Picard horn

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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 arXiv:astro-ph/0403597.

The space in question is the quotient of the upper half-plane model of hyperbolic 3-space by the group $\operatorname{PSL}_2(\mathbf{Z}[i])$ , which was first described in E. Picard, Sur un groupe de transformations des points de l'espace situés du même côté d'un plan, 1884, Bull. Soc. Math. France 12, 43-47.

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 $Υ 1$ 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.