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In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) is the value of the energy density of the vacuum of space. It was introduced by Albert Einstein as an addition to his theory of general relativity to "hold back gravity" and achieve a static universe, which was the accepted view at the time. Einstein abandoned the concept as his "greatest blunder" after Hubble's 1928 discovery that the distant galaxies are expanding away from each other, implying an overall expanding Universe (which is only detectable on the largest of scales). Surprisingly, the discovery of cosmic acceleration in 1998 has revived the need for a non-zero cosmological constant, this time to add a small acceleration to the ongoing expansion. As physicist Tony Rothman notes, "Einstein's equations do not specify the universe; rather they may be considered a general framework within which you can construct many different model universes."
The cosmological constant Λ appears in Einstein's field equation in the form of
where R and g describe the structure of spacetime, T pertains to matter and energy 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 moved onto the right-hand side of the equation, and defined with a proportionality factor of 8π: Λ = 8πρvac, where unit conventions of general relativity are used (otherwise factors of G and c would also appear). 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 the universe, as observed. (See dark energy and cosmic inflation for details.)
ΩΛ (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, and is estimated to be 0.692 ± 0.010, according to the recent Planck results released in 2013. 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, usually denoted w, which is the ratio of pressure that dark energy puts on the Universe to the energy per unit volume. This ratio is w = −1 for a true cosmological constant, and is generally different for alternative time-varying forms of vacuum energy such as quintessence.
Einstein included the cosmological constant as a term in his field equations for general relativity because he was dissatisfied that otherwise his equations did not allow, apparently, for a static universe: gravity would cause a universe which was initially at dynamic equilibrium to contract. To counteract this possibility, Einstein added the cosmological constant. However, soon after Einstein developed his static theory, observations by Edwin Hubble indicated that the universe appears to be expanding; this was consistent with a cosmological solution to the original general-relativity equations that had been found by the mathematician Friedmann, working on the Einstein equations of general-relativity. Einstein later reputedly referred to his failure to accept the validation of his equations; when they had predicted the expansion of the universe in theory, before it was demonstrated in observation of the cosmological red shift, as the "biggest blunder" of his life.[dubious ]
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. 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.
Observations announced in 1998 of distance–redshift relation for Type Ia supernovae indicated that the expansion of the universe is accelerating. When combined with measurements of the cosmic microwave background radiation these implied a value of , 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. In terms of Planck units, and as a natural dimensionless value, the cosmological constant, λ, is on the order of 10−122.
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).
A problem arises with inclusion of the cosmological constant in the standard model: i.e., the appearance of solutions with regions of discontinuities (see classification of discontinuities at typical matter density). Discontinuity also affects the past sign of the pressure of the cosmological constant, changing from the current negative pressure to attractive, with lookback towards the early Universe. Another investigation found the cosmological time, dt, diverges for any finite interval, ds, associated with an observer approaching the cosmological horizon, representing a physical limit to observation for the standard model when the cosmological term is included. This is a key requirement for a complete interpretation of astronomical observations, particularly pertaining to the nature of dark energy and the cosmological constant. All of these findings should be considered major shortcomings of the standard model, but only when the cosmological constant term is included.
Quantum field theory 
|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.
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 . 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!".
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.
One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987 following the anthropic principle. 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. In 1992 Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.
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, i.e. about three times the current value since determined.
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." Critics respond that, as the authors acknowledge in their paper, the model “entails ... the same degree of tuning required in any cosmological model”.
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- Carroll, Sean M., "The Cosmological Constant" (short), "The Cosmological Constant"(extended).
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- News story: More evidence for dark energy being the cosmological constant
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- Copeland, Ed; Merrifield, Mike. "Λ – Cosmological Constant". Sixty Symbols. Brady Haran for the University of Nottingham.