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{{Cosmology}}
In [[physical cosmology]], the '''cosmological constant''' (usually denoted by the Greek capital letter [[lambda]]: Λ) was proposed by [[Albert Einstein]] as a modification of his original theory of [[general relativity]] to achieve a [[Einstein's universe|stationary universe]]. Einstein abandoned the concept after the observation of the [[Hubble redshift]] indicated that the universe might not be stationary, as he had based his theory on the idea that the universe is unchanging.<ref name=Yale>{{Citation|last = Urry|first = Meg|year = 2008|title ="The Mysteries of Dark Energy"|series = Yale Science|publisher = [[Yale University]]|url = http://itunes.yale.edu}}</ref> However, the discovery of [[cosmic acceleration]] in the 1990s has renewed interest in a cosmological constant.

==Equation==
The cosmological constant Λ appears in [[Einstein field equations|Einstein's modified field equation]] in the form of
:<math>R_{\mu \nu} - {\textstyle 1 \over 2}R\,g_{\mu \nu} + \Lambda\,g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu}</math>
where ''[[Ricci tensor|R]]'' and ''[[metric tensor (general relativity)|g]]'' pertain to the structure of [[spacetime]], ''[[stress-energy tensor|T]]'' pertains to matter and energy (thought of as affecting that structure), and ''[[gravitational constant|G]]'' and ''[[speed of light|c]]'' are conversion factors which arise from using traditional units of measurement. When Λ is zero, this reduces to the original field equation of general relativity. When ''T'' is zero, the field equation describes empty space (the [[vacuum]]). Astronomical observations imply that the constant cannot exceed 10<sup>−46</sup>&nbsp;km<sup>−2</sup>.<ref>{{cite journal | author = Christopher S. Kochanek | title = "Is There a Cosmological Constant?" | journal = The Astrophysical Journal | year = 1996 | month = August | day = 1 | volume = 466 | issue = 2 | pages = 638–659 | doi = 10.1086/177538}}</ref>

The cosmological constant has the same effect as an intrinsic [[energy density]] of the vacuum, ''ρ<sub>vac</sub>'' (and an associated [[pressure]]). In this context it is commonly defined with a [[Proportionality (mathematics)|proportionality]] factor of 8<math>\pi</math>: Λ = 8<math>\pi</math>''ρ<sub>vac</sub>'', where modern unit conventions of general relativity are followed (otherwise factors of ''G'' and ''c'' would also appear). It is common to quote values of energy density directly, though still using the name "cosmological constant".

A positive vacuum energy density resulting from a cosmological constant implies a negative pressure, and vice versa. If the energy density is positive, the associated negative pressure will drive an accelerated expansion of empty space. (See [[dark energy]] and [[cosmic inflation]] for details.)

===Omega Lambda===
In lieu of the cosmological constant itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and the  [[critical density]] of the universe. This ratio is usually denoted <math>\Omega_{\Lambda}</math>. In a flat universe <math>\Omega_{\Lambda}</math> corresponds to the fraction of the energy density of the Universe due to the cosmological constant. Note that this definition is tied to the critical density of the present cosmological era: the critical density changes with [[cosmological time]], but the energy density due to the cosmological constant remains unchanged throughout the history of the universe.

===Equation of state===
Another ratio that is used by scientists is the [[Equation of state (cosmology)|equation of state]] which is the ratio of pressure that dark energy puts on the Universe to the energy per unit volume.<ref>{{cite journal |last=Hogan |first=Jenny |authorlink= |coauthors= |year=2007 |month= |title=Welcome to the Dark Side |journal=Nature |volume=448 |issue=7151 |pages=240–245 |doi=10.1038/448240a |url= |accessdate= |quote= }}</ref>

== History ==
Einstein included the cosmological constant as a term in his [[Einstein field equations|field equations]] for [[general relativity]] because he was dissatisfied that otherwise his equations did not allow, apparently, for a [[static universe]]: gravity would cause a universe which was initially at dynamic equilibrium to contract. To counteract this possibility, Einstein added the cosmological constant.<ref name=Yale/> However, soon after [[Albert Einstein|Einstein]] developed his static theory, observations by [[Edwin Hubble]] indicated that the universe appears to be expanding; this was consistent with a cosmological solution to the ''original'' general-relativity equations that had been found by the mathematician [[Alexander Alexandrovich Friedman|Friedman]].

It is now thought that adding the cosmological constant to Einstein's equations does not lead to a static universe at equilibrium because the [[equilibrium point|equilibrium]] is unstable: if the universe expands slightly, then the expansion releases [[vacuum energy]], which causes yet more expansion. Likewise, a universe which contracts slightly will continue contracting. 

