Picard horn: Difference between revisions
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A '''Picard horn''', also called the '''Picard topology''' or '''Picard model''', is a theoretical model for the |
A '''Picard horn''', also called the '''Picard topology''' or '''Picard model''', is a theoretical model for the |
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[[shape of the Universe]]. It is a horn topology, meaning it has [[hyperbolic geometry]] (the term "horn" is due to [[pseudosphere]] models of hyperbolic space). |
[[shape of the Universe]]. It is a horn topology, meaning it has [[hyperbolic geometry]] (the term "horn" is due to [[pseudosphere]] models of hyperbolic space). |
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Revision as of 22:20, 18 January 2011
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A Picard horn, also called the Picard topology or Picard model, is a theoretical model for the shape of the Universe. It is a horn topology, meaning it has hyperbolic geometry (the term "horn" is due to pseudosphere models of hyperbolic space).
The term was coined by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper Hyperbolic Universes with a Horned Topology and the CMB Anisotropy[1].
The space in question is the quotient of the upper half-plane model of hyperbolic 3-space by the group , which was first described by Emile Picard[2] in 1884[3].
![{\displaystyle \operatorname {PSL} _{2}(\mathbf {Z} [i])}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a5f6f313b6977b5c3b269fd159ea92c7f219133f)
A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Fritz Grunewald and Wolfgang Huntebrinker, A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp, Experiment. Math. Volume 5, Issue 1 (1996), 57-80. The same source calculates the first 80 eigenvalues of the Laplacian, tabulated on p. 72, where is a fundamental domain of the Picard space.
The model was created in an attempt to describe the microwave background radiation apparent in the universe, and has finite volume and useful spectral characteristics (the first several eigenvalues of the Laplacian are computed and in good accord with observation). In this model one end of the figure curves finitely into the bell of the horn. The curve along any side of horn is considered to be a negative curve. The other end extends to infinity.
References
- Sherriff, Lucy (2004-05-27). "Boffins trumpet horn shaped universe". The Register. Retrieved 2006-12-28.
- Battersby, Stephen (2004-04-15). "Big Bang glow hints at funnel-shaped Universe". New Scientist. Retrieved 2007-12-01.
- ^ Aurich, Ralf (2004). "Hyperbolic Universes with a Horned Topology and the CMB Anisotropy". Class.Quant.Grav. 21. Institute of Physics: 4901–4926. Archived from the original on 2004-10-14. Retrieved 2010-09-18.
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- ^ Picard, E. (1884-03-07). "Sur un groupe de transformations des points de l'espace situés du même côté d'un plan". Bulletin de la Société Mathématique de France (in French). 12: 43–37. Retrieved 2010-02-26.
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