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The Status of Cosmic Topology After Planck Data

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The Status of Cosmic Topology After Planck Data universe Review The Status of Cosmic Topology after Planck Data Jean-Pierre Luminet 1,2 Received: 19 November 2015; Accepted: 7 January 2016; Published: 15 January 2016 Academic Editors: Stephon Alexander, Jean-Michel Alimi, Elias C. Vagenas and Lorenzo Iorio 1 Laboratoire d’Astrophysique de Marseille (LAM), CNRS UMR 7326, F-13388 Marseille, France 2 LUTH (Observatoire de Paris), CNRS UMR 8102, F-92195 Meudon, France; [email protected] Abstract: In the last decade, the study of the overall shape of the universe, called Cosmic Topology, has become testable by astronomical observations, especially the data from the Cosmic Microwave Background (hereafter CMB) obtained by WMAP and Planck telescopes. Cosmic Topology involves both global topological features and more local geometrical properties such as curvature. It deals with questions such as whether space is finite or infinite, simply-connected or multi-connected, and smaller or greater than its observable counterpart. A striking feature of some relativistic, multi-connected small universe models is to create multiples images of faraway cosmic sources. While the last CMB (Planck) data fit well the simplest model of a zero-curvature, infinite space model, they remain consistent with more complex shapes such as the spherical Poincaré Dodecahedral Space, the flat hypertorus or the hyperbolic Picard horn. We review the theoretical and observational status of the field. Keywords: cosmology; general relativity; topology; cosmic microwave background; Planck telescope 1. Introduction The idea that the universe might have a non-trivial topology and, if sufficiently small in extent, display multiple images of faraway sources, was first discussed in 1900 by Karl Schwarzschild (see [1] for reference and English translation). With the advent of Einstein’s general relativity theory (see, e.g., the recent historical overview in [2]) and the discoveries of non-static universe models by Friedmann and Lemaître in the decade 1922–1931, the face of cosmology definitively changed. While Einstein’s cosmological model of 1917 described space as the simply-connected, positively curved hypersphere S3, de Sitter in 1917 and Lemaître in 1927 used the multi-connected projective sphere P3 (obtained by identifying opposite points of S3) for describing the spatial part of their universe models. In 1924, Friedmann [3] pointed out that Einstein’s equations are not sufficient for deciding if space is finite or infinite: Euclidean and hyperbolic spaces, which in their trivial (i.e., simply-connected) topology are infinite in extent, can become finite (although without an edge) if one identifies different points—an operation which renders the space multi-connected. This opens the way to the existence of phantom sources, in the sense that at a single point of space an object coexists with its multiple images. The whole problem of cosmic topology was thus posed, but as the cosmologists of the first half of 20th century had no experimental means at their disposal to measure the topology of the universe, the vast majority of them lost all interest in the question. A revival of interest in multi-connected cosmologies arose in the 1970s, under the lead of theorists who investigated several kinds of topologies (see [4] for an exhaustive review and references, in which the term “Cosmic Topology” was coined). However, most cosmologists either remained completely ignorant of the possibility, or regarded it as unfounded, until the 1990s when data on the CMB provided by space telescopes gave access to the largest possible volume of the observable universe. Since then, hundreds of articles have considerably enriched the field of theoretical and observational cosmology. Universe 2016, 2, 1; doi:10.3390/universe2010001 www.mdpi.com/journal/universe Universe 2016, 2, 1 2 of 9 Universe 2016, 2, 1 2 of 9 2. A A Theoretical Theoretical Reminder Reminder At veryvery large large scale scale our our Universe Universe seems seems to be to correctly be correctly described described by one by of theone Friedmann-Lemaître of the Friedmann‐ (hereafterLemaître (hereafter FL) models, FL) namely models, homogeneous namely homogeneous and isotropic and solutionsisotropic solutions of Einstein’s of Einstein’s equations, equations, of which theof which spatial the sections spatial havesections constant have constant curvature. curvature. They fall They into fall three into general three general classes, classes, according according to the signto the of sign curvature. of curvature. In most In most cosmological cosmological studies, studies, the spatial the spatial topology topology is assumed is assumed to be to that be that of the of correspondingthe corresponding