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<p><b>New page</b></p><div>{{Cosmology}}<br />
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The '''shape of the universe''' is a subject of investigation within [[cosmology]]. Cosmologists and astronomers describe the [[geometry]] of the [[Universe#Size of Universe and observable universe|observable universe]] which includes both [[#Local Geometry|local geometry]] and [[#Global Geometry|global geometry]] of the whole universe, which is in practice loosely termed [[topology]], even though strictly speaking it goes beyond topology.<br />
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The shape of the universe is not concerned with a geometry near to a dense [[mass]]. Rather, the geometries under investigation describe a universe where the average density of mass is evenly distributed. Notwithstanding that the universe is "weakly" [[homogeneity (physics)|inhomogeneous]] and [[anisotropy|anisotropic]] in the [[large-scale structure of the cosmos]], both astronomical and cosmological measurements determine the observable universe to be, on average, homogeneous, [[isotropic]], and expanding or [[accelerating universe|accelerating]]. Although the geometries within the observable universe are generated by the [[theory of relativity]] based on [[spacetime#Distance in spacetime|spacetime intervals]], local geometry turns out to be describable by the familiar geometries of three spatial dimensions. In turn, the local geometry, together with direct observation and further measurement, is used to constrain the possibilities for the global geometry to be a topology of three spatial dimensions.<br />
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One of the investigations within the study of local geometry is:<br />
*to evaluate the [[curvature]] of the geometry, currently constrained to be constant and very close to zero - corresponding to a "nearly" flat [[Curvature#Curvature of space|curvature of space]].<br />
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Two investigations within the study of global geometry are:<br />
*whether the universe is [[infinite]] in extent or is a [[compact space]]<br />
*whether the universe has a [[Simply connected space|simply or non-simply connected]] topology<br />
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== Local geometry ==<br />
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The '''local geometry''' is the geometry describing the observable universe. <br />
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Many astronomical observations show the observable universe to be homogeneous and isotropic and infer it to be accelerating.<br />
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Both astronomical measurements of [[supernova]] events and cosmological measurements of the structures in the ambient [[cosmic background radiation]], are well approximated by a [[Friedmann-LemaĆ®tre-Robertson-Walker]] model. The FLRW model describes a homogeneous, isotropic and accelerating universe in the context of [[General Relativity]]. An attraction of the FLRW is that the associated [[Friedman equations]] do yield a geometry of the universe as a function of well understood mathematics for [[fluid dynamics]]. Although stars and structures of mass can be introduced into an "almost FLRW" model, a strictly FLRW model is used to approximate the local geometry of the observable universe. <br />
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=== Curvature (local geometry) ===<br />
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The homogeneous and isotropic universe allows for a geometry with a constant curvature.<br />
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One aspect of local geometry to emerge from General Relativity and the FLRW model is that the [[Big Crunch|critical density]] is related to the curvature of the geometry of space. Expressing the ratio of ''average energy density of the universe'' over ''critical energy density'' as &Omega;, the curvature is given as:<br />
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:<math><br />
\mbox{curvature} =<br />
\begin{cases}<br />
\mbox{zero}, & \mbox{if the critical density, } \Omega, = 1<br />
\\ \mbox{positive}, & \mbox{if } \Omega > 1<br />
\\ \mbox{negative}, & \mbox{if } \Omega < 1<br />
\end{cases}<br />
</math><br />
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Astronomical measurements of both mass density of the universe and spacetime intervals using supernova events constrain the curvature to be zero or positive but near to zero. Cosmological measurements infer the curvature as close to zero - either zero or just positive or negative.<br />
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=== Feasible local geometries ===<br />
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The many feasible geometries of a constant curvature are categorized into three, according to the sign of the curvature:<br />
:<math> \mbox{local geometry} =<br />
\begin{cases}<br />
\mbox{flat}, & \mbox{if the curvature is exactly zero}<br />
\\ \mbox{spherical}, & \mbox{if the curvature is positive}<br />
\\ \mbox{hyperbolic}, & \mbox{if the curvature is negative}<br />
\end{cases}<br />
</math><br />
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An observable universe that is "nearly flat", from the measured curvature, allows for a simplification whereby the dynamic, accelerating dimension of the geometry can be separated and omitted by invoking [[comoving coordinates]]. Comoving coordinates, from a single [[frame of reference]], leave a static geometry of three spatial dimensions.<br />
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Of eight feasible geometries given by the [[geometrization conjecture]], the shape of the observable universe, or the local geometry, is in all likelihood described by one of the three "primitive" geometries:<br />
