View source for Raoult's law

{{Beyond the Standard Model|expanded=Evidence}}
{{Cosmology}}
[[File:080998 Universe Content 240 after Planck.jpg|thumb|right|250px|Estimated ratios of [[dark matter]] and dark energy (which may be the cosmological constant{{clarify|date=July 2013}}) in the universe.  Dark energy now dominates the energy of the universe, in contrast to earlier epochs when it was insignificant.]]
In [[physical cosmology|cosmology]], the '''cosmological constant''' (usually denoted by the Greek capital letter [[lambda]]: Λ) is the value of the energy density of the vacuum of space. It was introduced by [[Albert Einstein]] as an addition to his theory of [[general relativity]] to "hold back gravity" and achieve a [[static universe]], which was the accepted view at the time. Einstein abandoned the concept as his "greatest blunder" after Hubble's 1928 discovery that the distant galaxies are expanding away from each other, implying an overall expanding Universe (which is only detectable on the largest of scales).  Surprisingly, the discovery of [[cosmic acceleration]] in 1998 has revived the need for a non-zero cosmological constant, this time to add a small acceleration to the ongoing expansion. As physicist Tony Rothman notes, "Einstein's equations do not specify the universe; rather they may be considered a general framework within which you can construct many different model universes."

==Equation==
The cosmological constant Λ appears in [[Einstein field equations|Einstein's field equation]] in the form of
:<math>R_{\mu \nu} -\frac{1}{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]]'' describe the structure of [[spacetime]], ''[[stress-energy tensor|T]]'' pertains to matter and energy affecting that structure, and ''[[gravitational constant|G]]'' and ''[[speed of light|c]]'' are conversion factors that 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]]).

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 moved onto the right-hand side of the equation, and defined with a [[Proportionality (mathematics)|proportionality]] factor of 8{{pi}}: Λ = 8{{pi}}''ρ''<sub>vac</sub>, where unit conventions of general relativity are used (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 the universe, as observed. (See [[dark energy]] and [[cosmic inflation]] for details.)

===Ω<sub>Λ</sub> (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  [[Friedmann equations#Density parameter|critical density]] of the universe. This ratio is usually denoted Ω<sub>Λ</sub>, and is estimated to be {{nowrap|0.692 ± 0.010}}, according to the recent Planck results released in 2013.<ref>Collaboration, Planck, PAR Ade, N Aghanim, C Armitage-Caplan, M Arnaud, et al., Planck 2013 results. XVI. Cosmological parameters. arXiv preprint arXiv:1303.5076, 2013.</ref> In a flat universe Ω<sub>Λ</sub> 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]], usually denoted ''w'', 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= |pmid=17637630 |bibcode = 2007Natur.448..240H }}</ref> This ratio is {{nowrap|''w'' {{=}} &minus;1}} for a true cosmological constant, and is generally different for alternative time-varying forms of vacuum energy such as [[quintessence (physics)|quintessence]].

== History ==
[[Albert Einstein|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">{{Cite book|last = Urry|first = Meg|authorlink=Meg Urry|year = 2008|title =The Mysteries of Dark Energy|series = Yale Science|publisher = [[Yale University]]|url = http://itunes.yale.edu}}</ref> However, soon after 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 Friedmann|Friedmann]], working on the Einstein equations of general-relativity. Einstein later reputedly referred to his failure to accept the validation of his equations; when they had predicted the expansion of the universe in theory, before it was demonstrated in observation of the cosmological [[red shift]], as the "biggest blunder" of his life.<ref>Gamov, George (1970). ''My World Line''. Viking Press. p. 44. ISBN 978-0670503766</ref>{{dubious|Einstein's "biggest blunder"|reason=Single source of assertion has been called into doubt |date=August 2013}}<ref>{{cite web|last=Rosen|first=Rebecca J.|title=Einstein Likely Never Said One of His Most Oft-Quoted Phrases|url=http://www.theatlantic.com/technology/archive/2013/08/einstein-likely-never-said-one-of-his-most-oft-quoted-phrases/278508/|work=The Atlantic|publisher=The Atlantic Media Company|accessdate=10 August 2013}}</ref>

In fact 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. {{citation needed|date=December 2011}}

