Shape of the universe

Last updated

In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curvature). General relativity explains how spatial curvature (local geometry) is constrained by gravity. The global 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. 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).

Contents

Observational evidence (BOMMERANG Project, MAXIMA, Planck, WMAP) indicates that the observable universe is spatially flat. [1] [2] [3] [4] [5] It is unknown whether the universe is simply connected like euclidean space or multiply connected like a torus. [6]

Shape of the observable universe

The universe's structure can be examined from two angles:

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

The observable universe (of a given current observer) is a roughly spherical region extending about 46 billion light-years in every direction (from that observer, the observer being the current Earth, unless specified otherwise). [7] It appears older and more redshifted the deeper we look into space. 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.

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. Different mathematical models of the universe's global geometry can be constructed, all consistent with observations and general relativity. 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, similar to how a cuboid [ citation needed ] is longer in one dimension than the others. Scientists test these models by looking for novel implications – phenomena not yet observed but necessary if the model is accurate. For instance, a small closed universe would produce multiple images of the same object in the sky, though not necessarily of the same age. As of 2024, observational evidence indicates that the observable universe is spatially flat with an unknown global structure.

Curvature of the universe

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. 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°.

The local geometry of the universe is determined by whether the density parameter O is greater than, less than, or equal to 1. From top to bottom: a spherical universe with O > 1, a hyperbolic universe with O < 1, and a flat universe with O = 1. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of (local) space. End of universe.jpg
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 LCDM universe. Particle horizon: green, Hubble radius: blue, Event horizon: purple, Light cone: orange. Spacetime-diagram-flat-universe-proper-coordinates.png
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 (OL-Ok). Hyperbolic.universe.proper.coordinates.png
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). Big-crunch spacetime-diagram matter-dominated proper-distances.png
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 bend the curvature of spacetime and is used to determine what curvature the universe has by using a value called 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. Put another way,

Scientists could experimentally calculate Ω to determine the curvature 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: [8] [9]

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.

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. [10]

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.

Global universal structure

Global structure covers the geometry and the topology of the whole universe—both the observable universe and beyond. While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature. The universe is often taken to be a geodesic manifold, free of topological defects; relaxing either of these complicates the analysis considerably. A global geometry is a local geometry plus a topology. It follows that a topology alone does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries.

As stated in the introduction, investigations within the study of the global structure of the universe include:

Infinite or finite

One of the unresolved questions about the universe is whether it is infinite or finite in extent. [12] [13]

Ancient mythologies variously described the universe as finite. [14]

By way of the account of Diogenes Laërtius, for Leucippus [15] (c. 5th century B.C.) [16] the universe is spatially infinite. [15] Eudoxus (c. 380 B.C) [17] [a] in thought of motion considered the stars integral to a sphere. [19] [20] [21] The concept of Aristotle [22] [23] (384—322 B.C.E.), [24] concentric spheres [22] [23] existed outgoing from Earth, the furthest contained the stars and was sometimes termed the kosmos, [23] outside of which there was nothing; [25] [26] [23] neither any place, time, or void extracosmic. [26] [27]

From the concepts of [28] Aristotle [28] [29] [b] which became the mode for Ptolemy [33] [28] (2nd century AD [34] post [33] Ὑποθέσεις τῶν πλανωμένων [35] ) the preferred [33] general cosmology [17] into the Middle Ages was the cosmos was finite [28] because of Aristotelian cosmology. [33] Dante Alighieri, Paradiso, [36] (1308 - 1320) [37] conceived of a Ptolemaic understanding universe which explained the Earth was central to spheres the outer of which was the realm of God, the perception of all prominent medieval era thinkers. [36] Bradwardine (1344) and Oresme during the 14th century contested the Aristotlian view on the basis of infinite God. [38]

The advent of the heliocentric model produced in scientific thought the possibility of an infinite universe. [39] A Universe infinite in size, using Copernicus, explained by Thomas Digges in: A perfit description of the caelestiall orbs, published 1576, was an inceptual break from the tradition of the reality of a celestial outer realm known as Paradise. [40]

Einstein in consideration of his General Theory of Relativity [41] (1916) [38] demonstrated in Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie (1917) a finite universe. [41] The de Sitter infinite universe (1917) was caused by incompatbility of Relativity and Euclidean space. [38] Hilbert (1925) thought the universe was determined finite by elliptical geometry or infinite by Euclidean geometry [42] (i.e. flat). [43]

The factor which could determine from our position in the universe (and the 21st century) a scientific answer of which version of the universe is thought reality with regards to the geometry of the universe is: if positively curved is finite, if flat or negatively curved is infinite. [44] A finite universe is volumetrical, [45] [43] an infinite universe could encompass an infinity of space with a finite amount of matter. [45] 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".

