Cosmological constant

Sketch of the timeline of the Universe in the ΛCDM model. The accelerated expansion in the last third of the timeline represents the dark-energy dominated era.
Estimated ratios of dark matter and dark energy (which may be the cosmological constant) in the universe. According to current theories of physics, dark energy now dominates as the largest source of energy of the universe, in contrast to earlier epochs when it was insignificant.

In 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 originally introduced by Albert Einstein in 1917[1] 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 after Hubble's 1929 discovery that all galaxies outside the Local Group (the group that contains the Milky Way Galaxy) are moving away from each other, implying an overall expanding universe. From 1929 until the early 1990s, most cosmology researchers assumed the cosmological constant to be zero.

Since the 1990s, several developments in observational cosmology, especially the discovery of the accelerating universe from distant supernovas in 1998 (in addition to independent evidence from the cosmic microwave background and large galaxy redshift surveys), have shown that around 68% of the mass–energy density of the universe can be attributed to dark energy.[2] While dark energy is poorly understood at a fundamental level, the main required properties of dark energy are that it functions as a type of anti-gravity, it dilutes much more slowly than matter as the universe expands, and it clusters much more weakly than matter, or perhaps not at all. The cosmological constant is the simplest possible form of dark energy since it is constant in both space and time, and this leads to the current standard model of cosmology known as the Lambda-CDM model, which provides a good fit to many cosmological observations.


The cosmological constant appears in Einstein's field equation in the form

where the Ricci tensor/scalar R and the metric tensor g describe the structure of spacetime, the stress-energy tensor T describes the energy and momentum density and flux of the matter in that point in spacetime, and the universal constants G and c are conversion factors that arise from using traditional units of measurement. When Λ is zero, this reduces to the field equation of general relativity usually used in the mid-20th century. 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, ρvac (and an associated pressure). In this context, it is commonly moved onto the right-hand side of the equation, and defined with a proportionality factor of 8π: Λ = 8πρvac, where unit conventions of general relativity are used (otherwise factors of G and c would also appear, i.e. Λ = 8π(G/c2)ρvac = κρvac, where κ is Einstein's constant). It is common to quote values of energy density directly, though still using the name "cosmological constant", with convention 8πG = 1. The true dimension of Λ is a length−2.

Given the Planck (2018) values of ΩΛ = 0.6889±0.0056 and Ho = 67.66±0.42 (km/s)/Mpc = (2.1927664±0.0136)×10−18 s−1, Λ has the value of

[lower-alpha 1]

or 2.888×10−122 in reduced Planck units or 4.33×10−66 eV2 in natural units.

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.)

  1. Λ in m-2 units is evaluated as 3 * (H0/c)2 * ΩΛ

ΩΛ (Omega Lambda)

Instead 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, the tipping point for a sufficient density to stop the universe from expanding forever. This ratio is usually denoted ΩΛ, and is estimated to be 0.6889±0.0056, according to results published by the Planck Collaboration in 2018.[3]

In a flat universe, ΩΛ is the fraction of the energy of the universe due to the cosmological constant, i.e., what we would intuitively call the fraction of the universe that is made up of dark energy. Note that this value changes over time: the critical density changes with cosmological time, but the energy density due to the cosmological constant remains unchanged throughout the history of the universe: the amount of dark energy increases as the universe grows, while the amount of matter does not.

Equation of state

Another ratio that is used by scientists is the 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.[4] This ratio is w = 1 for a true cosmological constant, and is generally different for alternative time-varying forms of vacuum energy such as quintessence.


Einstein included the cosmological constant as a term in his 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 that was initially at dynamic equilibrium to contract. To counteract this possibility, Einstein added the cosmological constant.[5] 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 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.[6][7][8]

In fact, adding the cosmological constant to Einstein's equations does not lead to a static universe at equilibrium because the equilibrium is unstable: if the universe expands slightly, then the expansion releases vacuum energy, which causes yet more expansion. Likewise, a universe that contracts slightly will continue contracting.[9]:59

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.[5] The explanation of this small but positive value is an outstanding theoretical challenge (see the section below).

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 "epistemological" property that the universe is "self-gauging", and Erwin Schrödinger's pure-affine theory using a simple variational principle produced the field equation with a cosmological term.

