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Cosmological constant

## Summary

In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term Albert Einstein temporarily added to his field equations of general relativity. He later removed it. Much later it was revived and reinterpreted as the energy density of space, or vacuum energy, that arises in quantum mechanics. It is closely associated with the concept of dark energy.[1]

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.

Einstein originally introduced the constant in 1917[2] to counterbalance the effect of gravity and achieve a static universe, a notion which was the accepted view at the time. Einstein's theory was abandoned after the Hubble's confirmation of the expanding universe theory.[3] From the 1930s until the late 1990s, most physicists agreed with Einstein's theory, assuming the cosmological constant to be equal to zero.[4] That changed with the surprising discovery in 1998 that the expansion of the universe is accelerating, implying the possibility of a positive value for the cosmological constant.[5]

Since the 1990s, studies have shown that, assuming the cosmological principle, around 68% of the mass–energy density of the universe can be attributed to so-called dark energy.[6][7][8] The cosmological constant Λ is the simplest possible explanation for dark energy, and is used in the current standard model of cosmology known as the ΛCDM model.

According to quantum field theory (QFT) which underlies modern particle physics, empty space is defined by the vacuum state which is a collection of quantum fields. All these quantum fields exhibit fluctuations in their ground state (lowest energy density) arising from the zero-point energy present everywhere in space. These zero-point fluctuations should act as a contribution to the cosmological constant Λ, but when calculations are performed these fluctuations give rise to an enormous vacuum energy.[9] The discrepancy between theorized vacuum energy from quantum field theory and observed vacuum energy from cosmology is a source of major contention, with the values predicted exceeding observation by some 120 orders of magnitude, a discrepancy that has been called "the worst theoretical prediction in the history of physics".[10] This issue is called the cosmological constant problem and it is one of the greatest mysteries in science with many physicists believing that "the vacuum holds the key to a full understanding of nature".[11]

## History

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.[3] 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 reportedly 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 redshift—as his "biggest blunder".[12]

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.[13]

However, the cosmological constant remained a subject of theoretical and empirical interest. Empirically, the onslaught of cosmological data in recent 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, the so-called cosmological constant problem.

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.

## Sequence of events 1915–1998

• In 1915, Einstein publishes his equations of General Relativity, without a cosmological constant Λ.
• In 1917, Einstein adds the parameter Λ to his equations when he realizes that his theory implies a dynamic universe for which space is function of time. He then gives this constant a very particular value to force his Universe model to remain static and eternal (Einstein static universe), which he will later call "the greatest stupidity of his life".
• In 1922, the Russian physicist Alexander Friedmann mathematically shows that Einstein's equations (whatever Λ) remain valid in a dynamic universe.
• In 1927, the Belgian astrophysicist Georges Lemaître shows that the Universe is in expansion by combining General Relativity with some astronomical observations, those of Hubble in particular.
• In 1931, Einstein finally accepts the theory of an expanding universe and proposes, in 1932 with the Dutch physicist and astronomer Willem de Sitter, a model of a continuously expanding Universe with zero cosmological constant (Einstein-de Sitter space-time).
• In 1998, two teams of astrophysicists, one led by Saul Perlmutter, the other led by Brian Schmidt and Adam Riess, carried out measurements on distant supernovae which showed that the speed of galaxies' recession in relation to the Milky Way increases over time. The universe is in accelerated expansion, which requires having a strictly positive Λ. The universe would contain a mysterious dark energy producing a repulsive force that counterbalances the gravitational braking produced by the matter contained in the universe (see standard cosmological model).
For this work, Perlmutter (American), Schmidt (American-Australian), and Riess (American) jointly received the Nobel Prize in physics in 2011.

## Equation

Estimated ratios of dark matter and dark energy (which may be the cosmological constant[1]) 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.

The cosmological constant Λ appears in the Einstein field equations in the form

${\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}\,T_{\mu \nu },}$

where the Ricci tensor/scalar Rμν (R is the scalar curvature) and the metric tensor gμν describe the structure of spacetime, the stress-energy tensor Tμν describes the energy and momentum density and flux of the matter in that point in spacetime, and the universal constants of gravitation G and the speed of light c are conversion factors that arise when using traditional units of measurement . When Λ is zero, this reduces to the field equation of general relativity usually used in the 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 = 8 π ρvac , where unit conventions of general relativity are used (otherwise factors of G and c would also appear, i.e., Λ = 8 π ρvac G / c4 = κ ρvac , where κ is Einstein's rescaled version of the gravitational constant G). It is common to quote values of energy density directly, though still using the name "cosmological constant", using Planck units so that 8πG = 1. The true dimension of Λ is length−2.

