View source for Cosmological constant

{{Short description|Value representing energy density of space}}
[[File:CMB Timeline300 no WMAP.jpg|thumb|upright=1.8|Sketch of the [[Timeline of cosmological epochs|timeline of the Universe]] in the [[Lambda-CDM model|ΛCDM model]]. The accelerated expansion in the last third of the timeline represents the [[scale factor (cosmology)|dark-energy dominated era]].]]
{{Cosmology}}

In [[physical cosmology|cosmology]], the '''cosmological constant''' (usually denoted by the Greek capital letter [[lambda]]: {{math|Λ}}), alternatively called '''Einstein's cosmological constant''', 
is a [[coefficient]] that [[Albert Einstein]] initially added to his [[Einstein field equations| field equations]] of [[general relativity]]. He later removed it; however, much later it was revived to express the energy density of space, or [[vacuum energy]], that arises in [[quantum mechanics]]. It is closely associated with the concept of [[dark energy]].<ref name="CC Definition" />

Einstein introduced the constant in 1917<ref name=":0">{{harvp|Einstein|1917}}.</ref> to counterbalance the effect of gravity and achieve a [[static universe]], which was then assumed. Einstein's cosmological constant was abandoned after [[Edwin Hubble]] confirmed that the universe was expanding.<ref name="Rugh 2001 3"/> From the 1930s until the late 1990s, most physicists thought the cosmological constant to be zero.<ref name="Λ = 0?"/> That changed with the discovery in 1998 that the [[accelerating expansion of the universe|expansion of the universe is accelerating]], implying that the cosmological constant may have a positive value after all.<ref name="1998 Discovery"/>

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 dark energy.<ref name="Ellis 2009">{{cite journal|last=Ellis|first=G. F. R.|title=Dark energy and inhomogeneity|journal=Journal of Physics: Conference Series|volume=189|doi=10.1088/1742-6596/189/1/012011|year=2009|issue=1 |article-number=012011|bibcode=2009JPhCS.189a2011E |s2cid=250670331 |doi-access=free}}</ref><ref name="Colin et al">{{cite journal |last1=Colin |first1=Jacques |last2=Mohayaee |first2=Roya |last3=Rameez |first3=Mohamed |last4=Sarkar |first4=Subir |date=20 November 2019 |title=Evidence for anisotropy of cosmic acceleration |url=https://www.aanda.org/articles/aa/full_html/2019/11/aa36373-19/aa36373-19.html |journal=Astronomy and Astrophysics |volume=631 |pages=L13 |arxiv=1808.04597 |bibcode=2019A&A...631L..13C |doi=10.1051/0004-6361/201936373 |s2cid=208175643 |access-date=25 March 2022}}</ref><ref>{{harvp|Redd|2013}}</ref> The cosmological constant {{math|Λ}} is the simplest possible explanation for dark energy, and is used in the standard model of cosmology known as the [[Lambda-CDM model|ΛCDM model]].

According to [[quantum field theory]] (QFT), which underlies modern [[particle physics]], empty space is defined by the [[vacuum state]], which is composed of a collection of [[quantum fields]]. All these quantum fields exhibit fluctuations in their [[ground state]] (lowest energy density) arising from the [[zero-point energy]] existing everywhere in space. These zero-point fluctuations should contribute to the cosmological constant {{math|Λ}}, but actual calculations give rise to an enormous vacuum energy.<ref>{{harvp|Rugh|Zinkernagel|2001|p=1}}</ref> 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&nbsp;orders of magnitude, a discrepancy that has been called "the worst theoretical prediction in the history of physics!".<ref name="CC Problem"/> 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".<ref name="CC Problem 3"/>

== History ==
In the ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' (1687), [[Isaac Newton]] identified a central force proportional to distance as one of the "two principal cases of attraction" alongside the [[inverse-square law]] of gravity; in modern physics, this linear force law is mathematically equivalent to the force associated with a cosmological constant.<ref>{{Cite journal |last=Calder |first=Lucy |last2=Lahav |first2=Ofer |author-link2=Ofer Lahav |date=2008 |title=Dark energy: back to Newton? |url=https://academic.oup.com/astrogeo/article-lookup/doi/10.1111/j.1468-4004.2008.49113.x |journal=Astronomy & Geophysics |language=en |volume=49 |issue=1 |pages=1.13–1.18 |doi=10.1111/j.1468-4004.2008.49113.x |issn=1366-8781|arxiv=0712.2196 }}</ref><ref name=":1752">{{cite book |last=Rowlands |first=Peter |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=189–190 |doi=10.1142/q0108}}</ref>