Positive value

Observations announced in 1998 of distance–redshift relation for Type Ia supernovae[10][11] indicated that the expansion of the universe is accelerating. When combined with measurements of the cosmic microwave background radiation these implied a value of ΩΛ ≈ 0.7,[12] a result which has been supported and refined by more recent measurements.[13] There are other possible causes of an accelerating universe, such as quintessence, but the cosmological constant is in most respects the 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 10−52 m−2, in metric units. It is often expressed as 10−35 s−2 or 10−122[14] in other unit systems. The value is based on recent measurements of vacuum energy density, ,[15] or 10−47 GeV4, 10−29 g/cm3 in other unit systems.

As was only recently seen, by works of 't Hooft, Susskind[16] and others, a positive cosmological constant has surprising consequences, such as a finite maximum entropy of the observable universe (see the holographic principle).


Quantum field theory

A major outstanding problem is that most quantum field theories predict a huge value for the quantum vacuum. A common assumption is that the quantum vacuum is equivalent to the cosmological constant. Although no theory exists that supports this assumption, arguments can be made in its favor.[17]

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 ( in natural unit or in reduced Planck unit). As noted above, the measured cosmological constant is smaller than this by a factor of ~10−120. This discrepancy has been called "the worst theoretical prediction in the history of physics!".[18]

Some 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 cosmology from particle physics.

A possible solution is offered by light front quantization, a rigorous alternative to the usual second quantization method. Vacuum fluctuations do not appear in the Light-Front vacuum state.[19][20] This absence means that there is no contribution from QED, Weak interactions and QCD to the cosmological constant which is thus predicted to be zero in a flat space-time.[21] Unlike supersymmetric theories (discussed above), that light front quantization predict Λ=0 within the standard model of particle physics may not be a problem since the small non-zero value of the cosmological constant could originate for example from a slight curvature of the shape of the universe (which is not excluded within 0.4% (as of 2017)[22][23][24]) since a curved-space could modify the Higgs field zero-mode, thereby possibly producing a non-zero contribution to the cosmological constant.

Anthropic principle

One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987 following the anthropic principle.[25] 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 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 is 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.[26] In 1992, Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.[27]

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). 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,[28] i.e. about three times the current value since determined.

