Jump to content

Shape of the universe

From Wikipedia, the free encyclopedia

Determination of the shape of the universe is a problem of physical cosmology.

Observational evidence (BOOMERANG 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]

The observable universe

[edit]
See also: Distance measures (cosmology)

Human knowledge of the universe is restricted by the fact that any signal information from further than the cosmological horizon within the universe moving towards any position of human perception or telescope is non-available. [7]

Possible shapes of the universe

[edit]
Further information: Curvature § Space, and Flatness problem
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.

Einstein stated, by the law of gravitation, heavy masses curve space-time. [8] The curvature of spacetime is the same as the density parameter, [9] represented with Ω (omega). 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,

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

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:[10][11]

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.[12]

The Friedmann–Lemaître–Robertson–Walker (FLRW) model using Friedmann equations is commonly used to model the universe.

Global universal structure

[edit]

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

  • whether the universe is infinite or finite in extent,
  • whether the geometry of the global universe is flat, positively curved, or negatively curved
  • whether the topology is simply connected (for example, like a sphere) or else multiply connected (for example, like a torus).[13]

Infinite or finite

[edit]

One of the unresolved questions about the universe is whether it is infinite or finite in extent.[14][15] Answers within the 21st century depend on the current standard cosmological model.[16]

Ancient mythologies variously described the universe as finite.[17]

By way of the account of Diogenes Laërtius, for Leucippus[18] (c. 5th century BC)[19] the universe is spatially infinite.[18] Eudoxus (c. 380 BC)[20][a] in thought of motion considered the stars integral to a sphere.[22][23][24] The concept of Aristotle[25][26] (384–322 BC),[27] concentric spheres[25][26] existed outgoing from Earth, the furthest contained the stars and was sometimes termed the kosmos,[26] outside of which there was nothing;[28][29][26] neither any place, time, or void extracosmic.[29][30]

From the concepts of Aristotle[31][32][b] which became the mode for Ptolemy[36][31] (2nd century AD[37] post[36] Ὑποθέσεις τῶν πλανωμένων[38]) the preferred[36] general cosmology[20] into the Middle Ages was the cosmos was finite[31] because of Aristotelian cosmology.[36] Dante Alighieri, Paradiso,[39] (1308–1320)[40] 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.[39] Bradwardine (1344) and Oresme during the 14th century contested the Aristotlian view on the basis of infinite God.[41]

The advent of the heliocentric model produced in scientific thought the possibility of an infinite universe.[42] A Universe infinite in size, using Copernicus, explained by Thomas Digges in: A perfit description of the caelestiall orbs, published 1576, was a conceptual break from the tradition of the reality of a celestial outer realm known as Paradise.[43]

Einstein in consideration of his general theory of relativity[44] (1916)[41] demonstrated in Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie (1917) a finite universe.[44] The de Sitter infinite universe (1917) was caused by incompatibility of Relativity and Euclidean space.[41] Hilbert (1925) thought the universe was determined finite by elliptical geometry or infinite by Euclidean geometry[45] (i.e. flat).[46]

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.[47] A finite universe is volumetrical,[48][46] an infinite universe could encompass an infinity of space with a finite amount of matter.[48]

Observational methods

[edit]

In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed[49] and applied to cosmological observations.[50][51]

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.[52][53][54]

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.[49] 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.[49]

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.[55]

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.[56]

Universe with zero curvature

[edit]

A flat universe can have zero total energy.[57]

Universe with positive curvature

[edit]

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[50][58] and an optimal orientation on the sky for the model was estimated in 2008.[51]

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 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".[59]

See also

[edit]

Notes

[edit]
  1. ^ ἐν τρισὶν ἐτίθετ᾽ εἶναι σφαίραις, ὧν τὴν μὲν πρώτην τὴν τῶν ἀπλανῶν ἄστρων εἶναι [21]
  2. ^ Aristotle knew of the thoughts of Leucippus (and Democritus)[33] and considered the possibility of an infinite universe.[34][35]