The cosmological constant was originally introduced in [[Albert Einstein]]'s 1917 paper entitled ''Cosmological considerations in the General Theory of Relativity''.<ref name=":0" /> 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 for a [[static universe]]: gravity would cause a universe that was initially non-expanding to contract. To counteract this possibility, Einstein added the cosmological constant.<ref name="Rugh 2001 3" /> However, he 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".<ref>{{Cite journal |last=Kirshner |first=Robert P. |date=2004-01-06 |title=Hubble's diagram and cosmic expansion |journal=Proceedings of the National Academy of Sciences |language=en |volume=101 |issue=1 |pages=8–13 |doi=10.1073/pnas.2536799100 |doi-access=free |issn=0027-8424 |pmc=314128 |pmid=14695886}}</ref> Einstein's static universe is unstable against matter density perturbations.<ref>{{Cite journal |last=Eddington |first=A. S. |date=1930-05-09 |title=On the Instability of Einstein's Spherical World |journal=Monthly Notices of the Royal Astronomical Society |volume=90 |issue=7 |pages=668–678 |doi=10.1093/mnras/90.7.668 |doi-access=free |issn=0035-8711}}</ref> Furthermore, without the cosmological constant Einstein could have found the expansion of the universe before Hubble's observations.<ref name=":1">{{Cite journal |last=Hubble |first=Edwin |date=1929-03-15 |title=A relation between distance and radial velocity among extra-galactic nebulae |journal=Proceedings of the National Academy of Sciences |language=en |volume=15 |issue=3 |pages=168–173 |doi=10.1073/pnas.15.3.168 |doi-access=free |issn=0027-8424 |pmc=522427 |pmid=16577160|bibcode=1929PNAS...15..168H }}</ref>

In 1929, not long after Einstein developed his static theory, observations by [[Edwin Hubble]]<ref name=":1" /> 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|Alexander 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" (according to [[George Gamow]]).<ref name="Biggest Blunder" />

It transpired 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 that contracts slightly will continue contracting.<ref>{{harvp|Ryden|2003|p=59}}</ref>

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.<ref name="1998 Discovery"/> The explanation of this small but positive value is a remaining 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, [[Arthur 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]]",<ref>{{Cite book |last=Ryckman |first=Thomas |url=https://books.google.com/books?id=5Hw8DwAAQBAJ&pg=PA10 |title=The Reign of Relativity: Philosophy in Physics 1915-1925 |date=2005 |publisher=Oxford University Press |isbn=978-0-19-517717-6 |series= |location= |pages=10-11}}</ref> 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.

In 1990s, [[Saul Perlmutter]] at Lawrence Berkeley National Laboratory, [[Brian Schmidt]] of the Australian National University and [[Adam 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.<ref>{{Cite journal |last1=Riess |first1=Adam G. |last2=Filippenko |first2=Alexei V. |last3=Challis |first3=Peter |last4=Clocchiatti |first4=Alejandro |last5=Diercks |first5=Alan |last6=Garnavich |first6=Peter M. |last7=Gilliland |first7=Ron L. |last8=Hogan |first8=Craig J. |last9=Jha |first9=Saurabh |last10=Kirshner |first10=Robert P. |last11=Leibundgut |first11=B. |last12=Phillips |first12=M. M. |last13=Reiss |first13=David |last14=Schmidt |first14=Brian P. |last15=Schommer |first15=Robert A. |date=1998 |title=Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant |url=https://iopscience.iop.org/article/10.1086/300499 |journal=The Astronomical Journal |volume=116 |issue=3 |pages=1009–1038 |doi=10.1086/300499|arxiv=astro-ph/9805201 |bibcode=1998AJ....116.1009R }}</ref> Following this discovery, the cosmological constant was reinserted in the general relativity equations.

=== Sequence of events 1915–1998 ===
* In 1915, Einstein publishes his equations of general relativity, without a cosmological constant {{math|Λ}}.
* In 1917, Einstein adds the parameter {{math|Λ}} 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 physicist [[Alexander Friedmann]] mathematically shows that Einstein's equations (whatever {{math|Λ}}) remain valid in a dynamic universe.
* In 1927, the Belgian astrophysicist [[Georges 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 universe]]).<ref>{{Citation |last=Einstein |first=A. |title=Zum kosmologischen Problem der allgemeinen Relativitätstheorie |date=2005 |work=Albert Einstein: Akademie-Vorträge |pages=361–364 |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/3527608958.ch43 |access-date=2025-10-19 |publisher=John Wiley & Sons, Ltd |language=en |doi=10.1002/3527608958.ch43 |isbn=978-3-527-60895-9|url-access=subscription }}</ref><ref>{{Cite journal |last=O’Raifeartaigh |first=C. |last2=McCann |first2=B. |date=2014-02-01 |title=Einstein’s cosmic model of 1931 revisited: an analysis and translation of a forgotten model of the universe |url=https://doi.org/10.1140/epjh/e2013-40038-x |journal=The European Physical Journal H |language=en |volume=39 |issue=1 |pages=63–85 |doi=10.1140/epjh/e2013-40038-x |issn=2102-6467|arxiv=1312.2192 }}</ref> This is followed by another model, in 1932, with the Dutch physicist and astronomer [[Willem de Sitter]] in which both the cosmological constant and spatial curvature are set to zero (the [[Einstein–de Sitter universe]]).
* In 1998, two teams of astrophysicists, the [[Supernova Cosmology Project]] and the [[High-Z Supernova Search Team]], 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 {{math|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]]''). {{pb}} For this work, Perlmutter, Schmidt, and Riess jointly received the [[Nobel Prize in Physics]] in 2011.

