Editing Shape of the universe (section)

== Curvature of the universe ==
{{further|Curvature#Space| Flatness problem}}

The [[curvature of Riemannian manifolds|curvature]] is a quantity describing how the geometry of a space differs locally from 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){{snd}}a drawn triangle's angles add up to 180° and the [[Pythagorean theorem]] holds; such 3-dimensional space is locally modeled by Euclidean space {{math|'''E'''<sup>''3''</sup>}}.
# Positive curvature{{snd}}a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a [[N-sphere|3-sphere]] {{math|'''S'''<sup>''3''</sup>}}.
# Negative curvature{{snd}}a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a [[hyperbolic space]] {{math|'''H'''<sup>''3''</sup>}}.

Curved geometries are in the domain of [[non-Euclidean geometry]]. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a [[saddle]] or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.

[[File:End of universe.jpg|thumb|275px|The local geometry of the universe is determined by whether the [[density parameter#Density parameter|density parameter {{math|Ω}}]] is greater than, less than, or equal to 1. From top to bottom: a [[spherical geometry|spherical universe]] with {{math|Ω > 1}}, a [[hyperbolic geometry|hyperbolic universe]] with {{math|Ω < 1}}, and a [[Euclidean geometry|flat universe]] with {{math|Ω {{=}} 1}}. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of (local) space.]][[File:Spacetime-diagram-flat-universe-proper-coordinates.png|thumb|275px|[[Comoving and proper distances|Proper distance]] spacetime diagram of our flat [[Lambda-CDM_model|ΛCDM]] universe. [[Particle horizon]]: green, [[Hubble radius]]: blue, [[Event horizon]]: purple, [[Light cone]]: orange.]][[File:Hyperbolic.universe.proper.coordinates.png|thumb|275px|Hyperbolic universe with the same radiation and matter density parameters as ours, but with negative curvature instead of dark energy (Ω<sub>Λ</sub>→Ω<sub>k</sub>).]][[File:Big-crunch_spacetime-diagram_matter-dominated_proper-distances.png|thumb|275px|Closed universe without dark energy and with overcritical matter density, which leads to a [[Big Crunch]]. Neither the hyperbolic nor the closed examples have an Event horizon (here the purple curve is the cosmic Antipode).]]

[[General relativity]] explains that mass and energy bend the curvature of spacetime and is used to determine what curvature the universe has by using a value called the [[density parameter]], represented with Omega ({{math|Ω}}). The density parameter is the average density of the universe divided by the critical energy density, that is, the mass energy needed for a universe to be flat. Put another way,
* If {{math|Ω {{=}} 1}}, the universe is flat.
* If {{math|Ω > 1}}, there is positive curvature.
* If {{math|Ω < 1}}, there is negative curvature.

Scientists could experimentally calculate {{math|Ω}} to determine the curvature two ways. One is to count all the [[mass–energy equivalence|mass–energy]] in the universe and take its average density, then divide that average by the critical energy density. Data from the [[Wilkinson Microwave Anisotropy Probe]] (WMAP) as well as the [[Planck (spacecraft)|Planck spacecraft]] give values for the three constituents of all the mass–energy in the universe – normal mass ([[baryonic matter]] and [[dark matter]]), relativistic particles (predominantly [[photon]]s and [[neutrino]]s), and [[dark energy]] or the [[cosmological constant]]:<ref>{{cite web|title= Density Parameter, Omega|url= http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html|website= hyperphysics.phy-astr.gsu.edu|access-date= 2015-06-01}}</ref><ref>{{Cite journal |last1=Ade |first1=P. A. R. |last2=Aghanim |first2=N. |author-link2=Nabila Aghanim |last3=Armitage-Caplan |first3=C. |last4=Arnaud |first4=M. |last5=Ashdown |first5=M. |last6=Atrio-Barandela |first6=F. |last7=Aumont |first7=J. |last8=Baccigalupi |first8=C. |last9=Banday |first9=A. J. |last10=Barreiro |first10=R. B. |last11=Bartlett |first11=J. G. |last12=Battaner |first12=E. |last13=Benabed |first13=K. |last14=Benoît |first14=A. |last15=Bernard |first15=J.-P. |display-authors=3 |date=November 2014 |title=Planck 2013 results. XVI. Cosmological parameters |journal=[[Astronomy & Astrophysics]] |volume=571 |pages=A16 |arxiv=1303.5076 |bibcode=2014A&A...571A..16P |doi=10.1051/0004-6361/201321591 |issn=0004-6361 |s2cid=118349591 |collaboration=[[Planck Collaboration]] |last16=Bersanelli |first16=M. |last17=Bielewicz |first17=P. |last18=Bobin |first18=J. |last19=Bock |first19=J. J. |last20=Bonaldi |first20=A. |last21=Bond |first21=J. R. |last22=Borrill |first22=J. |last23=Bouchet |first23=F. R. |last24=Bridges |first24=M. |last25=Bucher |first25=M. |last26=Burigana |first26=C. |last27=Butler |first27=R. C. |last28=Calabrese |first28=E. |last29=Cappellini |first29=B.}}</ref>
: Ω<sub>mass</sub> ≈ {{val|0.315|0.018}}
: Ω<sub>relativistic</sub> ≈ {{val|9.24|e=−5}}
: Ω<sub>Λ</sub> ≈ {{val|0.6817|0.0018}}
: Ω<sub>total</sub> = Ω<sub>mass</sub> + Ω<sub>relativistic</sub> + Ω<sub>Λ</sub> = {{val|1.00|0.02}}
The actual value for critical density value is measured as ''ρ''<sub>critical</sub> = {{val|9.47|e=−27|u=kg.m-3}}. From these values, within experimental error, the universe seems to be spatially flat.

