Universe

=== Model of the universe based on general relativity ===
{{Main|Solutions of the Einstein field equations}}
{{See also|Big Bang|Ultimate fate of the universe}}
[[General relativity]] is the [[Differential geometry|geometric]] [[Theoretical physics|theory]] of [[gravitation]] published by [[Albert Einstein]] in 1915 and the current description of gravitation in [[modern physics]]. It is the basis of current [[Physical cosmology|cosmological]] models of the universe. General relativity generalizes [[special relativity]] and [[Newton's law of universal gravitation]], providing a unified description of gravity as a geometric property of [[space]] and [[Time in physics|time]], or spacetime. In particular, the [[curvature]] of spacetime is directly related to the [[energy]] and [[momentum]] of whatever [[matter]] and [[radiation]] are present.<ref name="zeilik_cosmology" />

The relation is specified by the [[Einstein field equations]], a system of [[partial differential equation]]s. In general relativity, the distribution of matter and energy determines the geometry of spacetime, which in turn describes the [[acceleration]] of matter. Therefore, solutions of the Einstein field equations describe the evolution of the universe. Combined with measurements of the amount, type, and distribution of matter in the universe, the equations of general relativity describe the evolution of the universe over time.<ref name="zeilik_cosmology" />

With the assumption of the [[cosmological principle]] that the universe is homogeneous and isotropic everywhere, a specific solution of the field equations that describes the universe is the [[metric (general relativity)|metric tensor]] called the [[Friedmann–Lemaître–Robertson–Walker metric]],
:<math>
ds^2 = -c^{2} dt^2 +
R(t)^2 \left( \frac{dr^2}{1-k r^2} + r^2 d\theta^2 + r^2 \sin^2 \theta \, d\phi^2 \right)
</math>
where (''r'', θ, φ) correspond to a [[spherical coordinate system]]. This metric has only two undetermined parameters. An overall [[dimensionless]] length [[scale factor (cosmology)|scale factor]] ''R'' describes the size scale of the universe as a function of time (an increase in ''R'' is the [[expansion of the universe]]),<ref>{{harvtxt|Raine|Thomas|2001|p=12}}</ref> and a curvature index ''k'' describes the geometry. The index ''k'' is defined so that it can take only one of three values: 0, corresponding to flat [[Euclidean geometry]]; 1, corresponding to a space of positive [[curvature]]; or −1, corresponding to a space of positive or negative curvature.<ref name="RaineThomas66" /> The value of ''R'' as a function of time ''t'' depends upon ''k'' and the [[cosmological constant]] ''Λ''.<ref name="zeilik_cosmology">{{cite book |title=Introductory Astronomy & Astrophysics |last1=Zeilik |first1=Michael |last2=Gregory |first2=Stephen A. |date=1998 |edition=4th |publisher=Saunders College Publishing |isbn=978-0-03-006228-5 |section=25-2}}</ref> The cosmological constant represents the energy density of the vacuum of space and could be related to dark energy.<ref name="peebles" /> The equation describing how ''R'' varies with time is known as the [[Friedmann equation]] after its inventor, [[Alexander Friedmann]].<ref>{{cite journal |author=Friedmann |first=A. |author-link=Alexander Friedmann |date=1922 |title=Über die Krümmung des Raumes |url=http://publikationen.ub.uni-frankfurt.de/files/16735/E001554876.pdf |url-status=live |journal=Zeitschrift für Physik |volume=10 |issue=1 |pages=377–386 |bibcode=1922ZPhy...10..377F |doi=10.1007/BF01332580 |s2cid=125190902 |archive-url=http://arquivo.pt/wayback/20160515100312/http%3A//publikationen.ub.uni%2Dfrankfurt.de/files/16735/E001554876.pdf |archive-date=May 15, 2016 |access-date=August 13, 2015}}</ref>

The solutions for ''R(t)'' depend on ''k'' and ''Λ'', but some qualitative features of such solutions are general. First and most importantly, the length scale ''R'' of the universe can remain constant ''only'' if the universe is perfectly isotropic with positive curvature (''k''=1) and has one precise value of density everywhere, as first noted by [[Albert Einstein]].<ref name="zeilik_cosmology" /> However, this equilibrium is unstable: if the density were slightly different from the needed value, at any place, the difference would be amplified over time.

Second, all solutions suggest that there was a [[gravitational singularity]] in the past, when ''R'' went to zero and matter and energy were infinitely dense. It may seem that this conclusion is uncertain because it is based on the questionable assumptions of perfect homogeneity and isotropy (the cosmological principle) and that only the gravitational interaction is significant. However, the [[Penrose–Hawking singularity theorems]] show that a singularity should exist for very general conditions. Hence, according to Einstein's field equations, ''R'' grew rapidly from an unimaginably hot, dense state that existed immediately following this singularity (when ''R'' had a small, finite value); this is the essence of the [[Big Bang]] model of the universe. Understanding the singularity of the Big Bang likely requires a [[quantum theory of gravity]], which has not yet been formulated.<ref>{{harvtxt|Raine|Thomas|2001|pp=122–123}}</ref>

Third, the curvature index ''k'' determines the sign of the curvature of constant-time spatial surfaces<ref name="RaineThomas66">{{harvtxt|Raine|Thomas|2001|p=66}}</ref> averaged over sufficiently large length scales (greater than about a billion [[light-year]]s). If ''k''=1, the curvature is positive and the universe has a finite volume.<ref name="RaineThomas70" /> A universe with positive curvature is often visualized as a [[3-sphere|three-dimensional sphere]] embedded in a four-dimensional space. Conversely, if ''k'' is zero or negative, the universe has an infinite volume.<ref name="RaineThomas70">{{harvtxt|Raine|Thomas|2001|p=70}}</ref> It may seem counter-intuitive that an infinite and yet infinitely dense universe could be created in a single instant when ''R'' = 0, but exactly that is predicted mathematically when ''k'' is nonpositive and the [[cosmological principle]] is satisfied. By analogy, an infinite plane has zero curvature but infinite area, whereas an infinite cylinder is finite in one direction and a [[torus]] is finite in both. A toroidal universe could behave like a normal universe with [[periodic boundary conditions]].

The [[ultimate fate of the universe]] is still unknown because it depends critically on the curvature index ''k'' and the cosmological constant ''Λ''. If the universe were sufficiently dense, ''k'' would equal +1, meaning that its average curvature throughout is positive and the universe will eventually recollapse in a [[Big Crunch]],<ref>{{harvtxt|Raine|Thomas|2001|p=84}}</ref> possibly starting a new universe in a [[Big Bounce]]. Conversely, if the universe were insufficiently dense, ''k'' would equal 0 or −1 and the universe would expand forever, cooling off and eventually reaching the [[Future of an expanding universe|Big Freeze]] and the [[heat death of the universe]].<ref name="zeilik_cosmology" /> Modern data suggests that the [[accelerated expansion|expansion of the universe is accelerating]]; if this acceleration is sufficiently rapid, the universe may eventually reach a [[Big Rip]]. Observationally, the universe appears to be flat (''k'' = 0), with an overall density that is very close to the critical value between recollapse and eternal expansion.<ref>{{harvtxt|Raine|Thomas|2001|pp=88, 110–113}}</ref>