== Equation ==
[[File:121236 NewPieChart320.png|thumb|Estimated ratios of [[dark 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 constant {{math|Λ}} appears in the [[Einstein field equations]] in the form
<math display=block>R_{\mu \nu} - \tfrac{1}{2} R\, g_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} ,</math>
where [[Ricci tensor|the Ricci tensor]] {{math|''R<sub>μν</sub>''}}, Ricci scalar {{math|''R''}} and the [[metric tensor (general relativity)|metric tensor]] {{math|''g<sub>μν</sub>''}} describe the structure of [[spacetime]], the [[stress–energy tensor]] {{math|''T<sub>μν</sub>''}} describes the energy density, momentum density and stress at that point in spacetime, and {{math|1=''κ'' = 8''πG''/''c''{{sup|4}}}}. The [[gravitational constant]] {{mvar|G}} and the [[speed of light]] {{mvar|c}} are universal constants. When {{math|Λ}} is zero, this reduces to the field equation of general relativity usually used in the 20th&nbsp;century. When {{math|''T<sub>μν</sub>''}} is zero, the field equation describes empty space (a [[vacuum]]).

The cosmological constant has the same effect as an intrinsic [[energy density]] of the vacuum, ''{{mvar|ρ}}''{{sub|vac}} (and an associated [[pressure]]). In this context, it is commonly moved to the right-hand side of the equation using {{math|1=Λ = ''κρ''{{sub|vac}}}}. It is common to quote values of energy density directly, though still using the name "cosmological constant". The dimension of {{math|Λ}} is generally understood as length{{sup|−2}}.

Using the Planck units, and the value evaluated in 2025 for the [[Hubble's law|Hubble constant]] ''{{mvar|H}}''<sub>0</sub> = {{val|76.5|2.2|u=(km/s)/Mpc}} = {{val|2.48|0.07|u=s-1|e=-18}},<ref>{{Cite journal |last1=Scolnic |first1=Daniel |author1-link=Daniel M. Scolnic|last2=Riess |first2=Adam G. |last3=Murakami |first3=Yukei S. |last4=Peterson |first4=Erik R. |last5=Brout |first5=Dillon |last6=Acevedo |first6=Maria |last7=Carreres |first7=Bastien |last8=Jones |first8=David O. |last9=Said |first9=Khaled |last10=Howlett |first10=Cullan |last11=Anand |first11=Gagandeep S. |date=2025-01-15 |title=The Hubble Tension in Our Own Backyard: DESI and the Nearness of the Coma Cluster |journal=The Astrophysical Journal Letters |volume=979 |issue=1 |pages=L9 |doi=10.3847/2041-8213/ada0bd |doi-access=free |arxiv=2409.14546 |bibcode=2025ApJ...979L...9S |issn=2041-8205}}</ref> {{math|Λ}} has the value of <math display="block">\begin{align}
\Lambda = 3\, \left( \frac{\,H_0\,}{c} \right)^2 \Omega_\Lambda &= 1.4657 \times 10^{-52}\ \text{m}^{-2} \\
&= 3.827 \times 10^{-123} \,l_{\text{P}}^{-2}
\end{align}</math> where <math display="inline">l_{\text{P}}</math> 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.)

=== Density parameter Ω ===
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 by [https://wmap.gsfc.nasa.gov/universe/uni_shape.html WMAP]), can be expressed as:

<math display="block">\Omega_\text{total} = \Omega_m + \Omega_{\Lambda} + \Omega_k = 1</math>

Where:

* Ω<sub>''m''</sub> is the matter density parameter (including both baryonic and dark matter);
* Ω<sub>Λ</sub> is the density parameter for dark energy (cosmological constant); and
* Ω<sub>''k''</sub> 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 the [[Friedmann equations#Density parameter|critical 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 the [[Planck Collaboration]] in 2018 and clearly mentioned on [https://wmap.gsfc.nasa.gov/universe/uni_matter.html WMAP].<ref name="Planck-20182">{{harvp|The Planck Collaboration|2020}}.</ref> 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 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.<ref>{{Cite web |last=Siegel |first=Ethan |title=Dark Energy May Not Be A Constant, Which Would Lead To A Revolution In Physics |url=https://www.forbes.com/sites/startswithabang/2019/01/31/dark-energy-may-not-be-a-constant-which-would-lead-to-a-revolution-in-physics/ |access-date=2023-09-10 |website=Forbes |language=en}}</ref>{{sfnp|Peebles|Ratra|2003}}<ref>{{Cite book |last=Davies |first=Paul |title=What's Eating the Universe?: And Other Cosmic Questions |publisher=Penguin Books Australia |year=2021 |isbn=978-0-14-199372-0 |language=en}}</ref>