With or without boundary

Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a disc, have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the 3-sphere and 3-torus, that have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact means that it is finite in extent ("bounded") and complete. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are both closed manifolds.

Observational methods

In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed [46] and applied to cosmological observations. [47] [48]

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. [49] [50] [51]

Curvature

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. [46] 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. [46] 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. [52]

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. [53] (i.e. positive curvature: K = +1, ΩK < 0, Ω > 1, negative curvature: K = −1, ΩK > 0, Ω < 1, zero curvature: K = 0, ΩK = 0, Ω = 1).

Universe with zero curvature

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. Flat universes that are finite in extent include the torus and Klein bottle. Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable. These are the Bieberbach manifolds. The most familiar is the aforementioned 3-torus universe.

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.

A flat universe can have zero total energy. [54]

Universe with positive curvature

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 [47] [55] and an optimal orientation on the sky for the model was estimated in 2008. [48]

Universe with negative curvature

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 are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood. 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". [56]

Curvature: open or closed

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. 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.

See also

Notes

  1. ↑ ἐν τρισὶν ἐτίθετ᾽ εἶναι σφαίραις, ὧν τὴν μὲν πρώτην τὴν τῶν ἀπλανῶν ἄστρων εἶναι [18]
  2. ↑ Aristotle knew of the thoughts of Leucippus (and Democritus) [30] and considered the possibility of an infinite universe. [31] [32]