Since it no longer seemed to be needed, Einstein called it the '"biggest blunder" of his life, and  abandoned the cosmological constant. However, the cosmological constant remained a subject of theoretical and empirical interest. Empirically, the onslaught of cosmological data in the past decades strongly suggests that our universe has a positive cosmological constant.<ref name=Yale/>  The explanation of this small but positive value is an outstanding theoretical challenge (''see the section below'').

Finally, it should be noted that some early generalizations of Einstein's gravitational theory, known as [[classical unified field theories]], either introduced a cosmological constant on theoretical grounds or found that it arose naturally from the mathematics. For example, Sir [[Arthur Stanley Eddington]] claimed that the cosmological constant version of the vacuum field equation expressed the "[[epistemology|epistemological]]" property that the universe is "self-[[gauge theory#A brief history|gauging]]", and [[Erwin Schrödinger]]'s pure-[[affine connection|affine]] theory using a simple [[History of variational principles in physics|variational principle]] produced the field equation with a cosmological term.

==Positive cosmological constant == 
Observations made in the late 1990s of distance–redshift relations indicate that the expansion of the universe is accelerating. When combined with measurements of the [[cosmic microwave background radiation]] these implied a value of <math>\Omega_{\Lambda} \simeq 0.7</math>,<ref>See e.g. {{cite journal |last=Baker |first=Joanne C. |authorlink= |coauthors=''et al.'' |year=1999 |month= |title=Detection of cosmic microwave background structure in a second field with the Cosmic Anisotropy Telescope |journal=Monthly Notices of the Royal Astronomical Society |volume=308 |issue=4 |pages=1173–1178 |doi=10.1046/j.1365-8711.1999.02829.x |url= |accessdate= |quote= }}</ref> a result which has been supported and refined by more recent measurements. There are other possible causes of an [[accelerating universe]], such as [[quintessence (physics)|quintessence]], but the cosmological constant is in most respects the most [[Occam's razor|economical]] solution. Thus, the current standard model of cosmology, the [[Lambda-CDM model]], includes the cosmological constant, which is measured to be on the order of 10<sup>−35&nbsp;</sup>s<sup>−2</sup>, or 10<sup>−47</sup>&nbsp;GeV<sup>4</sup>, or 10<sup>−29</sup>&nbsp;g/cm<sup>3</sup>,<ref>{{cite journal | last = Tegmark |first=Max |coauthors=''et al.'' | title = Cosmological parameters from SDSS and WMAP | journal = Physical Review D | year = 2004 | volume = 69 | issue = 103501 |doi=10.1103/PhysRevD.69.103501 | pages = 103501  }}</ref> or about 10<sup>−120</sup> in [[Planck units|reduced Planck units]].

== Cosmological constant problem ==
{{seealso|Vacuum catastrophe}}
{{unsolved|physics|Why doesn't the [[zero-point energy]] of [[vacuum]] cause a large cosmological constant? What cancels it out?}}
A major outstanding [[Unsolved problems in physics|problem]] is that most [[quantum field theory|quantum field theories]] predict a huge cosmological constant from the energy of the quantum [[vacuum fluctuation|vacuum]]. 

This conclusion follows from [[dimensional analysis]] and [[effective field theory]]. If the universe is described by an effective local quantum field theory down to the [[Planck scale]], then we would expect a cosmological constant of the order of <math>M_{\rm pl}^4</math>. As noted above, the measured cosmological constant is smaller than this by a factor of 10<sup>120</sup>. This discrepancy has been termed "the worst theoretical prediction in the history of physics!"<ref name=Hobson>{{cite book |page=187 |title=General Relativity: An introduction for physicists |publisher=Cambridge University Press |author=MP Hobson, GP Efstathiou & AN Lasenby |url=http://books.google.com/books?id=5dryXCWR7EIC&pg=PA187 |isbn=9780521829519 |year=2006 |edition=Reprinted with corrections 2007}}</ref>

Some [[supersymmetry|supersymmetric]] theories require a cosmological constant that is exactly zero, which further complicates things. This is the '''cosmological constant problem''', the worst problem of [[fine-tuning]] in [[physics]]: there is no known natural way to derive the tiny cosmological constant used in [[physical cosmology|cosmology]] from [[particle physics]].  