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|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 value==
Observations announced in 1998 of distance–redshift relation for [[Type Ia supernovae]]<ref>{{cite journal | author=Riess, A. et al. | title=Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant | year=1998 | month=September | journal=The Astronomical Journal | volume=116 | pages=1009–1038 | doi=10.1086/300499 | bibcode=1998AJ....116.1009R | issue=3|arxiv = astro-ph/9805201 }}</ref><ref>{{cite journal | author=Perlmutter, S. et al. | title=Measurements of Omega and Lambda from 42 High-Redshift Supernovae | year=1999 | month=June | journal=The Astrophysical Journal | volume=517 | issue=2 | pages=565–586 | doi=10.1086/307221 | bibcode=1999ApJ...517..565P|arxiv = astro-ph/9812133 }}</ref> indicated 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= |arxiv = astro-ph/9904415 |bibcode = 1999MNRAS.308.1173B }}</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 [[Occams Razor|simplest 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>−52&nbsp;</sup>m<sup>−2</sup>, in metric units.  Multiplied by other constants that appear in the equations, it is often expressed as 10<sup>−35&nbsp;</sup>s<sup>−2</sup>, 10<sup>−47</sup>&nbsp;GeV<sup>4</sup>, 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  |arxiv = astro-ph/0310723 |bibcode = 2004PhRvD..69j3501T }}</ref> In terms of [[Planck units]], and as a natural dimensionless value, the cosmological constant, λ, is on the order of 10<sup>−122</sup>.<ref>[[John D. Barrow]] [http://arxiv.org/abs/1105.3105 The Value of the Cosmological Constant]</ref>

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]]).

A problem arises with inclusion of the cosmological constant in the standard model: i.e., the appearance of solutions with regions of discontinuities (see ''[[classification of discontinuities]]'' at typical matter density).<ref name="Oztas">{{cite journal|author=A.M. Öztas and M.L. Smith|title=Elliptical Solutions to the Standard Cosmology Model with Realistic Values of Matter Density|journal=International Journal of Theoretical Physics|year=2006|volume=45|issue=5|pages=925–936|doi=10.1007/s10773-006-9082-7|bibcode = 2006IJTP...45..896O}}</ref> Discontinuity also affects the past sign of the pressure of the cosmological constant, changing from the current negative pressure to attractive, with lookback towards the early Universe. Another investigation found the cosmological time, ''dt'', diverges for any finite interval, ''ds'', associated with an observer approaching the cosmological horizon, representing a physical limit to observation for the standard model when the cosmological term is included. This is a key requirement for a complete interpretation of astronomical observations, particularly pertaining to the nature of dark energy and the cosmological constant.<ref name="Melia">{{cite journal|author=F. Melia and M. Abdelqadar|title=The Cosmological Spacetime|journal=International Journal of Modern Physics D|year=2009|volume=18|issue=12|pages=1889–1901|doi=10.1142/S0218271809015746|bibcode = 2009IJMPD..18.1889M|arxiv = 0907.5394 }}</ref> All of these findings should be considered major shortcomings of the standard model, but only when the cosmological constant term is included.

==Predictions==

===<span id="Cosmological constant problem"> Quantum field theory </span>===
{{see also|Vacuum catastrophe}}
{{unsolved|physics|Why can't the [[zero-point energy]] of the [[vacuum]] be interpreted as a cosmological constant? What causes the discrepancies?}}
A major outstanding [[Unsolved problems in physics|problem]] is that most [[quantum field theory|quantum field theories]] predict a huge value for the [[quantum fluctuation|quantum vacuum]]. A common assumption is that the [[quantum fluctuation|quantum vacuum]] is equivalent to the cosmological constant. Although no theory exists that supports this assumption, arguments can be made in its favor.<ref>{{cite journal | last=Rugh | first=S | title=The Quantum Vacuum and the Cosmological Constant Problem | journal=Studies in History and Philosophy of Modern Physics| volume=33 | issue=4 | pages=663–705 | url=http://philsci-archive.pitt.edu/398/ | doi=10.1016/S1355-2198(02)00033-3 | year=2001 | last2=Zinkernagel | first2=H. }}</ref>

Such arguments are usually based on [[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 called "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/?id=5dryXCWR7EIC&pg=PA187 |isbn=978-0-521-82951-9 |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]].