simply-connected simply‐connected space: space: the the hypersphere, hypersphere, Euclidean Euclidean space space or or 3D-hyperboloid, 3D‐hyperboloid, the firstfirst being finite finite and and the the other other two two infinite. infinite. However, However, there there is no is particular no particular reason reason for space for space to have to havea trivial a trivial topology: topology: the Einstein the Einstein field equations field equations are local are partial local differential partial differential equations equations which relate which the relatemetric the and metric its derivatives and its derivatives at a point to at the a point matter to‐energy the matter-energy contents of space contents at that of spacepoint. atTherefore, that point. to Therefore,a metric element to a metric solution element of Einstein solution field of Einstein equations, field there equations, are several, there if are not several, an infinite if not number, an infinite of number,compatible of topologies, compatible which topologies, are also which possible are also models possible for the models physical for theuniverse. physical For universe. example, Forthe 3 3 example,hypertorus the T hypertorus and the usual T3 and Euclidean the usual space Euclidean E are locally space Eidentical,3 are locally and identical, relativistic and cosmological relativistic cosmologicalmodels describe models them describe with the them same with FL equations, the same FL even equations, though eventhe former though is thefinite former in extent is finite while in extentthe latter while is infinite. the latter Only is infinite. the boundary Only the conditions boundary on conditions the spatial on coordinates the spatial coordinatesare changed. are The changed. multi‐ Theconnected multi-connected FL cosmological FL cosmological models modelsshare exactly share exactly the same the same kinematics kinematics and and dynamics dynamics as as the corresponding simply-connectedsimply‐connected ones; ones; in particular,in particular, the timethe evolutionstime evolutions of the scaleof the factor scale are factor identical. are identical.In FL models, the curvature of physical space (averaged on a sufficiently large scale) depends on theIn wayFL models, the total the energy curvature density of physical of the universe space (averaged may counterbalance on a sufficiently the kineticlarge scale) energy depends of the on the way the total energy density of the universe may counterbalance the kinetic energy of the expanding space. The normalized density parameter W0, defined as the ratio of the actual energy densityexpanding to the space. critical The value normalized that strictly density Euclidean parameter space Ω would0, defined require, as the characterizes ratio of the the actual present-day energy density to the critical value that strictly Euclidean space would require, characterizes the present‐day contents (matter, radiation and all forms of energy) of the universe. If W0 is greater than 1, then the contents (matter, radiation and all forms of energy) of the universe. If Ω0 is greater than 1, then the space curvature is positive and the geometry is spherical; if W0 is smaller than 1, the curvature is space curvature is positive and the geometry is spherical; if Ω0 is smaller than 1, the curvature is negative and geometry is hyperbolic; eventually W0 is strictly equal to 1 and space is locally Euclidean (currentlynegative and said geometry flat, although is hyperbolic; the term eventually can be misleading). Ω0 is strictly equal to 1 and space is locally Euclidean (currently said flat, although the term can be misleading). Figure 1.1.The The Illusion Illusion of theof Universalthe Universal Covering Covering Space. Space. In the caseIn the of a case 2D torus of a space, 2D torus the fundamental space, the domain,fundamental which domain, represents which real represents space, is the real interior space, ofis the a rectangle, interior of whose a rectangle, opposite whose edges opposite are identified. edges Theare identified. observer O The sees observer rays of light O sees from rays the of source light from S coming the source from severalS coming directions. from several He has directions. the illusion He of seeing distinct sources S ,S ,S , etc., distributed along a regular canvas which covers the UC has the illusion of seeing distinct1 2 sources3 S1, S2, S3, etc., distributed along a regular canvas which covers space—anthe UC space—an infinite plane.infinite plane. Independently of curvature, a much discussed question in the history of cosmology (and also Independently of curvature, a much discussed question in the history of cosmology (and also philosophy) is to know whether space is finite or infinite in extent. Of course no physical measure can philosophy) is to know whether space is finite or infinite in extent. Of course
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