One possible explanation for the small but non-zero value was noted by [[Steven Weinberg]] in 1987 following the [[anthropic principle]].<ref>{{cite journal | last=Weinberg | first=S | title=Anthropic Bound on the Cosmological Constant | journal=Phys. Rev. Lett. | volume=59 | pages=2607–2610 | url=http://adsabs.harvard.edu/abs/1987PhRvL..59.2607W | doi=10.1103/PhysRevLett.59.2607 | year=1987}}</ref> Weinberg explains that if the vacuum energy took different values in different domains of the universe, then observers would necessarily measure values similar to that which is observed: the formation of life-supporting structures would be suppressed in domains where the vacuum energy is much larger, and domains where the vacuum energy is much smaller would be comparatively rare. This argument depends crucially on the reality of a spatial distribution in the vacuum energy density. There is no evidence that the vacuum energy does vary, but it may be the case if, for example, the vacuum energy is (even in part) the potential of a scalar field such as the residual [[inflaton]] (also see [[Quintessence (physics)|quintessence]]). Critics note that these [[multiverse]] theories, when used as an explanation for fine-tuning, commit the [[inverse gambler's fallacy]].

As was only recently seen, by works of [[Gerardus 't Hooft|'t Hooft]], [[Leonard Susskind|Susskind]]<ref>[[Lisa Dyson]], [[Matthew Kleban]], [[Leonard Susskind]]: [http://arxiv.org/abs/hep-th/0208013 "Disturbing Implications of a Cosmological Constant"]</ref> and others, a positive cosmological constant has surprising consequences, such as a finite maximum [[entropy]] of the observable universe (see the [[holographic principle]]). 

More recent work has suggested the problem may be indirect evidence of a [[Cyclic model|cyclic universe]] predicted by [[string theory]]. With every cycle of the universe ([[Big Bang]] then eventually a [[Big Crunch]]) taking about a [[1000000000000 (number)|trillion]] (10<sup>12</sup>) years, "the amount of matter and radiation in the universe is reset, but the cosmological constant is not. Instead, the cosmological constant gradually diminishes over many cycles to the small value observed today."<ref>[http://www.newscientistspace.com/article.ns?id=dn9114&print=true 'Cyclic universe' can explain cosmological constant], NewScientistSpace, 4 May 2006</ref> Critics respond that, as the authors acknowledge in their paper, the model “entails tuning” to “the same degree of tuning required in any cosmological model.”<ref>Steinhardt and Turok, 1437</ref>

==de Sitter relativity==

''See main article [[de Sitter relativity]].''

In de Sitter relativity (which is an all-energy-scale applicable example of doubly special relativity), special relativity is modified so that the symmetry group is a de Sitter rather than Poincare group. It is thought de Sitter relativity will be more accurate than special relativity at high energies. The de Sitter group naturally incorporates an invariant length–parameter and results in a residual spacetime curvature even in the absence of matter or energy. This corresponds to a special relativity with a built-in cosmological constant and a correspondingly modified de Sitter general relativity.

==Facebook page==
On July 2, 2009, the Cosmological Constant joined the online community [[Facebook]].<ref>Facebook Cosmological Constant [http://www.facebook.com/profile.php?id=100000041405872&ref=name profile].</ref>

==See also==
<div class="references-small" style="-moz-column-count:2; column-count:2;">
*[[Big Bang]]
*[[Dark energy]]
*[[Einstein's universe]]
*[[Friedmann equations]]
*[[Higgs mechanism]]
*[[Lambdavacuum solution]]
*[[Naturalness (physics)]]
*[[Quantum electrodynamics]]
*[[Unruh effect]]
*[[Vacuum energy]]
*[[Zero-point energy]]
*[[Zero-point field]]

</div>

==References==
{{Reflist|2}}

==Further reading==
{{sisterlinks}}
* Michael, E., University of Colorado, Department of Astrophysical and Planetary Sciences, "The Cosmological Constant", ''[http://super.colorado.edu/~michaele/Lambda/lambda.html]''
* [[Ferguson, Kitty]] (1991). ''Stephen Hawking: Quest For A Theory of Everything'', Franklin Watts. ISBN 0-553-29895-X.
* {{cite journal
 | author = John D. Barrow and John K. Webb 
 | title = Inconstant Constants
 | url =http://www.sciam.com/article.cfm?articleID=0005BFE6-2965-128A-A96583414B7F0000&ref=sciam&chanID=sa006
 | journal = [[Scientific American]]
 | month = June | year = 2005
 }}

==External links==
* [[Sean M. Carroll|Carroll, Sean M.]], ''[http://pancake.uchicago.edu/~carroll/encyc/ "The Cosmological Constant"]'' (short), ''[http://www.livingreviews.org/lrr-2001-1 "The Cosmological Constant"]''(extended).
* [http://www.newscientistspace.com/article.ns?id=dn9114&print=true 'Cyclic universe' can explain cosmological constant].
*[http://news.bbc.co.uk/1/hi/sci/tech/6156110.stm News story: More evidence for dark energy being the cosmological constant]

[[Category:Physical cosmology]]
[[Category:General relativity]]
[[Category:Theories of gravitation]]

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