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Shape of the universe
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{{short descriptionThe local and global geometry of the universe}}{{RedirectEdge of the Universethe Bee Gees songEdge of the Universe (song)}}{{Multiple issues{{More citations neededdate=November 2007}}{{Expert neededdate=April 2017}}}}{{Cosmologycomp/struct}}The shape of the universe is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes general global properties of its shape as of a continuous object. The shape of the universe is related to general relativity, which describes how spacetime is curved and bent by mass and energy.Cosmologists distinguish between the observable universe and the global universe. The observable universe consists of the part of the universe that can, in principle, be observed by light reaching Earth within the age of the universe. It encompasses a region of space that currently forms a ball centered at Earth of estimated radius {{convert46.5e9lym}}. This does not mean the universe is 46.5 billion years old; instead the universe is measured to be 13.8 billion years old, but space itself has also expanded, causing the size of the observable universe to be larger than the distance traversible by light over the duration of its current age. Assuming an isotropic nature, the observable universe is similar for all contemporary vantage points.The global shape of the universe can be described with three attributes:BOOK, Tegmark, Max, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, 2014, Knopf, 9780307599803, 1,  the content below is remote from Wikipedia
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 Finite or infinite
 Flat (no curvature), open (negative curvature), or closed (positive curvature)
 Connectivity, how the universe is put together, i.e., simply connected space or multiply connected.
but the data are also consistent with other possible shapes, such as the socalled PoincarÃ© dodecahedral space and the Sokolovâ€“Starobinskii space (quotient of the upper halfspace model of hyperbolic space by 2dimensional lattice).
Shape of the observable universe
{{See alsoDistance measures (cosmology)}}As stated in the introduction, there are two aspects to consider: its local geometry, which predominantly concerns the curvature of the universe, particularly the observable universe, and
 its global geometry, which concerns the topology of the universe as a whole.
Curvature of the universe
The curvature is a quantity describing how the geometry of a space differs locally from the one of the flat space. The curvature of any locally isotropic space (and hence of a locally isotropic universe) falls into one of the three following cases: Zero curvature (flat); a drawn triangle's angles add up to 180Â° and the Pythagorean theorem holds; such 3dimensional space is locally modeled by Euclidean space {{mathE3}}.
 Positive curvature; a drawn triangle's angles add up to more than 180Â°; such 3dimensional space is locally modeled by a region of a 3sphere {{mathS3}}.
 Negative curvature; a drawn triangle's angles add up to less than 180Â°; such 3dimensional space is locally modeled by a region of a hyperbolic space {{mathH3}}.
 If {{mathÎ© {{=}} 1}}, the universe is flat
 If {{mathÎ© > 1}}, there is positive curvature
 if {{mathÎ© < 1}} there is negative curvature
Global universe structure
Global structure covers the geometry and the topology of the whole universeâ€”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 3space and hyperbolic 3space 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: 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 like a sphere or multiply connected, like a torusBOOK, Space and time in the modern universe, P.C.W.Davies, 1977, cambridge university press, 9780521291514,
Infinite or finite
One of the presently unanswered questions about the universe is whether it is infinite or finite in extent. For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled up with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. 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 {{mvard}}, there are points that are of a distance at least {{mvard}} apart. A finite universe is a bounded metric space, where there is some distance {{mvard}} such that all points are within distance {{mvard}} of each other. The smallest such {{mvard}} is called the diameter of the universe, in which case the universe has a welldefined "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 very 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 3sphere and 3torus, which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact basically 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 3sphere and 3torus are both closed manifolds.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.JOURNAL, Luminet, JeanPierre, JeanPierre Luminet, Marc, LachiÃ¨zeRey, Cosmic Topology, Physics Reports, 254, 3, 135â€“214, 1995, grqc/9605010, 10.1016/03701573(94)00085h, 1995PhR...254..135L, Many textbooks erroneously state that a flat universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe. For example, Euclidean space is flat, simply connected, and infinite, but the torus is flat, multiply connected, finite, and compact.