Using the values known in 2018 and Planck units for ΩΛ = 0.6889±0.0056 and the Hubble constant H0 = 67.66±0.42 (km/s)/Mpc = (2.1927664±0.0136)×10−18 s−1, Λ has the value of

{\displaystyle {\begin{aligned}\Lambda =3\,\left({\frac {\,H_{0}\,}{c}}\right)^{2}\Omega _{\Lambda }&=1.1056\times 10^{-52}\ {\text{m}}^{-2}\\&=2.888\times 10^{-122}\,l_{\text{P}}^{-2}\end{aligned}}}

where ${\displaystyle l_{\text{P}}}$  is the Planck length. 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.)

### ΩΛ (Omega sub 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 by ΩΛ and is estimated to be 0.6889±0.0056, according to results published by the Planck Collaboration in 2018.[14]

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, because the amount of dark energy increases as the universe grows but the amount of matter does not.[citation needed]

### 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.[15] This ratio is w = −1 for the cosmological constant used in the Einstein equations; alternative time-varying forms of vacuum energy such as quintessence generally use a different value. The value w = −1.028±0.032, measured by the Planck Collaboration (2018)[14] is consistent with −1, assuming w does not change over cosmic time.

## Positive value

Lambda-CDM, accelerated expansion of the universe. The time-line in this schematic diagram extends from the Big Bang/inflation era 13.7 Byr ago to the present cosmological time.

Observations announced in 1998 of distance–redshift relation for Type Ia supernovae[5] indicated that the expansion of the universe is accelerating, if one assumes the cosmological principle.[6][7] When combined with measurements of the cosmic microwave background radiation these implied a value of ΩΛ ≈ 0.7,[16] a result which has been supported and refined by more recent measurements[17] (and previous works too[18][19] ). If one assumes the cosmological principle, as in the case for all models using the Friedmann–Lemaître–Robertson–Walker metric, while there are other possible causes of an accelerating universe, such as quintessence, the cosmological constant is in most respects the simplest solution. Thus, the Lambda-CDM model, the current standard model of cosmology which uses the FLRW metric, 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 (by multiplication with c2, i.e. ≈1017 m2⋅s−2) or as 10−122 P−2 [20] (in units of the square of the Planck length, i.e. ≈10−70 m2). The value is based on recent measurements of vacuum energy density, ${\displaystyle \rho _{\text{vacuum}}=5.96\times 10^{-27}{\text{ kg/m}}^{3}=5.3566\times 10^{-10}{\text{ J/m}}^{3}=3.35{\text{ GeV/m}}^{3}}$ .[21] However, due to the Hubble tension and the CMB dipole, recently it has been proposed that the cosmological principle is no longer true in the late universe and that the FLRW metric breaks down,[22][23][24] so it is possible that observations usually attributed to an accelerating universe are simply a result of the cosmological principle not applying in the late universe.[6][7]

As was only recently seen, by works of 't Hooft, Susskind and others, a positive cosmological constant has surprising consequences, such as a finite maximum entropy of the observable universe (see the holographic principle).[25]

## Predictions

### Quantum field theory

Why does the zero-point energy of the quantum vacuum not cause a large cosmological constant? What cancels it out?

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.[26]

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 ${\displaystyle M_{\rm {pl}}^{2}}$  (${\displaystyle 1}$  in reduced Planck units). As noted above, the measured cosmological constant is smaller than this by a factor of ~10120. This discrepancy has been called "the worst theoretical prediction in the history of physics!"[10]

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.

No vacuum in the string theory landscape is known to support a metastable, positive cosmological constant, and in 2018 a group of four physicists advanced a controversial conjecture which would imply that no such universe exists.[27]

### Anthropic principle

One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987 following the anthropic principle.[28] 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.[29] In 1992, Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.[30]

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

### Failure to detect dark energy

An attempt to directly observe dark energy in a laboratory failed to detect a new force.[32] Inferring the presence of dark energy through its interaction with baryons in the cosmic microwave background has also led to a negative result,[33] although the current analyses have been derived only at the linear perturbation regime.

