Cosmological constant

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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. After the discovery of the Hubble redshift and the introduction of the expanding space paradigm, Einstein abandoned the concept. However, the discovery of cosmic acceleration in the 1990s has renewed interest in a cosmological constant.

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

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

where R and g pertain to the structure of spacetime, T pertains to matter (thought of as affecting that structure), and G and c are conversion factors which 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 can be thought of as the intrinsic energy density of the vacuum, ρ_vac. It is most commonly defined with a proportionality factor of 8π: Λ = 8πρ_vac, where modern unit conventions of general relativity are followed (otherwise factors of G and c would also appear). If modern conventions are adopted, Λ has dimensions of 1/distance², the same as the dimensions of mass-energy density. Because of the ambiguity in the definition of Λ, it is more common to quote values of energy density directly, though still using the name "cosmological constant".

Because of the way the cosmological constant appears in Einstein's field equation, a positive vacuum energy density 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.

In lieu of the cosmological constant, cosmologists often quote the ratio between the energy density due to the cosmological constant and the current critical density of the universe. This ratio is usually called $Ω Λ$ . In a flat universe $Ω Λ$ corresponds to the fraction of the energy density of the Universe which is associated with 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.

1 General relativity
2 Cosmological constant problem
3 See also
4 References
- 4.1 Further information

[edit] General relativity

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 dynamical 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 Friedman.

It is now thought that 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. These sorts of small contractions are inevitable, due to the uneven distribution of matter throughout the universe.

Since it no longer seemed to be needed, Einstein abandoned the cosmological constant and called it the "biggest blunder" of his life. (He may have been referring to his methodology rather than to the constant itself.) Ironically, the cosmological constant is still of interest, as observations made in the late 1990s of distance-redshift relations indicate that the expansion of the universe is accelerating. When combined with measurements of the cosmic microwave background radiation these implied a value of $\Omega_{\Lambda} \simeq 0.7$ ^[1], 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 most economical 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^-35s^-2, or 10^-47GeV⁴, or 10^-29g/cm³, or about 10^-120 in reduced Planck units.

[edit] Cosmological constant problem

Unsolved problems in physics: Why doesn't the zero-point energy of vacuum cause a large cosmological constant? What cancels it out?

A major outstanding problem is that most quantum field theories predict a huge cosmological constant from the energy of the quantum vacuum. This would need to be cancelled almost, but not exactly, by an equally large term of the opposite sign. 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 infinitesimal cosmological constant observed in cosmology from particle physics.

One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987^[2]. 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, and domains where the vacuum energy is much smaller would be comparatively rare. This argument depends crucially on the reality of a spatial distribution in the vacuum energy density. 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). Critics note that these multiverse theories, when used as an explanation for fine-tuning, commit the inverse gamblers fallacy.

As was only recently seen, by works of 't Hooft, Susskind^[3] and others, a positive cosmological constant has surprising consequences, such as a finite maximum entropy of the observable universe (see the holographic principle).

More recent work has suggested the problem may be indirect evidence of a cyclic universe predicted 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."^[4] 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.” ^[5]

[edit] See also

[edit] References

^ See e.g. Detection of cosmic microwave background structure in a second field with the Cosmic Anisotropy Telescope, Baker, Joanne C.; Grainge, Keith; Hobson, M. P.; Jones, Michael E.; Kneissl, R.; Lasenby, A. N.; O'Sullivan, C. M. M.; Pooley, Guy; Rocha, G.; Saunders, Richard; Scott, P. F.; Waldram, E. M., Monthly Notices of the Royal Astronomical Society, Volume 308, Issue 4, pp. 1173-1178
^ [Weinberg, S. 1987, "Anthropic Bound on the Cosmological Constant", PRL 59]
^ Lisa Dyson, Matthew Kleban, Leonard Susskind: "Disturbing Implications of a Cosmological Constant"
^ 'Cyclic universe' can explain cosmological constant, NewScientistSpace, 04 May 2006
^ [Steinhardt and Turok, 1437]

[edit] Further information

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