References

  1. ↑ Paul Preuss (26 April 2000). "Strong Evidence for Flat Universe Reported by BOMMERANG Project". Lawrence Berkeley National Laboratory. Archived 19 November 2025 at the Wayback Machine
  2. ↑ Paul Preuss (9 May 2000). "MAXIMA Project's Imaging of Early Universe Agrees it is Flat, But..." Lawrence Berkeley National Laboratory. Archived 8 May 2025 at the Wayback Machine
  3. ↑ Planck Collaboration (29 October 2014). esa (ed.). "Planck 2013 results. XVI. Cosmological parameters". esa. we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone Archived 17 December 2025 at the Wayback Machine
  4. ↑ Chris Barnes; et al. (24 January 2014). Colleen Kaiser (ed.). "WMAP: Accomplishments". NASA . Retrieved 16 March 2015. Nailed down the curvature of space to within 0.4% of "flat" Euclidean Archived 15 October 2025 at the Wayback Machine
  5. ↑ Biron, Lauren (7 April 2015). "Our flat universe". Symmetry Magazine. FermiLab/SLAC. Archived 14 November 2025 at the Wayback Machine
  6. ↑ Yashar Akrami; Stefano Anselmi; Craig J. Copi; Johannes R. Eskilt; Andrew H. Jaffe (April 2024). "Promise of Future Searches for Cosmic Topology". Physical Review Letters . 132 171501. arXiv: 2210.11426 . doi: 10.1103/PHYSREVLETT.132.171501 . ISSN   0031-9007. 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.
  7. ↑ Crane, Leah (29 June 2024). de Lange, Catherine (ed.). "How big is the universe, really?" . New Scientist. p. 31.
  8. ↑ "Density Parameter, Omega". hyperphysics.phy-astr.gsu.edu. Retrieved 2015-06-01.
  9. ↑ 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.
  10. ↑ 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.
  11. ↑ Davies, Paul (1977). Space and Time in the Modern Universe. Cambridge: Cambridge University Press. ISBN   978-0-521-29151-4.
  12. ↑ Paul Sutter (November 23, 2025). "Is the Universe Infinite?". Universe Today.
  13. ↑ Anna Moore; Sara Webb; Sam Baron; Tanya Hill; Kevin Orrman-Rossiter (11 August 2021). "Is space infinite? We asked 5 experts". Australia: Swinburne University of Technology.
  14. ↑ J. J. Callahan. "The Curvature of Space in a Finite Universe". Scientific American . Vol. 235, no. 2 (August 1976). Nature America, Inc.: ITHAKA.
  15. 1 2 Diogenes Laërtius. "BOOK IX.: III". The Lives and Opinions of Eminent Philosophers. Translated by C. D. Yonge Laërtius. Queen’s College, Belfast: G. Bell & Sons Ltd: gutenberg.org. p. 388. 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.{{cite book}}: CS1 maint: location (link)
  16. ↑ 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. Metaphysics Research Lab, CSLI. 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{{cite book}}: CS1 maint: location missing publisher (link)
  17. 1 2 Molly Read. "A Brief History". University of Wisconsin, Madison.
  18. ↑ 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
  19. ↑ "Metaphysics 12.1073b". Aristotle in 23 Volumes, Vols.17, 18. Translated by Hugh Tredennick. Cambridge, MA: Harvard University Press 1933: perseus.tufts.edu.
  20. ↑ 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).
  21. ↑ Matthias Tomczak. "Lecture 8". Flinders University: University of Maine Ocean Observing System.
  22. 1 2 Center for History of Physics. "The Greek Worldview Continuation of the Greek tradition". The American Institute of Physics.
  23. 1 2 3 4 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.
  24. ↑ Justin Humphreys. Aristotle (384 B.C.E.—322 B.C.E.). University of Pennsylvania: Internet Encyclopedia of Philosophy.
  25. ↑ 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.
  26. 1 2
    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.
    "Much Ado about Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution". Google Books. ISBN   0521229839.
  27. ↑ 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.
  28. 1 2 3 4 G. I. Naan. "On the Infinity of the Universe". Science & Society . 27 (2 (Spring, 1963)). Sage Publications, Inc: ITHAKA.
  29. ↑ Mohan Matthen; R. J. Hankinson (1993). "Aristotle's Universe: Its Form and Matter". Synthese . 96 (3). Kluwer Academic Publishing: Springer Nature: ITHAKA.
  30. ↑
    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.
  31. ↑ 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.
  32. ↑ Jacques A. Bailly. "Aristotle on the Infinite, Space, and Time: Mathematical: 1) Multitude". uvm.edu.
  33. 1 2 3 4 Alexander Jones (2015). "Greek Cosmology and Cosmogony". In Ruggles, C. (ed.). Handbook of Archaeoastronomy and Ethnoastronomy. Springer, New York, NY. doi:10.1007/978-1-4614-6141-8_154. ISBN   978-1-4614-6141-8.
  34. ↑ 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.
  35. ↑ 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.
  36. 1 2 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
  37. ↑ Beinecke Rare Book & Manuscript Library, Yale University Library. "Divina Commedia, MS 428". yale.edu.
  38. 1 2 3 DJF; John J O'Connor; Edmund F Robertson. "MacTutor: The Infinite Universe". st-andrews.ac.uk. a finite universe (Einstein had to include a cosmological constant to achieve this as he believed the universe was static Archived 2025-10-04 at the Wayback Machine
  39. ↑ 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. 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
  40. ↑ 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. p. 116-117. ISBN   0099443724 – via plus.maths.org/content/do-infinities-exist-nature-0 University of Cambridge.
  41. 1 2 Cormac O'Raifeartaigh (February 3, 2017). "Albert Einstein and the origins of modern cosmology". Physics Today . AIP. doi:10.1063/PT.5.9085. finite in content. However, the Einstein universe came at a price. In his analysis, Einstein found that a nonzero solution to the field equations could be obtained only if a new term was introduced... known as the cosmological constant
  42. ↑
    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)
  43. 1 2 Joseph Silk (2 May 2001). "Is the Universe finite or infinite? An interview with Joseph Silk". University of Oxford: European Space Agency.
  44. ↑ Dragan Huterer (May 18, 2023). "Is the universe infinite or finite? Or is it so close to infinite that for all practical purposes it is?". Astronomy . No. FEBRUARY 2012. University of Michigan, Ann Arbor: Firecrown Media, Chattanooga, TN.
  45. 1 2 Richard Swinburne (1968). "The Size and Geometry of the Universe". Space and Time. University of Hull, UK: Palgrave Macmillan London: Springer Nature. doi:10.1007/978-1-349-00581-9_15. ISBN   978-1-349-00581-9. size of the space in which those objects are situated. I shall call this the s-size...From the seventeenth century onward men believed without question that the s-size of the Universe was infinite.
  46. 1 2 3 Lachièze-Rey & Luminet 1995
  47. 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.
  48. 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.
  49. ↑ Boudewijn François Roukema; Bajtlik S.; Biesiada M.; Szaniewska A.; Jurkiewicz H. (March 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. ISSN   0004-6361. Zbl   1118.85330. Wikidata   Q68598777.
  50. ↑ 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. ISSN   0004-6361. Zbl   1177.85087. Wikidata   Q68676519.
  51. ↑ Jan J Ostrowski; Boudewijn F Roukema; Zbigniew P Buliński (30 July 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. ISSN   0264-9381. Zbl   1253.83052. Wikidata   Q96692451.{{cite journal}}: CS1 maint: article number as page number (link)
  52. ↑ 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.
  53. ↑ 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.
  54. ↑ 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.
  55. ↑ Dumé, Isabelle (8 October 2003). "Is the universe a dodecahedron?". Physics World .
  56. ↑ 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.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.