9801212. doi:10.1088/0264-9381/15/9/013.
  25. ↑ Planck Collaboration (2016). "Planck 2015 results. XVIII. Background geometry and topology". Astronomy & Astrophysics. 594. A18. arXiv:1502.01593. doi:10.1051/0004-6361/201525829.
  26. ↑ Vardanyan, Mihran; Trotta, Roberto; Silk, Joseph (21 July 2009). "How flat can you get? A model comparison perspective on the curvature of the Universe". Monthly Notices of the Royal Astronomical Society. 397 (1): 431–444. arXiv:0901.3354. Bibcode:2009MNRAS.397..431V. doi:10.1111/j.1365-2966.2009.14938.x. S2CID 15995519.
  27. ↑ Ellis, George F R. (2007), "Issues in the Philosophy of Cosmology", Philosophy of Physics, North-Holland, pp. 1183–1285, doi:10.1016/B978-044451560-5/50014-2, ISBN 978-0-444-51560-5, retrieved 2026-05-01
  28. ↑ A Universe From Nothing lecture by Lawrence Krauss at AAI. 2009. Archived from the original on 2021-12-15. Retrieved 17 October 2011 – via YouTube.
  29. 1 2 Luminet, Jean-Pierre; Weeks, Jeffrey R.; Riazuelo, Alain; Lehoucq, Roland; Uzan, Jean-Philippe (October 2003). "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background". Nature. 425 (6958): 593–595. arXiv:astro-ph/0310253. Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. ISSN 0028-0836. PMID 14534579. S2CID 4380713.
  30. ↑ Dumé, Isabelle (8 October 2003). "Is the universe a dodecahedron?". Physics World.
  31. 1 2 Lew, B.; Roukema, B.; Szaniewska, Agnieszka; Gaudin, Nicolas E. (May 2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data". Astronomy & Astrophysics. 482 (3): 747–753. arXiv:0801.0006. Bibcode:2008A&A...482..747L. doi:10.1051/0004-6361:20078777. ISSN 0004-6361. S2CID 1616362.
  32. ↑ Thurston, William P. (1997). Three-Dimensional Geometry and Topology. Vol. 1. Princeton University Press. ISBN 978-0-691-08304-9.
  33. ↑ Aurich, Ralf; Lustig, S.; Steiner, F.; Then, H. (2004). "Hyperbolic Universes with a Horned Topology and the CMB Anisotropy". Classical and Quantum Gravity. 21 (21): 4901–4926. arXiv:astro-ph/0403597. Bibcode:2004CQGra..21.4901A. doi:10.1088/0264-9381/21/21/010. S2CID 17619026.
  34. ↑ J. J. Callahan. "The Curvature of Space in a Finite Universe". Scientific American. Vol. 235, no. 2 (August 1976). Nature America, Inc.: ITHAKA. JSTOR 24950420.
  35. ↑ Sylvia Berryman (October 18, 2022). "Ancient Atomism: 2. Ancient Greek Atomism: 2.1 Leucippus and Democritus". In Edward N. Zalta (ed.). Stanford Encyclopedia of Philosophy. Leucippus held that there are an infinite number of atoms moving for all time in an infinite void, and that these can form into cosmic systems or kosmoi
  36. ↑ Diogenes Laërtius. "BOOK IX.: III". The Lives and Opinions of Eminent Philosophers. Translated by C. D. Yonge Laërtius. London: G. Bell & Sons Ltd. p. 388 – via Project Gutenberg. These are his doctrines in general; in particular detail, they are as follow: he says that the universe is infinite, as I have already mentioned; that of it, one part is a plenum, and the other a vacuum.
  37. 1 2 Molly Read. "A Brief History". University of Wisconsin, Madison.
  38. ↑ Ἀριστοτέλους. "Λ.1073β". Μετὰ τὰ Φυσικά. Clarendon Press. 1924: perseus.tufts.edu. Second paragraph, 1st and 2nd lines; verified translation via Google traductor. Greek original: via iask.ai/q/Aristotle-Metaphysics-Book-I-Greek-original-39nedig: Perseus Catalog (not available): www.physics.ntua.gr/mourmouras/greats/aristoteles/meta_ta_physica.pdf
  39. ↑
    "Metaphysics 12.1073b". Aristotle in 23 Volumes, Vols.17, 18. Translated by Hugh Tredennick. Cambridge, MA: Harvard University Press 1933: perseus.tufts.edu.
