Incosmology, thecosmological constant (usually denoted by the Greek capital letterlambda:Λ), alternatively calledEinstein's cosmological constant, is a coefficient thatAlbert Einstein initially added to his field equations ofgeneral relativity. He later removed it; however, much later it was revived to express the energy density of space, orvacuum energy, that arises inquantum mechanics. It is closely associated with the concept ofdark energy.[1]
Einstein introduced the constant in 1917[2] to counterbalance the effect of gravity and achieve astatic universe, which was then assumed. Einstein's cosmological constant was abandoned afterEdwin Hubble confirmed that the universe was expanding.[3] From the 1930s until the late 1990s, most physicists thought the cosmological constant to be zero.[4] That changed with the discovery in 1998 that theexpansion of the universe is accelerating, implying that the cosmological constant may have a positive value after all.[5]
Since the 1990s, studies have shown that, assuming thecosmological principle, around 68% of the mass–energy density of the universe can be attributed to dark energy.[6][7][8] The cosmological constantΛ is the simplest possible explanation for dark energy, and is used in the standard model of cosmology known as theΛCDM model.
According toquantum field theory (QFT), which underlies modernparticle physics, empty space is defined by thevacuum state, which is composed of a collection ofquantum fields. All these quantum fields exhibit fluctuations in theirground state (lowest energy density) arising from thezero-point energy existing everywhere in space. These zero-point fluctuations should contribute to the cosmological constantΛ, but actual calculations 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 thecosmological 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]
The cosmological constant was originally introduced inEinstein's 1917 paper entitledCosmological considerations in the General Theory of Relativity.[2] Einstein included the cosmological constant as a term in hisfield equations forgeneral relativity because he was dissatisfied that otherwise his equations did not allow for astatic universe: gravity would cause a universe that was initially non-expanding to contract. To counteract this possibility, Einstein added the cosmological constant.[3] However, Einstein was not happy about adding this cosmological term. He later stated that "Since I introduced this term, I had always a bad conscience. ... I am unable to believe that such an ugly thing is actually realized in nature".[12] Einstein's static universe is unstable against matter density perturbations.[13] Furthermore, without the cosmological constant Einstein could have found the expansion of the universe before Hubble's observations.[14]
In 1929, not long after Einstein developed his static theory, observations byEdwin Hubble[14] 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 mathematicianAlexander 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 cosmologicalredshift—as his "biggest blunder" (according toGeorge Gamow).[15]
It transpired that adding the cosmological constant to Einstein's equations does not lead to a static universe at equilibrium because theequilibrium is unstable: if the universe expands slightly, then the expansion releasesvacuum energy, which causes yet more expansion. Likewise, a universe that contracts slightly will continue contracting.[16]
However, the cosmological constant remained a subject of theoretical and empirical interest. Empirically, the cosmological data of recent decades strongly suggest that our universe has a positive cosmological constant.[5] The explanation of this small but positive value is a remaining theoretical challenge, the so-calledcosmological constant problem.
Some early generalizations of Einstein's gravitational theory, known asclassical unified field theories, either introduced a cosmological constant on theoretical grounds or found that it arose naturally from the mathematics. For example,Arthur Eddington claimed that the cosmological constant version of the vacuum field equation expressed the "epistemological" property that the universe is "self-gauging", andErwin Schrödinger's pure-affine theory using a simplevariational principle produced the field equation with a cosmological term.
In 1990s,Saul Perlmutter at Lawrence Berkeley National Laboratory,Brian Schmidt of the Australian National University andAdam Riess of the Space Telescope Science Institute were searching for type Ia supernovae. At that time, they expected to observe the deceleration of the supernovae caused by gravitational attraction of mass according to Einstein's gravitational theory. The first reports published in July 1997 from the Supernova Cosmology Project used the supernova observation to support such deceleration hypothesis. But soon they found that supernovae were accelerating away. Both teams announced this surprising result in 1998. It implied the universe is undergoing accelerating expansion. The cosmological constant is needed to explain such acceleration.[17] Following this discovery, the cosmological constant was reinserted in the general relativity equations.
