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COSMOLOGICAL CONSTANT

The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as part of his theory of general relativity to achieve a stationary universe. After the discovery of Hubble redshift and the introduction of the expanding space paradigm Einstein abandoned the concept. New discoveries in the 1990s have, however, renewed interest in a cosmological constant.

The units of Λ are 1/second2. The constant is proportional to the energy density of the vacuum ρvac:

\Lambda = {{8\pi G} \over {c^2}} \rho_{vac}

where:

The term can be positive, negative, or zero. It is the energy density of empty space: it can be thought of as the "cost" of having space. Because the cosmological constant has negative pressure, according to general relativity a positive cosmological constant — which means empty space has positive energy — causes the expansion of empty space to accelerate (see dark energy for details).

Cosmologists generally describe the cosmological constant in units where unity would correspond to the value of the cosmological constant which would give a closed universe in the absence of matter. This normalised cosmological constant is usually called ΩΛ. In a flat universe ΩΛ corresponds to the fraction of the energy density of the Universe which is associated with the cosmological constant.

Contents

General relativity

Einstein included the term in the equations for general relativity because he was dissatisfied that his equations apparently did not allow 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 theory, observations by Edwin Hubble indicated that the universe appears to be expanding. Also, a static-universe solution to the original field equations was discovered, so it was not necessary after all to add the extra term in order to achieve such a solution.

It is now thought that adding the cosmological constant to Einstein's equations does not lead to a 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.

Einstein abandoned the cosmological constant and called it the "biggest blunder" of his life. It is likely that he was referring more to his methodology 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.

Cosmological constant problem

Image:Question_dropshade.png
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 demonstrated 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). It should be noted that there are good reasons to be wary of excessive use of the anthropic principle and much current research is aimed at understanding the observed vacuum energy density by non-anthropic means.

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." [1]

See also

References

  1. ^ 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
  2. ^ [Weinberg, S. 1987, "Anthropic Bound on the Cosmological Constant", PRL 59]
  3. ^ Lisa Dyson, Matthew Kleban, Leonard Susskind: "Disturbing Implications of a Cosmological Constant"

Further information