=== Equation of state ===
Another ratio that is used by scientists is the [[Equation of state (cosmology)|equation of state]], usually denoted {{mvar|w}}, which is the ratio of pressure that dark energy puts on the universe to the energy per unit volume.<ref>{{harvp|Brumfiel|2007|p=246}}.</ref> This ratio is {{nowrap|{{mvar|w}} {{=}} −1}} for the cosmological constant used in the Einstein equations; alternative time-varying forms of vacuum energy such as [[quintessence (physics)|quintessence]] generally use a different value. The value {{mvar|w}} = {{val|-1.028|0.032}}, measured by the Planck Collaboration (2018)<ref name="Planck-2018">{{harvp|The Planck Collaboration|2020}}.</ref> is consistent with {{val|-1}}, assuming {{mvar|w}} does not change over cosmic time.

== Value ==
[[File:Lambda-Cold Dark Matter, Accelerated Expansion of the Universe, Big Bang-Inflation.jpg|thumb|upright=2.4|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]]<ref name="1998 Discovery"/> indicated that the expansion of the universe is accelerating, if one assumes the [[cosmological principle]].<ref name="Ellis 2009"/><ref name="Colin et al"/> When combined with measurements of the [[cosmic microwave background radiation]] these implied a value of {{math|Ω<sub>Λ</sub>}} ≈ 0.7,<ref>See e.g. {{harvp|Baker et al.|1999}}.</ref> a result which has been supported and refined by more recent measurements<ref>See for example Table 9 in {{harvp|The Planck Collaboration|2015a|p=27}}.</ref> (as well as previous works<ref>{{cite journal|bibcode=1992Ap&SS.191..107P|title=Inflation and compactification from Galaxy redshifts?|year=1992 |doi = 10.1007/BF00644200 |last1=Paál |first1=G. |last2=Horváth |first2=I. |last3=Lukács |first3=B. |journal=Astrophysics and Space Science |volume=191 |issue=1 |pages=107–124 |s2cid=116951785 }}</ref><ref>{{cite journal|bibcode=1994Ap&SS.222...65H|title=Once More on Quasar Periodicities|year=1994 |doi = 10.1007/BF00627083 |last1=Holba |first1=Ágnes |last2=Horváth |first2=I. |last3=Lukács |first3=B. |last4=Paál |first4=G. |journal=Astrophysics and Space Science |volume=222 |issue=1–2 |pages=65–83 |s2cid=118379051 }}</ref>). If one assumes the cosmological principle, as in the case for all models that use 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 [[Occam's razor|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 {{val|e=-52|u=m-2}}. It may be expressed as {{val|e=-35|u=s-2}} (multiplying by {{nowrap|{{math|''c''{{sup|2}}}} ≈ {{val|e=17|u=m<sup>2</sup>⋅s<sup>−2</sup>}}}}) or as 10<sup>−122</sup>&nbsp;''ℓ''{{sub|P}}<sup>−2</sup><ref>{{harvp|Barrow|Shaw|2011}}.</ref> (where ''ℓ''{{sub|P}} is the Planck length). The value is based on recent measurements of vacuum energy density, {{math|''ρ''{{sub|vac}}}} = {{val|5.96|e=-27|u=kg/m3}} ≘ {{val|5.3566|e=-10|u=J/m3}} = {{val|3.35|u=GeV/m3}}.<ref>Calculated based on the Hubble constant and {{math|Ω{{sub|Λ}}}} from {{harvp|The Planck Collaboration|2015b}}.</ref> 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,<ref name="Snowmass21">{{citation |last1=Abdalla |first1=Elcio |title=Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies |date=11 Mar 2022 |journal=Journal of High Energy Astrophysics |volume=34 |page=49 |arxiv=2203.06142v1 |bibcode=2022JHEAp..34...49A |doi=10.1016/j.jheap.2022.04.002 |s2cid=247411131 |display-authors=2 |last2=Abellán |first2=Guillermo Franco |last3=Aboubrahim |first3=Amin}}.</ref><ref name="FLRW breakdown">{{cite journal |last1=Krishnan |first1=Chethan |last2=Mohayaee |first2=Roya |last3=Colgáin |first3=Eoin Ó |last4=Sheikh-Jabbari |first4=M. M. |last5=Yin |first5=Lu |title=Does Hubble Tension Signal a Breakdown in FLRW Cosmology? |journal=Classical and Quantum Gravity |date=16 September 2021 |volume=38 |issue=18 |page=184001 |doi=10.1088/1361-6382/ac1a81 |arxiv=2105.09790 |bibcode=2021CQGra..38r4001K |s2cid=234790314 |issn=0264-9381}}</ref><ref name="Heinesen-MacPherson">{{cite journal |last1=Heinesen |first1=Asta |last2=Macpherson |first2=Hayley J. |date=15 July 2021 |title=Luminosity distance and anisotropic sky-sampling at low redshifts: A numerical relativity study |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.023525 |journal=Physical Review D |volume=104 |issue=2 |article-number=023525 |arxiv=2103.11918 |bibcode=2021PhRvD.104b3525M |doi=10.1103/PhysRevD.104.023525 |s2cid=232307363 |access-date=25 March 2022}}</ref> 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.<ref name="Ellis 2009"/><ref name="Colin et al"/>