    Todd Timberlake (12 May 2011). "Computer Program Detail Page: Spheres of Eudoxus". American Association of Physics Teachers & National Science Foundation-National Science Digital Library (ISKME).
    Matthias Tomczak. "Lecture 8". Flinders University: University of Maine Ocean Observing System.
  40. ↑ Center for History of Physics. "The Greek Worldview Continuation of the Greek tradition". The American Institute of Physics.
  41. ↑ Jan Edward Garrett (November 7, 2012). "Introduction to Aristotle's Celestial and Terrestrial Physics". Western Kentucky University. Sometimes this sphere is simply called the kosmos, i.e., universe or world. There is no "place" and nothing material beyond this sphere.
  42. ↑ David J. Furley (August 1978). "The Greek Theory of the Infinite Universe". Journal of the History of Ideas. 42 (4 (Oct. - Dec., 1981)). Cambridge: University of Pennsylvania Press: ITHAKA: 571–585. doi:10.2307/2709119. JSTOR 2709119.
  43. ↑
    Grant E (1981). "5 - The historical roots of the medieval concept of an infinite, extracosmic void space". Much Ado about Nothing Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution. Cambridge University Press. pp. 105–115. doi:10.1017/CBO9780511895326.008. ISBN 978-0-521-22983-8.
    Grant, Edward (29 May 1981). Much Ado about Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution. Cambridge University Press. ISBN 0-521-22983-9.
  44. ↑ Aristotle. "BOOK I. 9". DE CAELO (PDF). Translated by J. L. Stocks; H. B. Wallis. St John's College, Oxford University: Humphrey Milford 1922. p. 279, lines 13–16, footnote. It is therefore evident that there is also no place or void or time outside the heaven. For in every place body can be present; and void is said to be that in which the presence of body, though not actual, is possible; and time is the number of movement.
  45. ↑
    Ἀριστοτέλους. "A.985b". Μετὰ τὰ Φυσικά. Clarendon Press. 1924: perseus.tufts.edu. Greek original: via iask.ai/q/Aristotle-Metaphysics-Book-I-Greek-original-39nedig: Perseus Catalog (not available): www.physics.ntua.gr/mourmouras/greats/aristoteles/meta_ta_physica.pdf
    "Metaphysics 1.985b". Aristotle in 23 Volumes, Vols.17, 18. Translated by Hugh Tredennick. Cambridge, MA: Harvard University Press 1933: perseus.tufts.edu.
    Helge Kragh (2010). "Ancient Greek-Roman Cosmology: Infinite, Eternal, Finite, Cyclic, and Multiple Universes". Journal of Cosmology. 9. University of Aarhus. A spatially infinite world was another impossibility, for by its very nature the world – meaning the heavens – revolved in a circle, and Aristotle pointed out that such motion was impossible as it would lead to an infinite velocity. What was enclosed by the outermost sphere comprised everything.
    Jacques A. Bailly. "Aristotle on the Infinite, Space, and Time: Mathematical: 1) Multitude". uvm.edu.
  46. ↑
    CHRISTOPHER GRANEY (October 31, 2022). "Augustine, Aquinas, and Calvin on the Size of the Moon, Scripture, and "Following the Science"". Specola Vaticana: The Society of Catholic Scientists.
    David Juste (10 May 2025). "'Ptolemy, Planetary Hypotheses (Greek)'". Ptolemaeus Arabus et Latinus. Works. Bayerische Akademie der Wissenschaften – via Center for History of Physics: history.aip.org/exhibits/cosmology.
    Mohan Matthen; R. J. Hankinson (1993). "Aristotle's Universe: Its Form and Matter". Synthese. 96 (3). Kluwer Academic Publishing: Springer Nature: ITHAKA: 417–435. doi:10.1007/BF01064010. JSTOR 20117821.
  47. ↑ G. I. Naan (1963). "On the Infinity of the Universe". Science & Society. 27 (2 (Spring, 1963)). Sage Publications, Inc: ITHAKA: 176–202. doi:10.1177/003682376302700203. JSTOR 40400937.
  48. ↑ Alexander Jones (2015). "Greek Cosmology and Cosmogony". In Ruggles, C. (ed.). Handbook of Archaeoastronomy and Ethnoastronomy. Springer, New York, NY. pp. 1549–1553. doi:10.1007/978-1-4614-6141-8_154. ISBN 978-1-4614-6141-8.
