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2014-01-15T23:24:30Z

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A '''de Sitter universe''' is a [[physical cosmology|cosmological]] solution to [[Albert Einstein|Einstein]]'s field equations of [[General Relativity]] which is named after [[Willem de Sitter]]. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the [[cosmological constant]], thought to correspond to [[dark energy]] in our universe or the [[inflaton field]] in the [[early universe]]. According to the models of [[cosmic inflation|inflation]] and current observations of the [[accelerating universe]], the [[Lambda-CDM|concordance models of physical cosmology]] are converging on a consistent model where our universe was best described as a de Sitter universe at about a time <math>t = 10^{-33}</math> seconds after the fiducial [[Big Bang]] [[Gravitational singularity|singularity]], and far into the [[Ultimate fate of the universe|future]].

==Mathematical expression==

A de Sitter universe has no ordinary matter content but with a positive [[cosmological constant]] (<math>\Lambda</math>) which sets the expansion rate, <math>H</math>. A larger cosmological constant leads to a larger expansion rate:
<center><math>H \propto \sqrt{\Lambda}</math>,</center>
where the constants of proportionality depend on conventions.

It is common to describe a patch of this solution as an expanding universe of the [[Friedmann-Robertson-Walker metric|FLRW]] form where the scale factor is given by<ref>{{cite book|last=Adler|first=Ronald|title=Introduction to General Relativity|year=1965|publisher=McGraw-Hill|location=NY|pages=468|coauthors=Maurice Bazin;Menahem Schiffer}}</ref>
<center><math>a(t) = e^{Ht}</math>,</center>
where the constant <math>H</math> is the Hubble expansion rate and <math>t</math> is time. As in all FLRW spaces, <math>a(t)</math>, the [[scale factor (Universe)|scale factor]], describes the [[metric expansion of space|expansion of physical spatial distances]].

Unique to universes described by the FLRW metric, a de Sitter universe has a [[Hubble Law]] which is not only consistent through all space, but also through all time (since the [[deceleration parameter]] is equal to <math>q=-1</math>), thus satisfying the [[perfect cosmological principle]] that assumes isotropy and homogeneity throughout space and time. As a class of models with different values of the Hubble constant, the [[static universe]] that Einstein developed, and for which he invented the cosmological constant, can be considered a special case of the de Sitter universe where the expansion is [[fine tuning|finely tuned]] to just cancel out the collapse associated with the [[positive curvature]] associated with a non-zero [[matter density]]. There are ways to cast de Sitter space with static coordinates (see [[de Sitter space]]), so unlike other FLRW models, de Sitter space can be thought of as a static solution to [[Einstein's equations]] even though the [[geodesic]]s followed by observers necessarily diverge in the normal way expected from the expansion of physical spatial dimensions. As a model for the universe, de Sitter's solution was not considered viable for the [[observable universe|observed universe]] until models for [[cosmic inflation|inflation]] and [[dark energy]] were developed. Before then, it was assumed that the [[Big Bang]] implied only an acceptance of the weaker [[cosmological principle]] which holds isotropy true only for spatial extents but not temporal extents.<ref>{{cite book|last=Dodelson|first=Scott|title=Modern Cosmology|year=2003|publisher=Academic Press|location=San Diego|isbn=978-0-12-219141-1|edition=4. [print.].}}</ref>

==Potential for the Universe==
{{Unreferenced section|date=April 2011}}
{{Weasel|section|{{subst:April 2011}}|date=April 2011}}

Because our Universe entered the [[Dark Energy Dominated Era]] a few billion years ago, our universe is probably{{why|date=April 2013}} approaching a de Sitter universe in the infinite future. If the current [[cosmic acceleration|acceleration]] of our universe is due to a cosmological constant then as the universe continues to expand all of the matter and radiation will be diluted. Eventually there will be almost nothing left but the cosmological constant, and our universe will have become a de Sitter universe. This is called the "Big Empty".

==Relative expansion==
{{Unreferenced section|date=April 2011}}
{{Weasel|section|{{subst:April 2011}}|date=April 2011}}

The exponential expansion of the scale factor means that the physical distance between any two non-accelerating observers will eventually be growing faster than the [[speed of light]]. At this point those two observers will no longer be able to make contact. Therefore any observer in a de Sitter universe would see [[event horizon]]s beyond which that observer can never see nor learn any information. If our universe is approaching a de Sitter universe then eventually we will not be able to observe any [[galaxy|galaxies]] other than our own [[Milky Way]] (and any others in the gravitationally bound [[Local Group]], assuming they were to somehow survive to that time without merging).

==Modelling cosmic inflation==

Another application of de Sitter space is in the [[early universe]] during [[cosmic inflation]]. Many inflationary models are approximately de Sitter space and can be modelled by giving the Hubble parameter a mild time dependence. For simplicity, some calculations involving inflation in the early universe can be performed in de Sitter space rather than a more realistic inflationary universe. By using the de Sitter universe instead, where the expansion is truly exponential, there are many simplifications.

==See also==
* [[Cosmic inflation]]
* [[De Sitter space]] for more mathematical properties
* [[Deceleration parameter]]
* [[Causal patch]]

==References==
{{reflist}}


[[Category:Physical cosmology]]
[[Category:Exact solutions in general relativity]]
[[Category:Cosmic inflation]]

[[ru:Модель де Ситтера]]

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