As was only recently seen, by works of [[Gerard 't Hooft|'t Hooft]], [[Leonard Susskind|Susskind]] and others, a positive cosmological constant has surprising consequences, such as a finite maximum [[entropy]] of the observable universe (see ''[[Holographic principle]]'').<ref>{{harvp|Dyson|Kleban|Susskind|2002}}.</ref>

== Predictions ==

=== Quantum field theory <span id="Cosmological constant problem"></span> ===
{{see also|Cosmological constant problem}}
{{unsolved|physics|Why does the [[zero-point energy]] of the quantum vacuum not 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 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>{{harvp|Rugh|Zinkernagel|2001|p=?}}</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 display=inline>M_{\rm pl}^2</math> (<math display=inline>1</math> in reduced Planck units). 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="CC Problem"/>

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 (physics)|fine-tuning]] in [[physics]]: there is no known natural way to derive the tiny cosmological constant used in [[physical cosmology|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 [[Swampland (physics)|no such universe exists]].<ref>{{cite web |last1=Wolchover |first1=Natalie |title=Dark Energy May Be Incompatible With String Theory |url=https://www.quantamagazine.org/dark-energy-may-be-incompatible-with-string-theory-20180809/ |website=[[Quanta Magazine]] |publisher=Simons Foundation |access-date=2 April 2020 |date=9 August 2018}}</ref>

=== Anthropic principle ===
One possible explanation for the small but non-zero value was noted by [[Steven Weinberg]] in 1987 following the [[anthropic principle]].<ref>{{harvp|Weinberg|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. 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.<ref>{{harvp|Vilenkin|2006|pages=138–139}}.</ref> In 1992, Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.<ref>{{harvp|Weinberg|1992|p=182}}.</ref>

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 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>{{harvp|Vilenkin|2006|page=146}}.</ref> i.e. about three times the current value since determined.

=== Failure to detect dark energy ===
An attempt to directly observe and relate quanta or fields like the [[chameleon particle]] or the [[symmetron]] theory to dark energy, in a laboratory setting, failed to detect a new force.<ref>{{cite journal |last1=Sabulsky |first1=D. O. |last2=Dutta |first2=I. |last3=Hinds |first3=E. A. |last4=Elder |first4=B. |last5=Burrage |first5=C/ |last6=Copeland |first6=E. J. |year=2019 |title=Experiment to Detect Dark Energy Forces Using Atom Interferometry |journal=Physical Review Letters |volume=123 |issue=6 |article-number=061102 |arxiv=1812.08244 |bibcode=2019PhRvL.123f1102S |doi=10.1103/PhysRevLett.123.061102 |pmid=31491160 |s2cid=118935116}}</ref> Inferring the presence of dark energy through its interaction with baryons in the [[cosmic microwave background]] has also led to a negative result,<ref>{{cite journal |last1=Vagnozzi |first1=S. |last2=Visinelli |first2=L. |last3=Mena |first3=O. |last4=Mota |first4=D. |year=2020 |title=Do we have any hope of detecting scattering between dark energy and baryons through cosmology? |journal=Mon. Not. R. Astron. Soc. |volume=493 |issue=1 |page=1139 |arxiv=1911.12374 |bibcode=2020MNRAS.493.1139V |doi=10.1093/mnras/staa311 |doi-access=free}}</ref> 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).<ref>{{Cite journal |last=Ogonowski |first=Piotr |date=2023-01-09 |title=Proposed method of combining continuum mechanics with Einstein Field Equations |url=https://www.worldscientific.com/doi/10.1142/S0218271823500104 |journal=International Journal of Modern Physics D |volume=32 |issue=3 |language=en |pages=2350010–2350024 |doi=10.1142/S0218271823500104 |issn=0218-2718|arxiv=2212.13113 |bibcode=2023IJMPD..3250010O |s2cid=254778036 }}</ref>

== See also ==

{{div col}}
* [[Big Rip]]
* [[Higgs mechanism]]
* [[Lambdavacuum solution]]
* [[Hierarchy problem]]
* [[Quantum electrodynamics]]
* [[de Sitter invariant special relativity]]
* [[Unruh effect]]
{{div col end}}