In 1915, Einstein publishes his equations of general relativity, without a cosmological constantΛ.
In 1917, Einstein adds the parameterΛ to his equations when he realizes that his theory implies a dynamic universe for which space is a function of time. He then gives this constant a value that makes his Universe model remain static and eternal (Einstein static universe).
In 1922, the Russian physicistAlexander Friedmann mathematically shows that Einstein's equations (whateverΛ) remain valid in a dynamic universe.
In 1927, the Belgian astrophysicistGeorges Lemaître shows that the Universe is expanding by combining general relativity with astronomical observations, those of Hubble in particular.
In 1931, Einstein accepts the theory of an expanding universe and proposes, in 1931, a model of a continuously expanding universe with zero cosmological constant (the Friedmann-Einstein model).[18][19] This is followed by another model, in 1932, with the Dutch physicist and astronomerWillem de Sitter in which both the cosmological constant and spatial curvature are set to zero (the Einstein–de Sitter model).
In 1998, two teams of astrophysicists, theSupernova Cosmology Project and theHigh-Z Supernova Search Team, carried out measurements on distant supernovae which showed that the speed of galaxies' recession in relation to theMilky Way increases over time. The universe is in accelerated expansion, which requires having a strictly positiveΛ. 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 (seeStandard cosmological model). For this work, Perlmutter, Schmidt, and Riess jointly received theNobel Prize in Physics in 2011.
Estimated ratios ofdark matter and dark energy (which may be the cosmological constant) in the universe. This image is made by NASA using the 9 year WMAP data. This is the final WMAP release.
The cosmological constantΛ appears in theEinstein field equations in the formwherethe Ricci tensorRμν, Ricci scalarR and themetric tensorgμν describe the structure ofspacetime, thestress–energy tensorTμν describes the energy density, momentum density and stress at that point in spacetime, andκ = 8πG/c4. Thegravitational constantG and thespeed of lightc are universal constants. WhenΛ is zero, this reduces to the field equation of general relativity usually used in the 20th century. WhenTμν is zero, the field equation describes empty space (avacuum).
The cosmological constant has the same effect as an intrinsicenergy density of the vacuum,ρvac (and an associatedpressure). In this context, it is commonly moved to the right-hand side of the equation usingΛ =κρvac. It is common to quote values of energy density directly, though still using the name "cosmological constant". The dimension ofΛ is generally understood as length−2.
Using the Planck units, and the value evaluated in 2025 for theHubble constantH0 =76.5±2.2 (km/s)/Mpc =(2.48±0.07)×10−18 s−1,[20]Λ has the value of where is thePlanck 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. (SeeDark energy andCosmic inflation for details.)
The dimensionless density parameter Ω represents the ratio of the actual density of a component of the universe to the critical density. The total density parameter for a flat universe (concluded byWMAP), can be expressed as:
Where:
Ωm is the matter density parameter (including both baryonic and dark matter);
ΩΛ is the density parameter for dark energy (cosmological constant); and
Ωk describes the curvature of the universe which is 0 in a flat universe.
Instead of the cosmological constant itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and thecritical density of the universe, the tipping point for a sufficient density to stop the universe from expanding forever (the dark energy density parameter). This ratio is estimated to be 0.714, according to results published by thePlanck Collaboration in 2018 and clearly mentioned onWMAP.[21] More intuitively, this parameter could be described as the fraction of the universe that is made up of dark energy. Note that this value changes over time: The critical density changes withcosmological 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.[22][23][24]
Another ratio that is used by scientists is theequation of state, usually denotedw, which is the ratio of pressure that dark energy puts on the universe to the energy per unit volume.[25] This ratio isw = −1 for the cosmological constant used in the Einstein equations; alternative time-varying forms of vacuum energy such asquintessence generally use a different value. The valuew =−1.028±0.032, measured by the Planck Collaboration (2018)[26] is consistent with−1, assumingw does not change over cosmic time.
