Cosmological constant

Cosmological constant
Estimated ratios of dark matter and dark energy (which may be the cosmological constant) in the universe. Dark energy now dominates the energy of the universe, in contrast to earlier epochs when it was insignificant.

In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. Einstein abandoned the concept after the observation of the Hubble redshift indicated that the universe might not be stationary, as he had based his theory on the idea that the universe is unchanging.[1] However, the discovery of cosmic acceleration in 1998 has renewed interest in a cosmological constant.



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

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

where R and g pertain to the structure of spacetime, T pertains to matter and energy (thought of as affecting that structure), and G and c are conversion factors that arise from using traditional units of measurement. When Λ is zero, this reduces to the original field equation of general relativity. When T is zero, the field equation describes empty space (the vacuum).

The cosmological constant has the same effect as an intrinsic energy density of the vacuum, ρvac (and an associated pressure). In this context it is commonly defined with a proportionality factor of 8π: Λ = 8πρvac, where unit conventions of general relativity are used (otherwise factors of G and c would also appear). It is common to quote values of energy density directly, though still using the name "cosmological constant".

A positive vacuum energy density resulting from a cosmological constant implies a negative pressure, and vice versa. If the energy density is positive, the associated negative pressure will drive an accelerated expansion of empty space. (See dark energy and cosmic inflation for details.)

Omega Lambda

In lieu of the cosmological constant itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and the critical density of the universe. This ratio is usually denoted ΩΛ. In a flat universe ΩΛ corresponds to the fraction of the energy density of the Universe due to the cosmological constant. Note that this definition is tied to the critical density of the present cosmological era: the critical density changes with cosmological time, but the energy density due to the cosmological constant remains unchanged throughout the history of the universe.

Equation of state

Another ratio that is used by scientists is the equation of state which is the ratio of pressure that dark energy puts on the Universe to the energy per unit volume.[2]


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 which was initially at dynamic equilibrium to contract. To counteract this possibility, Einstein added the cosmological constant.[1] 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 Friedman. Einstein later referred to his failure to predict the expansion of the universe from theory, before it was proven by observation of the cosmological red shift, as the "biggest blunder" of his life.

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 which contracts slightly will continue contracting.

However, the cosmological constant remained a subject of theoretical and empirical interest. Empirically, the onslaught of cosmological data in the past decades strongly suggests that our universe has a positive cosmological constant.[1] The explanation of this small but positive value is an outstanding theoretical challenge (see the section below).

Finally, it should be noted that some early generalizations of Einstein's gravitational theory, known as classical unified field theories, either introduced a cosmological constant on theoretical grounds or found that it arose naturally from the mathematics. For example, Sir Arthur Stanley Eddington claimed that the cosmological constant version of the vacuum field equation expressed the "epistemological" property that the universe is "self-gauging", and Erwin Schrödinger's pure-affine theory using a simple variational principle produced the field equation with a cosmological term.

Positive value

From the observed galaxy distribution in 1992 Paal et al.[3] was the first who suggested non zero cosmological constant. Two years later in another paper[4] they suggested \Omega_{\Lambda} \simeq 2/3. Observations announced in 1998 of distance–redshift relation for Type Ia supernovae[5][6] indicated that the expansion of the universe is accelerating. When combined with measurements of the cosmic microwave background radiation these implied a value of \Omega_{\Lambda} \simeq 0.7,[7] a result which has been supported and refined by more recent measurements. There are other possible causes of an accelerating universe, such as quintessence, but the cosmological constant is in most respects the simplest solution. Thus, the current standard model of cosmology, the Lambda-CDM model, includes the cosmological constant, which is measured to be on the order of 10−52 m−2, in metric units. Multiplied by other constants that appear in the equations, it is often expressed as 10−35 s−2, 10−47 GeV4, 10−29 g/cm3,[8].

As was only recently seen, by works of 't Hooft, Susskind[9] and others, a positive cosmological constant has surprising consequences, such as a finite maximum entropy of the observable universe (see the holographic principle).


Quantum field theory

Unsolved problems in physics
Why can't the zero-point energy of the vacuum be interpreted as a cosmological constant? What causes the discrepancies?

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. [10]

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 M_{\rm pl}^4. As noted above, the measured cosmological constant is smaller than this by a factor of 10−120. This discrepancy has been called "the worst theoretical prediction in the history of physics!".[11]

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.

Anthropic principle

One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987 following the anthropic principle.[12] Weinberg explains that if the vacuum energy took different values in different domains of the universe, then observers would necessarily measure values similar to that which is observed: the formation of life-supporting structures would be suppressed in domains where the vacuum energy is much larger. Specifically, if the vacuum energy is negative and its absolute value is substantially larger than it appears to be in the observed universe (say, a factor of 10 larger), holding all other variables (e.g. matter density) constant, that would mean that the universe is closed; furthermore, its lifetime would be shorter than the age of our universe, possibly too short for the intelligent life to form. On the other hand, a universe with a large positive cosmological constant would expand too fast, preventing galaxy formation. According to Weinberg, domains where the vacuum energy are compatible with life would be comparatively rare. Using this argument, Weinberg predicted that the cosmological constant would have a value of less than a hundred times the currently accepted value.[13] In 1992 Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.[14]

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 the 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 - predicting a large number of "parallel" universes, possibly 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,[15] i.e. about three times the current value since determined.