== References ==
=== Footnotes ===
{{reflist|30em|refs=
<ref name="CC Definition">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." {{harvp|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 ({{math|Λ}}) or quintessence (Q), are leading candidates." {{harvp|Caldwell|2002|p=2}}
</ref>
<ref name="Rugh 2001 3">{{harvp|Rugh|Zinkernagel|2001|p=3}}</ref>
<ref name="Λ = 0?">On the Cosmological Constant being thought to have zero value see for example:
* "Since the cosmological upper bound on {{abs|{{angle bracket|''ρ''}} + ''λ''/8''πG''}} was vastly less than any value expected from particle theory, most particle theorists simply assumed that for some unknown reason this quantity was zero." {{harvp|Weinberg|1989|p=3}}
* "An epochal astronomical discovery would be to establish by convincing observation that Λ is nonzero." {{harvp|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&nbsp;<&nbsp;10<sup>−120</sup>&nbsp;Planck units) that many particle physicists suspected that some fundamental principle must force its value to be precisely zero." {{harvp|Barrow|Shaw|2011|p=1}}
* "The only other natural value is Λ&nbsp;=&nbsp;0. If Λ really is tiny but not zero, it adds a most stimulating though enigmatic clue to physics to be discovered." {{harvp|Peebles|Ratra|2003|p=333}}
</ref>
<ref name="1998 Discovery">See for example:
* "This is the independent result of two teams. [[Supernova Cosmology Project]] ({{harvp|Perlmutter et al.|1999}}; also see {{harvp|Perlmutter et al.|1998}}) and the [[High-Z Supernova Search Team]] ({{harvp|Riess et al.|1998}}; also see {{harvp|Schmidt et al.|1998}})" {{harvp|Weinberg|2015|p=376}}
</ref>
<ref name="Biggest Blunder">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 {{harvs|txt|last1=Gamow|year1=1956|year2=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&nbsp;1956." {{harvp|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". {{harvp|O'Raifeartaigh|Mitton|2018}}
</ref>
<ref name="CC Problem">See for example:
* "This gives an answer about 120&nbsp;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!" {{harvp|Hobson|Efstathiou|Lasenby|2006|p=187}}
* "This, as we will see later, is approximately 120&nbsp;orders of magnitude larger than what is allowed by observation." {{harvp|Carroll|Press|Turner|1992|p=503}}
* "Theoretical expectations for the cosmological constant exceed observational limits by some 120&nbsp;orders of magnitude." {{harvp|Weinberg|1989|p=1}}</ref>
<ref name="CC Problem 3">See for example:
* "the vacuum holds the key to a full understanding of nature" {{harvp|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." {{harvp|Hobson|Efstathiou|Lasenby|2006|p=188}}</ref>
}}

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* {{cite journal |last1=Riess |first1=A.G. |last2=Filippenko |first2=A.V. |last3=Challis |first3=P. |last4=Clocchiatti |first4=A. |last5=Diercks |first5=A. |last6=Garnavich |first6=P.M. |last7=Gilliland |first7=R.L. |last8=Hogan|first8=C.J. |last9=Jha|first9=S. |last10=Kirshner |first10=R.P. |last11=Leibundgut|first11=B. |last12=Phillips |first12=M.M. |last13=Reiss |first13=D. |last14=Schmidt |first14=B.P. |last15=Schommer |first15=R.A. |last16=Smith |first16=R.C. |last17=Spyromilio |first17=J. |last18=Stubbs |first18=C. |last19=Suntzeff |first19=N.B. |last20=Tonry |first20=J. |title=Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant |journal=The Astronomical Journal |volume=116 |issue=3 |year=1998 |pages=1009–1038 |issn=0004-6256 |doi=10.1086/300499|bibcode=1998AJ....116.1009R |arxiv=astro-ph/9805201 |s2cid=15640044 |ref={{harvid|Riess et al.|1998}} }}
* {{cite journal |last1=Schmidt |first1=B.P. |last2=Suntzeff |first2=N.B. |last3=Phillips |first3=M.M. |last4=Schommer |first4=R.A. |last5=Clocchiatti |first5=A. |last6=Kirshner |first6=R.P. |last7=Garnavich |first7=P. |last8=Challis|first8=P. |last9=Leibundgut |first9=B. |last10=Spyromilio |first10=J. |last11=Riess |first11=A.G. |last12=Filippenko |first12=A.V. |last13=Hamuy |first13=M. |last14=Smith |first14=R. C. |last15=Hogan |first15=C. |last16=Stubbs |first16=C. |last17=Diercks |first17=A.|last18=Reiss |first18=D. |last19=Gilliland |first19=R. |last20=Tonry |first20=J. |last21=Maza |first21=J. |last22=Dressler |first22=A. |last23=Walsh |first23=J. |last24=Ciardullo |first24=R. |title=The High-Z Supernova Search: Measuring Cosmic Deceleration and Global Curvature of the Universe Using Type Ia Supernovae |journal=The Astrophysical Journal |volume=507 |issue=1 |year=1998 |pages=46–63 |issn=0004-637X |doi=10.1086/306308|bibcode=1998ApJ...507...46S |arxiv=astro-ph/9805200 |s2cid=15762698 |ref={{harvid|Schmidt et al.|1998}} }}
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{{refend}}

