Editing Shape of the universe (section)

== Global universal structure ==
Global structure covers the [[geometry]] and 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. The universe is often taken to be a [[geodesic manifold]], free of [[topological defect#Cosmological defects|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 3-space and [[hyperbolic geometry|hyperbolic 3-space]] 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 [[infinity|infinite]] or finite in extent,
* whether the geometry of the global universe is flat, positively curved, or negatively curved, and,
* whether the topology is simply connected (for example, like a [[sphere]]) or else multiply connected (for example, like a [[torus]]).<ref>{{Cite book |last=Davies |first=Paul |title=Space and Time in the Modern Universe |url=https://books.google.com/books?id=SZI5AAAAIAAJ |date=1977 |publisher=[[Cambridge University Press]] |isbn=978-0-521-29151-4 |location=Cambridge}}</ref>

=== Infinite or finite ===
One of the 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 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 [[bounded metric space|boundedness]]. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance {{mvar|d}}, there are points that are of a distance at least {{mvar|d}} apart. A finite universe is a bounded metric space, where there is some distance {{mvar|d}} such that all points are within distance {{mvar|d}} of each other. The smallest such {{mvar|d}} is called the diameter of the universe, in which case the universe has a well-defined "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 [[disk (mathematics)|disc]], have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is 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 [[3-sphere]] and [[3-torus]], that have no edges. Mathematically, these spaces are referred to as being [[compact space|compact]] without boundary. The term compact means that it is finite in extent ("bounded") and [[complete metric space|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 3-sphere and 3-torus are both closed manifolds.

==== Observational methods ====
In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed<ref name="Luminet1995" /> and applied to cosmological observations.<ref name="Nat03" /><ref name="RBSG08" />

In the 2000s and 2010s, it was shown that, since the universe is inhomogeneous as shown in the [[observable universe#Large-scale structure|cosmic web of large-scale structure]], acceleration effects measured on local scales in the patterns of the movements of galaxies should, in principle, reveal the global topology of the universe.<ref name="RBBSJ2007">{{cite Q|Q68598777}}</ref><ref name="RR09">{{cite Q|Q68676519}}</ref><ref name="ORB12">{{cite Q|Q96692451}}</ref>

=== Curvature ===
The curvature of the universe places constraints on the topology. If the spatial geometry is [[spherical 3-manifold|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.<ref name="Luminet1995">{{harvnb|Lachièze-Rey|Luminet|1995}}</ref> Many textbooks erroneously state that a flat or hyperbolic universe implies an infinite universe; however, the correct statement is that a flat universe that is also [[simply connected]] implies an infinite universe.<ref name="Luminet1995" /> For example, Euclidean space is flat, simply connected, and infinite, but there are [[torus#Flat torus|tori]] that are flat, multiply connected, finite, and compact (see [[flat torus]]).

In general, [[Riemannian geometry#Local to global theorems|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 [[geometrization conjecture|Thurston geometries]].

The latest research shows that even the most powerful future experiments (like the [[Square Kilometre Array|SKA]]) will not be able to distinguish between a flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10<sup>−4</sup>. If the true value of the cosmological curvature parameter is larger than 10<sup>−3</sup> we will be able to distinguish between these three models even now.<ref>{{Cite journal |last1=Vardanyan |first1=Mihran |last2=Trotta |first2=Roberto |last3=Silk |first3=Joseph |date=21 July 2009 |title=How flat can you get? A model comparison perspective on the curvature of the Universe |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=397 |issue=1 |pages=431–444 |arxiv=0901.3354 |bibcode=2009MNRAS.397..431V |doi=10.1111/j.1365-2966.2009.14938.x |s2cid=15995519 |doi-access=free}}</ref>

Final results of the ''Planck'' mission, released in 2018, show the cosmological curvature parameter, {{nowrap|1=1 − Ω = Ω<sub>''K''</sub> = −''Kc''<sup>2</sup>/''a''<sup>2</sup>''H''<sup>2</sup>}}, to be {{val|0.0007|0.0019}}, consistent with a flat universe.<ref>{{Cite journal |last1=Aghanim |first1=N. |author-link1=Nabila Aghanim |last2=Akrami |first2=Y. |last3=Ashdown |first3=M. |last4=Aumont |first4=J. |last5=Baccigalupi |first5=C. |last6=Ballardini |first6=M. |last7=Banday |first7=A. J. |last8=Barreiro |first8=R. B. |last9=Bartolo |first9=N. |last10=Basak |first10=S. |last11=Battye |first11=R. |last12=Benabed |first12=K. |last13=Bernard |first13=J.-P. |last14=Bersanelli |first14=M. |last15=Benoit-Levy |first15=A. |display-authors=3 |date=September 2020 |title=Planck 2018 results: VI. Cosmological parameters |journal=[[Astronomy & Astrophysics]] |volume=641 |pages=A6 |arxiv=1807.06209 |bibcode=2020A&A...641A...6P |doi=10.1051/0004-6361/201833910 |issn=0004-6361 |s2cid=119335614 |collaboration=[[Planck Collaboration]] |last16=Bernard |first16=J. P. |last17=Bersanelli |first17=M. |last18=Bielewicz |first18=P. |last19=Bonaldi |first19=A. |last20=Bonavera |first20=L. |last21=Bond |first21=J. R. |last22=Borrill |first22=J. |last23=Bouchet |first23=F. R. |last24=Boulanger |first24=F. |last25=Bucher |first25=M. |last26=Burigana |first26=C. |last27=Butler |first27=R. C. |last28=Calabrese |first28=E. |last29=Cardoso |first29=J. F.}}</ref> (i.e. positive curvature: {{nowrap|1=''K'' = +1}}, {{nowrap|Ω<sub>''K''</sub> < 0}}, {{nowrap|Ω > 1}}, negative curvature: {{nowrap|1=''K'' = −1}}, {{nowrap|Ω<sub>''K''</sub> > 0}}, {{nowrap|Ω < 1}}, zero curvature: {{nowrap|1=''K'' = 0}}, {{nowrap|1=Ω<sub>''K''</sub> = 0}}, {{nowrap|1=Ω = 1}}).