Cyclic model

More recent work has suggested the problem may be indirect evidence of a cyclic universe possibly as allowed by string theory. With every cycle of the universe (Big Bang then eventually a Big Crunch) taking about a trillion (1012) years, "the amount of matter and radiation in the universe is reset, but the cosmological constant is not. Instead, the cosmological constant gradually diminishes over many cycles to the small value observed today."[16] Critics respond that, as the authors acknowledge in their paper, the model “entails tuning” to “the same degree of tuning required in any cosmological model.”[17]

de Sitter relativity

In de Sitter relativity (which is an all-energy-scale applicable example of doubly special relativity), special relativity is modified so that the symmetry group is a de Sitter rather than Poincaré group. It is thought de Sitter relativity will be more accurate than special relativity at high energies. The de Sitter group naturally incorporates an invariant length–parameter and results in a residual spacetime curvature even in the absence of matter or energy. This corresponds to a special relativity with a built-in cosmological constant and a correspondingly modified de Sitter general relativity.

See also


  1. ^ a b c Urry, Meg (2008). The Mysteries of Dark Energy. Yale Science. Yale University. 
  2. ^ Hogan, Jenny (2007). "Welcome to the Dark Side". Nature 448 (7151): 240–245. Bibcode 2007Natur.448..240H. doi:10.1038/448240a. PMID 17637630. 
  3. ^ . Bibcode 1992Ap&SS.191..107P. doi:10.1007/BF00644200. 
  4. ^ . Bibcode 1994Ap&SS.222...65H. doi:10.1007/BF00627083. 
  5. ^ Riess, A. et al. (September 1998). "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant". The Astronomical Journal 116 (3): 1009–1038. arXiv:astro-ph/9805201. Bibcode 1998AJ....116.1009R. doi:10.1086/300499. 
  6. ^ Perlmutter, S. et al. (June 1999). "Measurements of Omega and Lambda from 42 High-Redshift Supernovae". The Astrophysical Journal 517 (2): 565–586. arXiv:astro-ph/9812133. Bibcode 1999ApJ...517..565P. doi:10.1086/307221. 
  7. ^ See e.g. Baker, Joanne C.; et al. (1999). "Detection of cosmic microwave background structure in a second field with the Cosmic Anisotropy Telescope". Monthly Notices of the Royal Astronomical Society 308 (4): 1173–1178. arXiv:astro-ph/9904415. Bibcode 1999MNRAS.308.1173B. doi:10.1046/j.1365-8711.1999.02829.x. 
  8. ^ Tegmark, Max; et al. (2004). "Cosmological parameters from SDSS and WMAP". Physical Review D 69 (103501): 103501. arXiv:astro-ph/0310723. Bibcode 2004PhRvD..69j3501T. doi:10.1103/PhysRevD.69.103501. 
  9. ^ Lisa Dyson, Matthew Kleban, Leonard Susskind: "Disturbing Implications of a Cosmological Constant"
  10. ^ Rugh, S (2001). "The Quantum Vacuum and the Cosmological Constant Problem". Studies in History and Philosophy of Modern Physics 33 (4): 663–705. doi:10.1016/S1355-2198(02)00033-3. 
  11. ^ MP Hobson, GP Efstathiou & AN Lasenby (2006). General Relativity: An introduction for physicists (Reprinted with corrections 2007 ed.). Cambridge University Press. p. 187. ISBN 9780521829519. 
  12. ^ Weinberg, S (1987). "Anthropic Bound on the Cosmological Constant". Phys. Rev. Lett. 59 (22): 2607–2610. Bibcode 1987PhRvL..59.2607W. doi:10.1103/PhysRevLett.59.2607. PMID 10035596. 
  13. ^ Alexander Vilenkin, Many Worlds in One: The Search for Other Universes, ISBN 9780809095230, page 138/9
  14. ^ Weinberg, Steven (1993). Dreams of a Final Theory: the search for the fundamental laws of nature. Vintage Press. p. 182. ISBN 0099223910. 
  15. ^ Alexander Vilenkin, Many Worlds in One: The Search for Other Universes, ISBN 9780809095230, page 146, which references Vilenkin' Predictions from quantum cosmology, Physical Review Letters, vol 74, page 846 (1995)
  16. ^ 'Cyclic universe' can explain cosmological constant, NewScientistSpace, 4 May 2006
  17. ^ Steinhardt and Turok, 1437

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