## References

### Footnotes

1. ^ a b It may well be that dark energy is explained by a static cosmological constant, or that this mysterious energy is not constant at all and has changed over time, as in the case with quintessence, see for example:
• "Physics invites the idea that space contains energy whose gravitational effect approximates that of Einstein's cosmological constant, Λ; nowadays the concept is termed dark energy or quintessence." Peebles & Ratra (2003), p. 1
• "It would then appear that the cosmological fluid is dominated by some sort of fantastic energy density, which has negative pressure, and has just begun to play an important role today. No convincing theory has yet been constructed to explain this state of affairs, although cosmological models based on a dark energy component, such as the cosmological constant (Λ) or quintessence (Q), are leading candidates." Caldwell (2002), p. 2
2. ^ Einstein (1917)
3. ^ a b
4. ^ On the Cosmological Constant being thought to have zero value see for example:
• "Since the cosmological upper bound on ${\displaystyle \left|\left\langle \rho \right\rangle +\lambda /8\pi G\right|}$  was vastly less than any value expected from particle theory, most particle theorists simply assumed that for some unknown reason this quantity was zero." Weinberg (1989), p. 3
• "An epochal astronomical discovery would be to establish by convincing observation that Λ is nonzero." Carroll, Press & Turner (1992), p. 500
• "Before 1998, there was no direct astronomical evidence for Λ and the observational upper bound was so strong ( Λ < 10−120 Planck units) that many particle physicists suspected that some fundamental principle must force its value to be precisely zero." Barrow & Shaw (2011), p. 1
• "The only other natural value is Λ = 0. If Λ really is tiny but not zero, it adds a most stimulating though enigmatic clue to physics to be discovered." Peebles & Ratra (2003), p. 333
5. ^ a b c See for example:
6. ^ a b c Ellis, G. F. R. (2009). "Dark energy and inhomogeneity". Journal of Physics: Conference Series. 189: 012011. doi:10.1088/1742-6596/189/1/012011. S2CID 250670331.
7. ^ a b c Jacques Colin; Roya Mohayaee; Mohamed Rameez; Subir Sarkar (20 November 2019). "Evidence for anisotropy of cosmic acceleration". Astronomy and Astrophysics. 631: L13. arXiv:1808.04597. doi:10.1051/0004-6361/201936373. S2CID 208175643. Retrieved 25 March 2022.
8. ^ Redd (2013)
9. ^
10. ^ a b See for example:
• "This gives an answer about 120 orders of magnitude higher than the upper limits on Λ set by cosmological observations. This is probably the worst theoretical prediction in the history of physics!" Hobson, Efstathiou & Lasenby (2006), p. 187
• "This, as we will see later, is approximately 120 orders of magnitude larger than what is allowed by observation." Carroll, Press & Turner (1992), p. 503
• "Theoretical expectations for the cosmological constant exceed observational limits by some 120 orders of magnitude." Weinberg (1989), p. 1
11. ^ See for example:
• "the vacuum holds the key to a full understanding of nature" Davies (1985), p. 104
• "The theoretical problem of explaining the cosmological constant is one of the greatest challenges of theoretical physics. It is most likely that we require a fully developed theory of quantum gravity (perhaps superstring theory) before we can understand Λ." Hobson, Efstathiou & Lasenby (2006), p. 188
12. ^ There is some debate over whether Einstein labelled the cosmological constant his "biggest blunder", with all references being traced back to a single person: George Gamow. (See Gamow (1956, 1970).) For example:
• "Astrophysicist and author Mario Livio can find no documentation that puts those words into Einstein's mouth (or, for that matter, his pen). Instead, all references eventually lead back to one man—physicist George Gamow—who reported Einstein's use of the phrase in two sources: His posthumously published autobiography My World Line (1970) and a Scientific American article from September 1956." Rosen (2013)
• " We also find it quite plausible that Einstein made such a statement to Gamow in particular. We conclude that there is little doubt that Einstein came to view the introduction of the cosmological constant a serious error, and that it is very plausible that he labelled the term his "biggest blunder" on at least one occasion". O'Raifeartaigh & Mitton (2018), p. 1
13. ^ Ryden (2003), p. 59
14. ^ a b Planck Collaboration (2018)
15. ^ Brumfiel (2007), p. 246
16. ^ See e.g. Baker et al. (1999)
17. ^ See for example Table 9 in The Planck Collaboration (2015a), p. 27
18. ^ "Inflation and compactification from Galaxy redshifts?". Bibcode:1992Ap&SS.191..107P. doi:10.1007/BF00644200. {{cite journal}}: Cite journal requires |journal= (help)
19. ^ "Once More on Quasar Periodicities". Bibcode:1994Ap&SS.222...65H. doi:10.1007/BF00627083. {{cite journal}}: Cite journal requires |journal= (help)
20. ^ Barrow & Shaw (2011)
21. ^ Calculated based on the Hubble constant and ${\displaystyle \Omega _{\Lambda }}$  from The Planck Collaboration (2015b)
22. ^ Elcio Abdalla; Guillermo Franco Abellán; et al. (11 Mar 2022), Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies, arXiv:2203.06142v1
23. ^ Krishnan, Chethan; Mohayaee, Roya; Colgáin, Eoin Ó; Sheikh-Jabbari, M. M.; Yin, Lu (16 September 2021). "Does Hubble Tension Signal a Breakdown in FLRW Cosmology?". Classical and Quantum Gravity. 38 (18): 184001. arXiv:2105.09790. Bibcode:2021CQGra..38r4001K. doi:10.1088/1361-6382/ac1a81. ISSN 0264-9381. S2CID 234790314.
24. ^ Asta Heinesen; Hayley J. Macpherson (15 July 2021). "Luminosity distance and anisotropic sky-sampling at low redshifts: A numerical relativity study". Physical Review D. 104 (2): 023525. arXiv:2103.11918. doi:10.1103/PhysRevD.104.023525. S2CID 232307363. Retrieved 25 March 2022.
25. ^ Dyson, Kleban & Susskind (2002)
26. ^
27. ^ Wolchover, Natalie (9 August 2018). "Dark Energy May Be Incompatible With String Theory". Quanta Magazine. Simons Foundation. Retrieved 2 April 2020.
28. ^ Weinberg (1987)
29. ^ Vilenkin (2006), pp. 138–139
30. ^ Weinberg (1992), p. 182
31. ^ Vilenkin (2006), p. 146
32. ^ D. O. Sabulsky; I. Dutta; E. A. Hinds; B. Elder; C. Burrage; E. J. Copeland (2019). "Experiment to Detect Dark Energy Forces Using Atom Interferometry". Physical Review Letters. 123 (6): 061102. arXiv:1812.08244. Bibcode:2019PhRvL.123f1102S. doi:10.1103/PhysRevLett.123.061102. PMID 31491160. S2CID 118935116.
33. ^ S. Vagnozzi; L. Visinelli; O. Mena; D. Mota (2020). "Do we have any hope of detecting scattering between dark energy and baryons through cosmology?". Mon. Not. R. Astron. Soc. 493 (1): 1139. arXiv:1911.12374. Bibcode:2020MNRAS.493.1139V. doi:10.1093/mnras/staa311.

