The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) occurs in Einstein's theory of general relativity. The units of Λ are 1/second². The constant is proportional to the energy density of the vacuum ρ:

where:

• π is pi
• G is the gravitational constant
• c is the speed of light in vacuum

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).

Einstein included the term in the equations for general relativity because he was dissatisfied that his equations do not allow for a stationary universe. Gravity would cause a universe which was initially at dynamical equilibrium to begin to contract. To counteract the contraction, Einstein added the cosmological constant. However, soon after Einstein developed his theory, observations by Edwin Hubble indicated that the universe is not at equilibrium, but rather is expanding. Morover, 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. Ironically, the cosmological constant is still of interest, as observations made in the late 1990's of distance-redshift relations indicate that the universe is accelerating. These observations can be explained very well by assuming a very small positive cosmological constant in Einstein's equations. 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 Planck units.

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 does not help. 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. Some physicists, such as Steven Weinberg, think the delicate balance of quantum vacuum energy is best explained by the anthropic principle.

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.)

Reference

• Carroll, Sean M., "The Cosmological Constant" (short), "The Cosmological Constant"(extended).
• hep-th/0208013 Lisa Dyson, Matthew Kleban, Leonard Susskind: "Disturbing Implications of a Cosmological Constant"