Cosmological constant
In cosmology, the cosmological constant, alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general relativity. He later removed it; however, much later it was revived to express the energy density of space, or vacuum energy, that arises in quantum mechanics. It is closely associated with the concept of dark energy.
Einstein introduced the constant in 1917 to counterbalance the effect of gravity and achieve a static universe, which was then assumed. Einstein's cosmological constant was abandoned after Edwin Hubble confirmed that the universe was expanding. From the 1930s until the late 1990s, most physicists thought the cosmological constant to be zero. That changed with the discovery in 1998 that the expansion of the universe is accelerating, implying that the cosmological constant may have a positive value after all.
Since the 1990s, studies have shown that, assuming the cosmological principle, around 68% of the mass–energy density of the universe can be attributed to dark energy. The cosmological constant is the simplest possible explanation for dark energy, and is used in the standard model of cosmology known as the ΛCDM model.
According to quantum field theory, which underlies modern particle physics, empty space is defined by the vacuum state, which is composed of a collection of quantum fields. All these quantum fields exhibit fluctuations in their ground state arising from the zero-point energy existing everywhere in space. These zero-point fluctuations should contribute to the cosmological constant, but actual calculations give rise to an enormous vacuum energy. The discrepancy between theorized vacuum energy from quantum field theory and observed vacuum energy from cosmology is a source of major contention, with the values predicted exceeding observation by some 120 orders of magnitude, a discrepancy that has been called "the worst theoretical prediction in the history of physics!". This issue is called the cosmological constant problem and it is one of the greatest mysteries in science with many physicists believing that "the vacuum holds the key to a full understanding of nature".
History
In the Principia, Isaac Newton identified a central force proportional to distance as one of the "two principal cases of attraction" alongside the inverse-square law of gravity; in modern physics, this linear force law is mathematically equivalent to the force associated with a cosmological constant.The cosmological constant was originally introduced in Albert Einstein's 1917 paper entitled Cosmological considerations in the General Theory of 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 for a static universe: gravity would cause a universe that was initially non-expanding to contract. To counteract this possibility, Einstein added the cosmological constant. However, he was not happy about adding this cosmological term. He later stated that "Since I introduced this term, I had always a bad conscience.... I am unable to believe that such an ugly thing is actually realized in nature". Einstein's static universe is unstable against matter density perturbations. Furthermore, without the cosmological constant Einstein could have found the expansion of the universe before Hubble's observations.
In 1929, not long 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 Alexander Friedmann, working on the Einstein equations of general relativity. Einstein reportedly 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 redshift—as his "biggest blunder".
It transpired 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 that contracts slightly will continue contracting.
However, the cosmological constant remained a subject of theoretical and empirical interest. Empirically, the cosmological data of recent decades strongly suggest that our universe has a positive cosmological constant. The explanation of this small but positive value is a remaining theoretical challenge, the so-called cosmological constant problem.
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, Arthur 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.
In 1990s, Saul Perlmutter at Lawrence Berkeley National Laboratory, Brian Schmidt of the Australian National University and Adam Riess of the Space Telescope Science Institute were searching for type Ia supernovae. At that time, they expected to observe the deceleration of the supernovae caused by gravitational attraction of mass according to Einstein's gravitational theory. The first reports published in July 1997 from the Supernova Cosmology Project used the supernova observation to support such deceleration hypothesis. But soon they found that supernovae were accelerating away. Both teams announced this surprising result in 1998. It implied the universe is undergoing accelerating expansion. The cosmological constant is needed to explain such acceleration. Following this discovery, the cosmological constant was reinserted in the general relativity equations.
Sequence of events 1915–1998
- In 1915, Einstein publishes his equations of general relativity, without a cosmological constant.
- In 1917, Einstein adds the parameter to his equations when he realizes that his theory implies a dynamic universe for which space is a function of time. He then gives this constant a value that makes his Universe model remain static and eternal.
- In 1922, the Russian physicist Alexander Friedmann mathematically shows that Einstein's equations remain valid in a dynamic universe.
- In 1927, the Belgian astrophysicist Georges Lemaître shows that the Universe is expanding by combining general relativity with astronomical observations, those of Hubble in particular.
- In 1931, Einstein accepts the theory of an expanding universe and proposes, in 1931, a model of a continuously expanding universe with zero cosmological constant. This is followed by another model, in 1932, with the Dutch physicist and astronomer Willem de Sitter in which both the cosmological constant and spatial curvature are set to zero.
- In 1998, two teams of astrophysicists, the Supernova Cosmology Project and the High-Z Supernova Search Team, carried out measurements on distant supernovae which showed that the speed of galaxies' recession in relation to the Milky Way increases over time. The universe is in accelerated expansion, which requires having a strictly positive. The universe would contain a mysterious dark energy producing a repulsive force that counterbalances the gravitational braking produced by the matter contained in the universe. For this work, Perlmutter, Schmidt, and Riess jointly received the Nobel Prize in Physics in 2011.
Equation
where the Ricci tensor, Ricci scalar and the metric tensor describe the structure of spacetime, the stress–energy tensor describes the energy density, momentum density and stress at that point in spacetime, and. The gravitational constant and the speed of light are universal constants. When is zero, this reduces to the field equation of general relativity usually used in the 20th century. When is zero, the field equation describes empty space.
The cosmological constant has the same effect as an intrinsic energy density of the vacuum, '. In this context, it is commonly moved to the right-hand side of the equation using. It is common to quote values of energy density directly, though still using the name "cosmological constant". The dimension of is generally understood as length.
Using the Planck units, and the value evaluated in 2025 for the Hubble constant '0 = =, has the value of where is the Planck length. 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.
Density parameter Ω
The dimensionless density parameter Ω represents the ratio of the actual density of a component of the universe to the critical density. The total density parameter for a flat universe, can be expressed as:Where:
- Ωm is the matter density parameter ;
- ΩΛ is the density parameter for dark energy ; and
- Ωk describes the curvature of the universe which is 0 in a flat universe.