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 flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10âˆ’4. If the true value of the cosmological curvature parameter is larger than 10âˆ’3 we will be able to distinguish between these three models even now.JOURNAL, 0901.3354, 2009MNRAS.397..431V, 10.1111/j.13652966.2009.14938.x, How flat can you get? A model comparison perspective on the curvature of the Universe, Monthly Notices of the Royal Astronomical Society, 397, 1, 431â€“444, 2009, Vardanyan, Mihran, Trotta, Roberto, Silk, Joseph, Results of the Planck mission released in 2015 show the cosmological curvature parameter, Î©K, to be 0.000Â±0.005, consistent with a flat universe.JOURNAL, 1502.01589, Planck 2015 results. XIII. Cosmological parameters, Planck Collaboration, Ade, P. A. R., Aghanim, N., Arnaud, M., Ashdown, M., Aumont, J., Baccigalupi, C., Banday, A. J., Barreiro, R. B., Bartlett, J. G., Bartolo, N., Battaner, E., Battye, R., Benabed, K., Benoit, A., BenoitLevy, A., Bernard, J.P., Bersanelli, M., Bielewicz, P., Bonaldi, A., Bonavera, L., Bond, J. R., Borrill, J., Bouchet, F. R., Boulanger, F., Bucher, M., Burigana, C., Butler, R. C., Calabrese, E., Cardoso, J.F., 29, 2016, 10.1051/00046361/201525830, 594, Astronomy & Astrophysics, A13, 2016A&A...594A..13P,{{anchorFlat universe}} Universe with zero curvature
In a universe with zero curvature, the local geometry is flat. The most obvious 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 3manifolds, of which 6 are orientable and 4 are nonorientable. These are the Bieberbach manifolds. The most familiar is the aforementioned 3torus 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.A flat universe can have zero total energy.Universe with positive curvature
File:universecolor.jpgthumbrightUniverse in an expanding sphere. The galaxies farthest away are moving fastest and hence experience length contraction and so become smaller to an observer in the centre.]]A positively curved universe is described by elliptic geometry, and can be thought of as a threedimensional hypersphere, or some other spherical 3manifold (such as the PoincarÃ© dodecahedral space), all of which are quotients of the 3sphere.PoincarÃ© dodecahedral space is a positively curved space, colloquially described as "soccerballshaped", as it is the quotient of the 3sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by JeanPierre Luminet and colleagues in 2003JOURNAL, Luminet, JeanPierre, JeanPierre Luminet, Jeff, Weeks, Alain, Riazuelo, Roland, Lehoucq, JeanPhillipe, Uzan, Dodecahedral space topology as an explanation for weak wideangle temperature correlations in the cosmic microwave background, 425, 6958, 593â€“5Nature (journal)>Nature, 20031009, astroph/0310253, 10.1038/nature01944,  bibcode= 2003Natur.425..593L, "Is the universe a dodecahedron?", article at PhysicsWeb. and an optimal orientation on the sky for the model was estimated in 2008.JOURNAL, Roukema, Boudewijn  author3=Agnieszka Szaniewska, Nicolas E. Gaudin, A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data, Astronomy and Astrophysics, 482  pages= 747, 2008, 0801.0006, 10.1051/00046361:20078777, 2008A&A...482..747L, Universe with negative curvatureA hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a threedimensional analog of an infinitely extended saddle shape. There are a great variety of hyperbolic 3manifolds, 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 threedimensional 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 "funnelshaped".JOURNAL, Aurich, Ralf, Lustig, S., Steiner, F., Then, H., Hyperbolic Universes with a Horned Topology and the CMB Anisotropy, Classical and Quantum Gravity, 21, 21, 4901â€“4926, 2004, 10.1088/02649381/21/21/010, 2004CQGra..21.4901A, astroph/0403597,Curvature: open or closedWhen cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive. 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 Friedmannâ€“LemaÃ®treâ€“Robertsonâ€“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.Milne model ("spherical" expanding)If one applies Minkowski spacebased special relativity to expansion of the universe, without resorting to the concept of a curved spacetime, then one obtains the Milne model. Any spatial section of the universe of a constant age (the proper time elapsed from the Big Bang) will have a negative curvature; this is merely a pseudoEuclidean geometric fact analogous to one that concentric spheres in the flat Euclidean space are nevertheless curved.Spatial geometry of this model is an unbounded hyperbolic space.The entire universe is contained within a light cone, namely the future cone of the Big Bang. For any given moment {{matht > 0}} of coordinate time (assuming the Big Bang has {{matht {{=}} 0}}), the entire universe is bounded by a sphere of radius exactly {{mathcâ€‰t}}.The apparent paradox of an infinite universe contained within a sphere is explained with length contraction: the galaxies farther away, which are travelling away from the observer the fastest, will appear thinner.This model is essentially a degenerate FLRW for {{mathÎ© {{=}} 0}}. It is incompatible with observations that definitely rule out such a large negative spatial curvature. However, as a background in which gravitational fields (or gravitons) can operate, due to diffeomorphism invariance, the space on the macroscopic scale, is equivalent to any other (open) solution of Einstein's field equations.See also{{Div col}}
References{{Reflist}}External links

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