==== Universe with zero curvature <span class="anchor" id="Flat universe"></span> ====
In a universe with zero curvature, the local geometry is [[geometrization conjecture#Euclidean geometry E3|flat]]. The most familiar such 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 3-manifolds, of which 6 are orientable and 4 are non-orientable. These are the [[flat manifold|Bieberbach manifolds]]. The most familiar is the aforementioned [[three-torus model of the universe|3-torus universe]].

In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion [[asymptote|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 in the sense that space will continue expanding forever.

A flat universe can have [[zero-energy universe|zero total energy]].<ref>{{cite AV media |date=2009 |title=A Universe From Nothing lecture by Lawrence Krauss at AAI |url=https://www.youtube.com/watch?v=7ImvlS8PLIo |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211215/7ImvlS8PLIo |archive-date=2021-12-15 |access-date=17 October 2011 |via=[[YouTube]]}}{{cbignore}}</ref>

==== Universe with positive curvature ====
A positively curved universe is described by [[elliptic geometry]], and can be thought of as a three-dimensional [[hypersphere]], or some other [[spherical 3-manifold]] (such as the [[Poincaré dodecahedral space]]), all of which are [[Quotient space (topology)|quotient]]s of the 3-sphere.

[[Homology sphere#Cosmology|Poincaré dodecahedral space]] is a positively curved space, colloquially described as "soccerball-shaped", as it is the [[Quotient space (topology)|quotient]] of the 3-sphere by the [[binary icosahedral group]], which is very close to [[icosahedral symmetry]], the symmetry of a soccer ball. This was proposed by [[Jean-Pierre Luminet]] and colleagues in 2003<ref name="Nat03">{{Cite journal |last1=Luminet |first1=Jean-Pierre |author-link1=Jean-Pierre Luminet |last2=Weeks |first2=Jeffrey R. |last3=Riazuelo |first3=Alain |last4=Lehoucq |first4=Roland |last5=Uzan |first5=Jean-Philippe |date=October 2003 |title=Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background |journal=[[Nature (journal)|Nature]] |volume=425 |issue=6958 |pages=593–595 |arxiv=astro-ph/0310253 |bibcode=2003Natur.425..593L |doi=10.1038/nature01944 |issn=0028-0836 |pmid=14534579 |s2cid=4380713}}</ref><ref name="physwebLum03">{{Cite web |last=Dumé |first=Isabelle |date=8 October 2003 |title=Is the universe a dodecahedron? |url=https://physicsworld.com/a/is-the-universe-a-dodecahedron/ |website=[[Physics World]] |language=en-GB}}</ref> and an optimal orientation on the sky for the model was estimated in 2008.<ref name="RBSG08">{{Cite journal |last1=Lew |first1=B. |last2=Roukema |first2=B. |last3=Szaniewska |first3=Agnieszka |last4=Gaudin |first4=Nicolas E. |date=May 2008 |title=A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data |journal=[[Astronomy & Astrophysics]] |volume=482 |issue=3 |pages=747–753 |arxiv=0801.0006 |bibcode=2008A&A...482..747L |doi=10.1051/0004-6361:20078777 |issn=0004-6361 |s2cid=1616362}}</ref>

==== Universe with negative curvature ====
A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There are a great variety of [[hyperbolic 3-manifold]]s, 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 three-dimensional 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 "funnel-shaped".<ref name="Aurich0403597">{{cite journal |last= Aurich |first= Ralf|author2=Lustig, S. |author3=Steiner, F. |author4=Then, H. |title= Hyperbolic Universes with a Horned Topology and the CMB Anisotropy |journal= [[Classical and Quantum Gravity]] |volume= 21 |issue= 21 |pages= 4901–4926 |date= 2004 |doi= 10.1088/0264-9381/21/21/010 |bibcode= 2004CQGra..21.4901A |arxiv= astro-ph/0403597|s2cid= 17619026}}</ref>

==== Curvature: open or closed ====
When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive, respectively. 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 [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–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.