Another way to measure Ω is to do so geometrically by measuring an angle across the observable universe. This can be done by using the [[CMB]] and measuring the power spectrum and temperature [[anisotropy]]. For instance, one can imagine finding a gas cloud that is not in thermal equilibrium due to being so large that light speed cannot propagate the thermal information. Knowing this propagation speed, we then know the size of the gas cloud as well as the distance to the gas cloud, we then have two sides of a triangle and can then determine the angles. Using a method similar to this, the [[BOOMERanG experiment]] has determined that the sum of the angles to 180° within experimental error, corresponding to {{nowrap|Ω<sub>total</sub> ≈ {{val|1.00|0.12}}}}.<ref>{{Cite journal |last1=de Bernardis |first1=P. |last2=Ade |first2=P. A. R. |last3=Bock |first3=J. J. |last4=Bond |first4=J. R. |last5=Borrill |first5=J. |last6=Boscaleri |first6=A. |last7=Coble |first7=K. |last8=Crill |first8=B. P. |last9=De Gasperis |first9=G. |last10=Farese |first10=P. C. |last11=Ferreira |first11=P. G. |last12=Ganga |first12=K. |last13=Giacometti |first13=M. |last14=Hivon |first14=E. |last15=Hristov |first15=V. V. |display-authors=3 |date=April 2000 |title=A flat Universe from high-resolution maps of the cosmic microwave background radiation |journal=[[Nature (journal)|Nature]] |language=en |volume=404 |issue=6781 |pages=955–959 |arxiv=astro-ph/0004404 |bibcode=2000Natur.404..955D |doi=10.1038/35010035 |issn=0028-0836 |pmid=10801117 |s2cid=4412370 |last16=Iacoangeli |first16=A. |last17=Jaffe |first17=A. H. |last18=Lange |first18=A. E. |last19=Martinis |first19=L. |last20=Masi |first20=S. |last21=Mason |first21=P. V. |last22=Mauskopf |first22=P. D. |last23=Melchiorri |first23=A. |last24=Miglio |first24=L. |last25=Montroy |first25=T. |last26=Netterfield |first26=C. B. |last27=Pascale |first27=E. |last28=Piacentini |first28=F. |last29=Pogosyan |first29=D. |last30=Prunet |first30=S.}}</ref>

These and other astronomical measurements constrain the spatial curvature to be very close to zero, although they do not constrain its sign. This means that although the local geometries of spacetime are generated by the theory of relativity based on [[spacetime interval]]s, we can approximate ''3-space'' by the familiar [[Euclidean geometry]].

The [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker (FLRW) model]] using [[Friedmann equations]] is commonly used to model the universe. The FLRW model provides a curvature of the universe based on the mathematics of [[fluid dynamics]], that is, modeling the matter within the universe as a perfect fluid. 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. Another way of saying this is that, if all forms of [[dark energy]] are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed (rather than the distortions caused by 'dense' objects such as galaxies). This assumption is justified by the observations that, while the universe is "weakly" [[homogeneity (physics)|inhomogeneous]] and [[anisotropic]] (see the [[large-scale structure of the cosmos]]), it is on average homogeneous and [[isotropic]] when analyzed at a sufficiently large spatial scale.