==== Secondary literature: news, popular science articles & books ====
{{refbegin|30em}}
* {{cite journal |last1=Abbott |first1=Larry |s2cid=30023659 |author-link1=Larry Abbott |title=The Mystery of the Cosmological Constant |journal=Scientific American |volume=258 |issue=5 |year=1988 |pages=106–113 |issn=0036-8733 |doi=10.1038/scientificamerican0588-106 |bibcode=1988SciAm.258e.106A}}
* {{cite journal |last1=Barrow |first1=J. D. |last2=Webb |first2=J. K. |title=Inconstant Constants |journal=Scientific American |volume=292 |issue=6 |year=2005 |pages=56–63 |issn=0036-8733 |doi=10.1038/scientificamerican0605-56 |pmid=15934653 |bibcode=2005SciAm.292f..56B |url=http://phys.unsw.edu.au/~jkw/hons/barr_webb_sciam11.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://phys.unsw.edu.au/~jkw/hons/barr_webb_sciam11.pdf |archive-date=2022-10-09 |url-status=live}}
* {{cite journal |last1=Brumfiel |first1=G. |title=A constant problem |journal=Nature |volume=448 |issue=7151 |year=2007 |pages=245–248 |issn=0028-0836 |doi=10.1038/448245a |pmid=17637631 |bibcode=2007Natur.448..245B |s2cid=4428576 |url=https://hosting.astro.cornell.edu/academics/courses/astro1109/readings/AConstantProblem.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://hosting.astro.cornell.edu/academics/courses/astro1109/readings/AConstantProblem.pdf |archive-date=2022-10-09 |url-status=live |doi-access=free}}
* {{cite book |last1=Davies |first1=P. C. W. |author-link1=Paul Davies |title=Superforce: The Search for a Grand Unified Theory of Nature |date=1985 |publisher=Simon and Schuster |location=New York |isbn=978-0-671-47685-4 |url=https://books.google.com/books?id=Bna5p4vJtucC |lccn=84005473 |oclc=12397205}}
* {{cite journal |last1=Hogan |first1=J. |title=Welcome to the dark side |journal=Nature |volume=448 |issue=7151 |year=2007 |pages=240–245 |issn=0028-0836 |doi=10.1038/448240a |pmid=17637630 |url=https://pa.brown.edu/articles/Nature_Vol448_20070719_UnseenUniverse.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://pa.brown.edu/articles/Nature_Vol448_20070719_UnseenUniverse.pdf |archive-date=2022-10-09 |url-status=live |bibcode=2007Natur.448..240H |s2cid=4415960 |doi-access=free}}
* {{cite journal |last1=O'Raifeartaigh |first1=C. |last2=Mitton |first2=S. |title=Einstein's "biggest blunder" – interrogating the legend |journal=Physics in Perspective |volume=20 |issue=4 |pages=318–341 |date=2018 |arxiv=1804.06768 |doi=10.1007/s00016-018-0228-9 |s2cid=119097586}}
* {{cite web |last1=Redd |first1=N. T. |title=What is Dark Energy? |url=https://www.space.com/20929-dark-energy.html |website=space.com |archive-url=https://archive.today/20160519162257/http://www.space.com/20929-dark-energy.html |archive-date=19 May 2016 |date=2013 |access-date=28 October 2018 |url-status=live}}
* {{cite web |last1=Rosen |first1=R. J. |title=Einstein Likely Never Said One of His Most Oft-Quoted Phrases |url=https://www.theatlantic.com/technology/archive/2013/08/einstein-likely-never-said-one-of-his-most-oft-quoted-phrases/278508/ |website=theatlantic.com |publisher=The Atlantic |archive-url=https://archive.today/20130810062508/http://www.theatlantic.com/technology/archive/2013/08/einstein-likely-never-said-one-of-his-most-oft-quoted-phrases/278508/ |archive-date=10 August 2013 |date=2013 |access-date=6 March 2017 |url-status=live}}
{{refend}}