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#### Secondary literature: review articles, monographs and textbooks

• Barrow, J. D.; Shaw, D. J. (2011). "The value of the cosmological constant". General Relativity and Gravitation. 43 (10): 2555–2560. arXiv:1105.3105. Bibcode:2011GReGr..43.2555B. doi:10.1007/s10714-011-1199-1. ISSN 0001-7701. S2CID 55125081.
• Caldwell, R. R. (2002). "A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state". Physics Letters B. 545 (1–2): 23–29. arXiv:astro-ph/9908168. Bibcode:2002PhLB..545...23C. doi:10.1016/S0370-2693(02)02589-3. ISSN 0370-2693. S2CID 9820570.
• Carroll, S. M.; Press, W. H.; Turner, E. L. (1992). "The Cosmological Constant" (PDF). Annual Review of Astronomy and Astrophysics. 30 (1): 499–542. Bibcode:1992ARA&A..30..499C. doi:10.1146/annurev.aa.30.090192.002435. ISSN 0066-4146. PMC 5256042. PMID 28179856.
• Hobson, M. P.; Efstathiou, G. P.; Lasenby, A. N. (2006). General Relativity: An Introduction for Physicists (2014 ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-82951-9. LCCN 2006277059. OCLC 903178203.
• Joyce, A.; Jain, B.; Khoury, J.; Trodden, M. (2015). "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. ISSN 0370-1573. S2CID 119187526.
• Peebles, P. J. E.; Ratra, B. (2003). "The Cosmological Constant and Dark Energy". Reviews of Modern Physics. 75 (2): 559–606. arXiv:astro-ph/0207347. Bibcode:2003RvMP...75..559P. doi:10.1103/RevModPhys.75.559. ISSN 0034-6861. S2CID 118961123.
• 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. S2CID 9007190.
• Ryden, B. S. (2003). Introduction to Cosmology. San Francisco: Addison-Wesley. ISBN 978-0-8053-8912-8. LCCN 2002013176. OCLC 50478401.
• Vilenkin, A. (2006). Many worlds in one: The Search For Other Universes. New York: Hill and Wang. ISBN 978-0-8090-9523-0. LCCN 2005027057. OCLC 799428013.
• Weinberg, S. (1989). "The Cosmological Constant Problem" (PDF). Reviews of Modern Physics. 61 (1): 1–23. Bibcode:1989RvMP...61....1W. doi:10.1103/RevModPhys.61.1. hdl:2152/61094. ISSN 0034-6861.
• Weinberg, S. (1992). Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature. New York: Pantheon Books. ISBN 978-0-679-74408-5. LCCN 93030534. OCLC 319776354.
• Weinberg, S. (2015). Lectures on Quantum Mechanics (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-11166-0. LCCN 2015021123. OCLC 910664598.