References

[edit]
  1. ^ Paul Preuss (26 April 2000). "Strong Evidence for Flat Universe Reported by BOMMERANG Project". Lawrence Berkeley National Laboratory. Archived from the original on 19 November 2025. Retrieved 26 December 2025.
  2. ^ Paul Preuss (9 May 2000). "MAXIMA Project's Imaging of Early Universe Agrees it is Flat, But..." Lawrence Berkeley National Laboratory. Archived from the original on 8 May 2025. Retrieved 26 December 2025.
  3. ^ Planck Collaboration (29 October 2014). esa (ed.). "Planck 2013 results. XVI. Cosmological parameters". esa. Archived from the original on 17 December 2025. Retrieved 26 December 2025. we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone
  4. ^ Chris Barnes; et al. (24 January 2014). Colleen Kaiser (ed.). "WMAP: Accomplishments". NASA. Archived from the original on 15 October 2025. Retrieved 16 March 2015. Nailed down the curvature of space to within 0.4% of "flat" Euclidean
  5. ^ Biron, Lauren (7 April 2015). "Our flat universe". Symmetry Magazine. FermiLab/SLAC. Archived from the original on 14 November 2025. Retrieved 8 April 2015.
  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. ^ Konstantinos Dimopoulos (2020). "Dynamics and Content of the Universe 2.2 The Universe expansion 2.2.1. Hubble-Lamaitre law". Introduction to Cosmic Inflation and Dark Energy. Taylor & Francis. p. 10-11. ISBN 9781351174855. SIgnals from beyond DH cannot reach us - is called the cosmological horizon - it is evident that there is more of the Universe beyond the horizon
  8. ^ Richard Feynman. Michael A. Gottlieb; Rudolf Pfeiffer (eds.). "42–1 Curved spaces with two dimensions". California Institute of Technology. Einstein had a different interpretation of the law of gravitation. According to him, space and time—which must be put together as space-time—are curved near heavy masses.
  9. ^ Fatima Zaidouni (2019). "The source of curvature" (PDF). Department of Physics and Astronomy, University of Rochester (pas.rochester.edu). 1 Introduction In this paper, we are building the understanding of the relation between spacetime curvature and matter energy density. As introduced previously, they are equivalent
  10. ^ "Density Parameter, Omega". hyperphysics.phy-astr.gsu.edu. Retrieved 2015-06-01.
  11. ^ 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.
  12. ^ 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.
  13. ^ Davies, Paul (1977). Space and Time in the Modern Universe. Cambridge: Cambridge University Press. ISBN 978-0-521-29151-4.
  14. ^ Paul Sutter (November 23, 2025). "Is the Universe Infinite?". Universe Today. Archived from the original on 23 November 2025. Retrieved 15 December 2025.
  15. ^ 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. Archived from the original on 15 November 2025. Retrieved 15 December 2025.
  16. ^ Gregory J Galloway; Marcus AKhuri; Eric Woolgar (2022). "The topology of general cosmological models" (PDF). Class. Quantum Grav. 39 (195004). Institute of Physics. doi:10.1088/1361-6382/ac75e1. Abstract Is the Universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the Universe is assumed to be homogeneous and isotropic.
  17. ^ 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.
  18. ^ a b 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)
  19. ^ 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)
  20. ^ a b Molly Read. "A Brief History". University of Wisconsin, Madison.
  21. ^ 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
  22. ^ "Metaphysics 12.1073b". Aristotle in 23 Volumes, Vols.17, 18. Translated by Hugh Tredennick. Cambridge, MA: Harvard University Press 1933: perseus.tufts.edu.
  23. ^ 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).
  24. ^ Matthias Tomczak. "Lecture 8". Flinders University: University of Maine Ocean Observing System.
  25. ^ a b Center for History of Physics. "The Greek Worldview Continuation of the Greek tradition". The American Institute of Physics.
  26. ^ a b c d 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.
  27. ^ Justin Humphreys. Aristotle (384 B.C.E.—322 B.C.E.). University of Pennsylvania: Internet Encyclopedia of Philosophy.
  28. ^ 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.
  29. ^ a b
    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. ISBN 0521229839.
  30. ^ 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.
  31. ^ a b c 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.
  32. ^ 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.
  33. ^
    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.
  34. ^ 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.
  35. ^ Jacques A. Bailly. "Aristotle on the Infinite, Space, and Time: Mathematical: 1) Multitude". uvm.edu.
  36. ^ a b c d 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.
  37. ^ 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.
  38. ^ 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.
  39. ^ a b 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
  40. ^ Beinecke Rare Book & Manuscript Library, Yale University Library (23 March 2021). "Divina Commedia, MS 428". yale.edu.
  41. ^ a b c 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 finite universe (Einstein had to include a cosmological constant to achieve this as he believed the universe was static
  42. ^ 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
  43. ^ 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.
  44. ^ a b 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. 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
  45. ^
    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)
  46. ^ a b Joseph Silk (2 May 2001). "Is the Universe finite or infinite? An interview with Joseph Silk". University of Oxford: European Space Agency.
  47. ^ 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.
  48. ^ a b Richard Swinburne (1968). "The Size and Geometry of the Universe". Space and Time. University of Hull, UK: Palgrave Macmillan London: Springer Nature. pp. 280–295. 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.
  49. ^ a b c Lachièze-Rey & Luminet 1995
  50. ^ a b 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.
  51. ^ a b 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.
  52. ^ 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.
  53. ^ 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.
  54. ^ 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)
  55. ^ 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.
  56. ^ 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.
  57. ^ 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.
  58. ^ Dumé, Isabelle (8 October 2003). "Is the universe a dodecahedron?". Physics World.
  59. ^ 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.
[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Shape_of_the_universe&oldid=1348585712"