==== Secondary literature: review articles, monographs and textbooks ====
{{refbegin|30em}}
* {{cite journal |last1=Barrow |first1=J. D. |last2=Shaw |first2=D. J. |title=The value of the cosmological constant |journal=General Relativity and Gravitation |volume=43 |issue=10 |year=2011 |pages=2555–2560 |issn=0001-7701 |doi=10.1007/s10714-011-1199-1 |bibcode=2011GReGr..43.2555B |arxiv=1105.3105 |s2cid=55125081}}
* {{cite journal |last1=Caldwell |first1=R. R. |title=A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state |journal=Physics Letters B |volume=545 |issue=1–2 |year=2002 |pages=23–29 |issn=0370-2693 |doi=10.1016/S0370-2693(02)02589-3 |bibcode=2002PhLB..545...23C |arxiv=astro-ph/9908168 |s2cid=9820570}}
* {{cite journal |last1=Carroll |first1=S. M. |author-link1=Sean M. Carroll |last2=Press |first2=W. H. |author-link2=William H. Press |last3=Turner |first3=E. L. |title=The Cosmological Constant |journal=Annual Review of Astronomy and Astrophysics |volume=30 |issue=1 |year=1992 |pages=499–542 |issn=0066-4146 |doi=10.1146/annurev.aa.30.090192.002435 |pmid=28179856 |pmc=5256042 |bibcode=1992ARA&A..30..499C |url=https://preposterousuniverse.com/wp-content/uploads/cpt92.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://preposterousuniverse.com/wp-content/uploads/cpt92.pdf |archive-date=2022-10-09 |url-status=live}}
* {{cite book |last1=Hobson |first1=M. P. |last2=Efstathiou |first2=G. P. |last3=Lasenby |first3=A. N. |title=General Relativity: An Introduction for Physicists |date=2006 |publisher=Cambridge University Press |location=Cambridge |isbn=978-0-521-82951-9 |edition=2014 |url=https://books.google.com/books?id=5dryXCWR7EIC |oclc=903178203 |lccn=2006277059}}
* {{cite journal |last1=Joyce |first1=A. |last2=Jain |first2=B. |last3=Khoury |first3=J. |last4=Trodden |first4=M. |title=Beyond the cosmological standard model |journal=Physics Reports |volume=568 |year=2015 |pages=1–98 |issn=0370-1573 |doi=10.1016/j.physrep.2014.12.002 |bibcode=2015PhR...568....1J |arxiv=1407.0059 |s2cid=119187526}}
* {{cite journal |last1=Peebles |first1=P. J. E. |author-link1=Jim Peebles |last2=Ratra |first2=B. |author-link2=Bharat Ratra |title=The Cosmological Constant and Dark Energy |journal=Reviews of Modern Physics |volume=75 |issue=2 |year=2003 |pages=559–606 |issn=0034-6861 |doi=10.1103/RevModPhys.75.559 |bibcode=2003RvMP...75..559P |arxiv=astro-ph/0207347 |s2cid=118961123}}
* {{cite journal |last1=Rugh |first1=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 |doi=10.1016/S1355-2198(02)00033-3 |date=2001 |last2=Zinkernagel |first2=H. |bibcode=2002SHPMP..33..663R |arxiv=hep-th/0012253 |s2cid=9007190}}
* {{cite book |last1=Ryden |first1=B. S. |title=Introduction to Cosmology |date=2003 |publisher=Addison-Wesley |location=San Francisco |isbn=978-0-8053-8912-8 |oclc=50478401 |lccn=2002013176}}
* {{cite book |last1=Vilenkin |first1=A. |author-link1=Alexander Vilenkin |title=Many worlds in one: The Search For Other Universes |date=2006 |publisher=Hill and Wang |location=New York |isbn=978-0-8090-9523-0 |oclc=799428013 |lccn=2005027057 |url-access=registration |url=https://archive.org/details/manyworldsinoneo00vile}}
* {{cite journal |last1=Weinberg |first1=S. |author-link1=Steven Weinberg |title=The Cosmological Constant Problem |journal=Reviews of Modern Physics |volume=61 |issue=1 |year=1989 |pages=1–23 |issn=0034-6861 |doi=10.1103/RevModPhys.61.1 |bibcode=1989RvMP...61....1W |url=https://repositories.lib.utexas.edu/bitstream/2152/61094/1/Weinberg_1989.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://repositories.lib.utexas.edu/bitstream/2152/61094/1/Weinberg_1989.pdf |archive-date=2022-10-09 |url-status=live |hdl=2152/61094 |s2cid=122259372 |hdl-access=free}}
* {{cite book |last1=Weinberg |first1=S. |author-link1=Steven Weinberg |title=Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature. |date=1992 |publisher=Pantheon Books |location=New York |isbn=978-0-679-74408-5 |lccn=93030534 |oclc=319776354}}
* {{cite book |last1=Weinberg |first1=S. |author-link1=Steven Weinberg |title=Lectures on Quantum Mechanics |date=2015 |publisher=Cambridge University Press |location=Cambridge |isbn=978-1-107-11166-0 |edition=2nd |lccn=2015021123 |oclc=910664598}}
{{refend}}

== External links ==
{{Sister project links}}
* Michael, E., University of Colorado, Department of Astrophysical and Planetary Sciences, "[https://web.archive.org/web/20070103073616/http://super.colorado.edu/~michaele/Lambda/lambda.html The Cosmological Constant]"
* [[Sean M. Carroll|Carroll, Sean M.]], ''[https://web.archive.org/web/20011201205845/http://pancake.uchicago.edu/~carroll/encyc/ "The Cosmological Constant"]'' (short), ''[https://web.archive.org/web/20040620123430/http://www.livingreviews.org/lrr-2001-1 "The Cosmological Constant"]''(extended).
* [https://news.bbc.co.uk/2/hi/science/nature/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]]|author2=Merrifield, Mike}}

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