  49. ↑ Leeds Centre for Dante Studies & the Devers Program in Dante Studies at the University of Notre Dame. "Paradiso". University of Leeds. Ptolemaic understanding of the universe (after Ptolemy, an Alexandrian polymath of the second century A.D.). This was broadly shared by all mediaeval thinkers
  50. ↑ Beinecke Rare Book & Manuscript Library, Yale University Library (23 March 2021). "Divina Commedia, MS 428". yale.edu.
  51. 1 2 3 DJF; John J O'Connor; Edmund F Robertson. "MacTutor: The Infinite Universe". st-andrews.ac.uk. Archived from the original on 4 October 2025. Retrieved 16 December 2025. a
  52. ↑ Sun Kwok (22 October 2021). "Is the Universe Finite?: Abstract". Our Place in the Universe - II The Scientific Approach to Discovery (1 ed.). University of British Columbia: Springer Nature Switzerland AG. Bibcode:2021opus.book.....K. doi:10.1007/978-3-030-80260-8. ISBN 978-3-030-80260-8. After the development of the heliocentric theory, the hypothesis of the daily rotation of the celestial sphere was replaced by the hypothesis of the rotation of the Earth. This removes the need for the stars to lie at the same distance and rotate together, which in turn opens the possibility that the stars may have different distances from Earth and that the Universe could be infinite in size
  53. ↑ John D. Barrow (2005). "chapter seven Is the Universe Infinite?". The Infinite Book: A Short Guide to the Boundless, Timeless and Endless (reprint ed.). Jonathon Cape, Vintage Books, Random House. pp. 116–117. ISBN 0-09-944372-4 – via plus.maths.org/content/do-infinities-exist-nature-0 University of Cambridge.
  54. ↑ Cormac O'Raifeartaigh (February 3, 2017). "Albert Einstein and the origins of modern cosmology". Physics Today (2) 12150. AIP. Bibcode:2017PhT..2017b2150O. doi:10.1063/PT.5.9085.
    "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie"
    The outcome of those deliberations was Einstein's "Cosmological considerations" paper of 1917. His ingenious breakthrough was to postulate that we inhabit a universe of closed spatial geometry. Relativity could deliver a satisfactory model of the known universe if it was assumed that the cosmos had the geometry of a three-dimensional sphere—unbounded spatially, yet finite in content.
  55. ↑
    David Hilbert (5 June 2012). "On the infinite" (PDF). Cambridge University Press: lawrencecpaulson. p. 186. Einstein has shown that euclidean geometry must be abandoned...all the results of astronomy are perfectly compatible with the postulate that the universe is elliptical.
    Paul Benacerraf; Hilary Putnam, eds. (1926). "Über das Unendliche - On the infinite DAVID HILBERT (in: Philosophy of mathematics)". Mathematische Annalen. 95. Translated by Erna Putnam; Gerald J. Massey (2nd ed.). Göttingen: Berlin: (Cambridge London New York New Rochelle Melbourne Sydney): Springer Verlag: (Cambridge University Press: math.dartmouth.edu. German language title: jamesrmeyer.com/infinite/hilbert-uber-das-unendliche)
  56. ↑ Joseph Silk (2 May 2001). "Is the Universe finite or infinite? An interview with Joseph Silk". University of Oxford: European Space Agency.
  57. ↑ Boudewijn François Roukema; Bajtlik S.; Biesiada M.; Szaniewska A.; Jurkiewicz H. (2007). "A weak acceleration effect due to residual gravity in a multiply connected universe". Astronomy & Astrophysics. 463 (3): 861–871. arXiv:astro-ph/0602159. Bibcode:2007A&A...463..861R. doi:10.1051/0004-6361:20064979. Zbl 1118.85330. Wikidata Q68598777.
  58. ↑ Boudewijn François Roukema; Rozanski P. T. (2009). "The residual gravity acceleration effect in the Poincare dodecahedral space". Astronomy & Astrophysics. 502: 27–35. arXiv:0902.3402. Bibcode:2009A&A...502...27R. doi:10.1051/0004-6361/200911881. Zbl 1177.85087. Wikidata Q68676519.
  59. ↑ Jan J Ostrowski; Boudewijn F Roukema; Zbigniew P Buliński (2012). "A relativistic model of the topological acceleration effect". Classical and Quantum Gravity. 29 (16) 165006. arXiv:1109.1596. doi:10.1088/0264-9381/29/16/165006. Zbl 1253.83052. Wikidata Q96692451.
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