:*3-dimensional [[Euclidean geometry]], generally annotated as ''E''<sup>''3''</sup><br />
:*3-dimensional [[spherical geometry]] with a small curvature, often annotated as ''S''<sup>''3''</sup><br />
:*3-dimensional [[hyperbolic geometry]] with a small curvature, often annotated as ''H''<sup>''3''</sup><br />
Even if the universe is not exactly flat, the curvature is close enough to zero to place the [[radius]] beyond the horizon of the observable universe.<br />
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== Global geometry ==<br />
'''Global geometry''' covers the geometry, in particular the topology, of the whole universe - 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. For a flat local geometry, the scale of any properties of the topology is arbitrary and may or may not be directly detectable. For spherical and hyperbolic geometries, the probability of detection of the topology by direct observation depends on the curvature. Using the radius of the curvature as a scale, a small curvature of the local geometry, with a corresponding scale greater than the observable horizon, makes the topology difficult to detect. A spherical geometry may well have a radius that can be detected. In a hyperbolic geometry the radius scale is unlikely to be within the observable horizon.<br />
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=== Compactness of the global shape ===<br />
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A '''compact space''' is a general topological definition that encompasses the more applicable notion of a [[Metric space#Boundedness and compactness|bounded metric space]]. If the 3-[[manifold]] of a spatial section of the universe is compact then, as on a sphere, distances extended far enough "in the same direction" will return to the start point and the space will have a definable "volume" or "scale". If the geometry of the universe is not compact, then it is infinite in extent with infinite paths of constant direction that, generally do not return and the space has no definable volume, such as the [[Euclidean plane]].<br />
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If the local geometry is spherical, the topology is compact. Otherwise, a flat or a hyperbolic local geometry, the topology can be either compact or infinite. Even given that the local geometry is nearly flat, the universe may still be bounded - as in a three dimensional version of a [[Cylinder (geometry)|cylinder]] that is also a flat geometry.<br />
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One of the endeavors in the analysis of data from the [[WMAP|Wilkinson Microwave Anisotropy Probe (WMAP)]] is to detect multiple "back-to-back" images of the distant universe in the cosmic background radiation. Assuming the light has enough time since its origin to travel around a bounded universe, multiple images may be observed. While current results and analysis do not rule out a bounded topology, if the universe is bounded then the curvature is extremely small.<br />
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=== Connectedness of the global manifold ===<br />
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A '''simply connected space''' is all of one piece, such as a sphere. A "multiply connected", or more strictly a '''non-simply connected''' space has circle-shaped "holes" or "handles". If the global geometry is non-simply connected then at some points in the geometry, near a the junction of a "handle", paths of light may reach an observer by two routes, a "closed path" - a path through the main body and a path via the handle.<br />
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The probability of detecting a multiply connected topology depends not only on the scale of the topology but also the degree of complication in the topology. The observer may have a privileged position near a closed path. In a hyperbolic local geometry, a non-simply connected space is unlikely to be detected unless the observer is near a closed path.<br />
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A second endeavor in the analysis of data from the WMAP is to separate any multiple images in the cosmic background radiation from potential "back-to-back" multiple images due to a compact space. Current results eliminate many forms of non-simply connected topologies implying that either we are not near a closed light path or the global geometry is simply connected.<br />
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== See also ==<br />
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[[Theorema Egregium]] The "remarkable theorem" discovered by [[Gauss]] that the inhabitants of a [[line (mathematics)|line]], a '''surface''' or any other [[differentiable manifold]] can [[Differential geometry and topology#Intrinsic versus extrinsic|intrinsically]] measure the curvature of the space that they inhabit.<br />
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[[String Theory#Extra dimensions|Extra dimensions in String Theory]] for 10 or 11 extra space-like dimensions all with a ''compact'' topology.<br />
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==External links==<br />
* [http://arXiv.org/abs/astro-ph/0310253 Oct 2003 - Poincar&eacute; dodecahedral model - WMAP statistics] model predictions for a spherical local geometry <br />
* [http://arXiv.org/abs/astro-ph/0402608 Feb 2004 - Poincar&eacute; dodecahedral model - 11 degree matched circles]<br />
* [http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html Universe is Finite, "Soccer Ball"-Shaped, Study Hints.] Possible wrap-around dodecahedral shape of the universe.<br />
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[[Category:Cosmology]]<br />
[[Category:Differential geometry]]<br />
[[es:Forma del universo]]<br />
[[fr:forme de l'Univers]]<br />
[[pl:Kszta&#322;t Wszech&#347;wiata]]<br />
[[simple:Shape of the universe]]<br />
[[sl:Oblika Vesolja]]</div>Todd