===Anthropic principle===
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 | bibcode=1987PhRvL..59.2607W | doi=10.1103/PhysRevLett.59.2607 | year=1987 | pmid=10035596 | issue=22}}</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. Specifically, if the vacuum energy is negative and its absolute value is substantially larger than it appears to be in the observed universe (say, a factor of 10 larger), holding all other variables (e.g. matter density) constant, that would mean that the universe is closed; furthermore, its lifetime would be shorter than the age of our universe, possibly too short for the intelligent life to form. On the other hand, a universe with a large positive cosmological constant would expand too fast, preventing galaxy formation. According to Weinberg, domains where the vacuum energy are compatible with life would be comparatively rare. Using this argument, Weinberg predicted that the cosmological constant would have a value of less than a hundred times the currently accepted value.<ref>[[Alexander Vilenkin]], ''Many Worlds in One: The Search for Other Universes'', ISBN 978-0-8090-9523-0, pp. 138–9</ref> In 1992 Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.<ref name="SW1993">{{cite book|last=Weinberg |first=Steven|title=Dreams of a Final Theory: the search for the fundamental laws of nature|year=1993|publisher=Vintage Press|page=182|isbn=0-09-922391-0}}</ref>

This argument depends on a lack of a variation of the distribution (spatial or otherwise) in the vacuum energy density, as would be expected if dark energy were the cosmological constant. 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]]). Another theoretical approach that deals with the issue is that of [[multiverse]] theories, which predict a large number of "parallel" universes with different laws of physics and/or values of fundamental constants. Again, the anthropic principle states that we can only live in one of the universes that is compatible with some form of intelligent life. Critics claim that these theories, when used as an explanation for fine-tuning, commit the [[inverse gambler's fallacy]].

In 1995 Weinberg's argument was refined by [[Alexander Vilenkin]] to predict a value for the cosmological constant that was only ten times the matter density,<ref>[[Alexander Vilenkin]], ''Many Worlds in One: The Search for Other Universes'', ISBN 978-0-8090-9523-0, p. 146, which references Vilenkin' [http://prl.aps.org/abstract/PRL/v74/i6/p846_1 ''Predictions from quantum cosmology''], Physical Review Letters, vol 74, p. 846 (1995)</ref> i.e. about three times the current value since determined.

===Cyclic model===
More recent work has suggested the problem may be indirect evidence of a [[Cyclic model|cyclic universe]] possibly as allowed 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.newscientist.com/article/dn9114-cyclic-universe-can-explain-cosmological-constant.html 'Cyclic universe' can explain cosmological constant], NewScientistSpace, 4 May 2006</ref> Critics respond that, as the authors acknowledge in their paper, the model “entails ... the same degree of tuning required in any cosmological model”.<ref name = "Steinhardt & Turok 2002">{{cite journal
 | last = Steinhardt | first = P. J. | authorlink = Paul Steinhardt | coauthors = [[Neil Turok|Turok, N.]]
 | title = A Cyclic Model of the Universe | journal = [[Science (journal)|Science]]
 | volume = 296 | issue = 5572 | pages = 1436–1439
 | date = 2002-04-25 | url = http://www.sciencemag.org/content/296/5572/1436.short
 | doi = 10.1126/science.1070462 | bibcode = 2002Sci...296.1436S | arxiv = hep-th/0111030v2 | accessdate = 2012-04-29 | pmid=11976408}}</ref>

==See also==
<div class="references-small" style="-moz-column-count:2; column-count:2;">
* [[Higgs mechanism]]
* [[Lambdavacuum solution]]
* [[Naturalness (physics)]]
* [[Quantum electrodynamics]]
* [[de Sitter relativity]]
* [[Unruh effect]]
</div>

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

==Further reading==
* 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==
{{Sister project 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]
* [http://www.scholarpedia.org/article/Cosmological_constant Cosmological constant] article from [[Scholarpedia]]
* {{cite web|last=Copeland|first=Ed|title=Λ – Cosmological Constant|url=http://www.sixtysymbols.com/videos/cosmological.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|coauthors=Merrifield, Mike}}

{{Einstein}}

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