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 forType Ia supernovae[5] indicated that the expansion of the universe is accelerating, if one assumes thecosmological principle.[6][7] When combined with measurements of thecosmic microwave background radiation these implied a value ofΩΛ ≈ 0.7,[27] a result which has been supported and refined by more recent measurements[28] (as well as previous works[29][30]). If one assumes the cosmological principle, as in the case for all models that use theFriedmann–Lemaître–Robertson–Walker metric, while there are other possible causes of anaccelerating universe, such as quintessence, the cosmological constant is in most respects thesimplest 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 of10−52 m−2. It may be expressed as10−35 s−2 (multiplying byc2 ≈1017 m2⋅s−2) or as 10−122ℓP−2[31] (whereℓP is the Planck length). The value is based on recent measurements of vacuum energy density,ρvac =5.96×10−27 kg/m3 ≘5.3566×10−10 J/m3 =3.35 GeV/m3.[32] However, due to theHubble tension and theCMB 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,[33][34][35] 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 maximumentropy of the observable universe (seeHolographic principle).[36]
A major outstandingproblem is that mostquantum field theories predict a huge value for thequantum vacuum. A common assumption is that thequantum vacuum is equivalent to the cosmological constant. Although no theory exists that supports this assumption, arguments can be made in its favor.[37]
Such arguments are usually based ondimensional analysis andeffective field theory. If the universe is described by an effective local quantum field theory down to thePlanck scale, then we would expect a cosmological constant of the order of ( 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]
Somesupersymmetric theories require a cosmological constant that is exactly zero, which further complicates things. This is the cosmological constant problem, the worst problem offine-tuning inphysics: there is no known natural way to derive the tiny cosmological constant used incosmology fromparticle physics.
No vacuum in thestring 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 thatno such universe exists.[38]
One possible explanation for the small but non-zero value was noted bySteven Weinberg in 1987 following theanthropic principle.[39] 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.[40] In 1992, Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.[41]
This argument depends on the vacuum energy density being constant throughout spacetime, 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 residualinflaton (also seeQuintessence). Another theoretical approach that deals with the issue is that ofmultiverse 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 theinverse gambler's fallacy.
In 1995, Weinberg's argument was refined byAlexander Vilenkin to predict a value for the cosmological constant that was only ten times the matter density,[42] i.e. about three times the current value since determined.
An attempt to directly observe and relate quanta or fields like thechameleon particle or thesymmetron theory to dark energy, in a laboratory setting, failed to detect a new force.[43] Inferring the presence of dark energy through its interaction with baryons in thecosmic microwave background has also led to a negative result,[44] although the current analyses have been derived only at the linear perturbation regime. It is also possible that the difficulty in detecting dark energy is due to the fact that the cosmological constant describes an existing, known interaction (e.g. electromagnetic field).[45]
^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
^On the Cosmological Constant being thought to have zero value see for example:
"Since the cosmological upper bound on |⟨ρ⟩ +λ/8πG| 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
"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
"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
^There is some debate over whether Einstein labelled the cosmological constant his "biggest blunder", with most 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 autobiographyMy World Line (1970) and aScientific American article from September 1956."Rosen (2013)
" On the other hand, the science historians Cormac O'Raifeartaigh and Simon Mitton point out that two contemporaries of Gamow, Ralph Alpher and John Wheeler, both reported hearing Einstein use the term "biggest blunder" on at least one occasion".O'Raifeartaigh & Mitton (2018)
Perlmutter, S.; Aldering, G.; Valle, M. Della; Deustua, S.; Ellis, R.S.; Fabbro, S.; Fruchter, A.; Goldhaber, G.; Groom, D.E.; Hook, I.M.; Kim, A.G.; Kim, M.Y.; Knop, R. A.; Lidman, C.; McMahon, R.G.; Nugent, P.; Pain, R.; Panagia, N.; Pennypacker, C. R.; Ruiz-Lapuente, P.; Schaefer, B.; Walton, N. (1998). "Discovery of a supernova explosion at half the age of the Universe".Nature.391 (6662):51–54.arXiv:astro-ph/9712212.Bibcode:1998Natur.391...51P.doi:10.1038/34124.ISSN0028-0836.S2CID4329577.
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