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Cosmological constant

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. However, the discovery of cosmic acceleration in the 1990s has renewed interest in a cosmological constant.

Equation

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). Astronomical observations imply that the constant cannot exceed 10⁻⁴⁶ km⁻².

The cosmological constant has the same effect as an intrinsic energy density of the vacuum, ρ_vac (and an associated pressure). In this context it is 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). 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, 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

Omega_{Lambda}

. In a flat universe

Omega_{Lambda}

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.

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.

History

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 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.

Since it no longer seemed to be needed, Einstein abandoned the cosmological constant and called it the "biggest blunder" of his life. 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. See the section on the observation of a Positive Cosmological Constant below. The explanation of this small but positive value is an outstanding theoretical challenge. See the section on the Cosmological Constant Problem 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 cosmological constant

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

, 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

A major outstanding problem is that most quantum field theories predict a huge cosmological constant from the energy of the quantum vacuum.

This conclusion follows from dimensional analysis and effective field theory. If the universe is described by an effective local quantum field theory till 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 10¹²⁰.

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, 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 gambler's fallacy.

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).

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. 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.”