See also


  1. Einstein, A (1917). "Kosmologische Betrachtungen zur allgemeinen Relativitaetstheorie". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 142–152.
  2. What is Dark Energy?,, 1 May 2013
  3. Collaboration, Planck, PAR Ade, N Aghanim, C Armitage-Caplan, M Arnaud, et al., Planck 2018 results. VI. Cosmological parameters. arXiv preprint 1807.06209v1 , 17 July 2018.
  4. Hogan, Jenny (2007). "Welcome to the Dark Side". Nature. 448 (7151): 240–245. Bibcode:2007Natur.448..240H. doi:10.1038/448240a. PMID 17637630.
  5. 1 2 Urry, Meg (2008). The Mysteries of Dark Energy. Yale Science. Yale University.
  6. Gamov, George (1970). My World Line. Viking Press. p. 44. ISBN 978-0670503766
  7. Rosen, Rebecca J. "Einstein Likely Never Said One of His Most Oft-Quoted Phrases". The Atlantic. The Atlantic Media Company. Retrieved 10 August 2013.
  8. "it is very plausible that he labelled the term his 'biggest blunder' on at least one occasion."
  9. Barbara Sue Ryden (2003). Introduction to cosmology. Addison-Wesley. ISBN 978-0-8053-8912-8.
  10. Riess, A.; et al. (September 1998). "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant". The Astronomical Journal. 116 (3): 1009–1038. arXiv:astro-ph/9805201. Bibcode:1998AJ....116.1009R. doi:10.1086/300499.
  11. Perlmutter, S.; et al. (June 1999). "Measurements of Omega and Lambda from 42 High-Redshift Supernovae". The Astrophysical Journal. 517 (2): 565–586. arXiv:astro-ph/9812133. Bibcode:1999ApJ...517..565P. doi:10.1086/307221.
  12. See e.g. Baker, Joanne C.; et al. (1999). "Detection of cosmic microwave background structure in a second field with the Cosmic Anisotropy Telescope". Monthly Notices of the Royal Astronomical Society. 308 (4): 1173–1178. arXiv:astro-ph/9904415. Bibcode:1999MNRAS.308.1173B. doi:10.1046/j.1365-8711.1999.02829.x.
  13. See for example Table 9 in The Planck Collaboration (20 September 2016). "Planck 2015 results I. Overview of products and scientific results". Astronomy & Astrophysics. 594: A1. arXiv:1502.01582. Bibcode:2016A&A...594A...1P. doi:10.1051/0004-6361/201527101.
  14. The Value of the Cosmological Constant
  15. Calculated based on the Hubble constant and from Planck 2015 results. XIII.
  16. Lisa Dyson, Matthew Kleban, Leonard Susskind: "Disturbing Implications of a Cosmological Constant"
  17. Rugh, S; Zinkernagel, H. (2001). "The Quantum Vacuum and the Cosmological Constant Problem". Studies in History and Philosophy of Modern Physics. 33 (4): 663–705. arXiv:hep-th/0012253. Bibcode:2002SHPMP..33..663R. doi:10.1016/S1355-2198(02)00033-3.
  18. MP Hobson; GP Efstathiou; AN Lasenby (2006). General Relativity: An introduction for physicists (Reprinted with corrections 2007 ed.). Cambridge University Press. p. 187. ISBN 978-0-521-82951-9.
  19. H. Leutwyler, J.R. Klauder, L. Streit. Quantum field theory on lightlike slabs, Nuovo Cim. A66 (1970) 536 DOI: 10.1007/BF02826338
  20. A. Casher and L. Susskind. Chiral magnetism (or magnetohadrochironics) Phys. Rev. D9 (1974) 436 DOI: 10.1103/PhysRevD.9.436
  21. S. J. Brodsky and R. Shrock. Condensates in Quantum Chromodynamics and the Cosmological Constant. Proc.Nat.Acad.Sci. 108 (2011) 45-50, [arXiv:0905.1151].
  22. "Will the Universe expand forever?". NASA. 24 January 2014. Retrieved 16 March 2015.
  23. "Our universe is Flat". FermiLab/SLAC. 7 April 2015.
  24. Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science. LXXIV1: 30.
  25. Weinberg, S (1987). "Anthropic Bound on the Cosmological Constant". Phys. Rev. Lett. 59 (22): 2607–2610. Bibcode:1987PhRvL..59.2607W. doi:10.1103/PhysRevLett.59.2607. PMID 10035596.
  26. Alexander Vilenkin, Many Worlds in One: The Search for Other Universes, ISBN 978-0-8090-9523-0, pp. 138–9
  27. Weinberg, Steven (1993). Dreams of a Final Theory: the search for the fundamental laws of nature. Vintage Press. p. 182. ISBN 978-0-09-922391-7.
  28. Alexander Vilenkin, Many Worlds in One: The Search for Other universes, ISBN 978-0-8090-9523-0, p. 146, which references Vilenkin' Predictions from quantum cosmology, Physical Review Letters, vol 74, p. 846 (1995)
  • Michael, E., University of Colorado, Department of Astrophysical and Planetary Sciences, "The Cosmological Constant"
  • Ferguson, Kitty (1991). Stephen Hawking: Quest For A Theory of Everything, Franklin Watts. ISBN 0-553-29895-X.
  • John D. Barrow; John K. Webb (June 2005). "Inconstant Constants". Scientific American.
  • Austin Joyce; Bhuvnesh Jain; Justin Khoury; Mark Trodden (2014). "Beyond the Cosmological Standard Model". Physics Reports. 568: 1–98. arXiv:1407.0059. Bibcode:2015PhR...568....1J. doi:10.1016/j.physrep.2014.12.002.
  • Cosmological constant (astronomy) at Encyclopædia Britannica
  • Carroll, Sean M., "The Cosmological Constant" (short), "The Cosmological Constant"(extended).
  • News story: More evidence for dark energy being the cosmological constant
  • Cosmological constant article from Scholarpedia
  • Copeland, Ed; Merrifield, Mike. "Λ – Cosmological Constant". Sixty Symbols. Brady Haran for the University of Nottingham.
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