Cosmological constant

In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that

Einstein originally introduced the constant in 1917[2] to counterbalance the effect of gravity and achieve a static universe, a notion that was the accepted view at the time. Einstein's cosmological constant was abandoned after Edwin Hubble's confirmation that the universe was expanding.^[3] From the 1930s until the late 1990s, most physicists agreed with Einstein's choice of setting the cosmological constant to zero.^[4] 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.^[5]

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 so-called dark energy.^[6][7][8] The cosmological constant Λ is the simplest possible explanation for dark energy, and is used in the current standard model of cosmology known as the ΛCDM model.

According to

The cosmological constant was originally introduced in Einstein's 1917 paper entitled “The cosmological considerations in the General Theory of Reality”.[2] 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.^[3] However, Einstein 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".^[12]

In 1929, not long after Einstein developed his static theory, observations by

However, the cosmological constant remained a subject of theoretical and empirical interest. Empirically, the cosmological data of recent decades strongly suggests that our universe has a positive cosmological constant.[5] 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 supernovas. By that time, they expected to observe the deceleration of the supernovas caused by the gravitation attraction of mass according to Einstein’s gravitational theory. The first reports published in July 1997 from Supernova Cosmology Project used the supernova observation to support such deceleration hypothesis. But soon they found that supernovas were flying away in an accelerating manner. In 1998, both teams announced this surprising result. It implied the Universe is under accelerating expansion. The cosmological constant is needed to explain such acceleration.^[15] After this discovery, the cosmological constant was put back to the equation of general relativity.

Sequence of events 1915–1998

• In 1915, Einstein publishes his equations of general relativity, without a cosmological constant Λ.
• In 1917, Einstein adds the parameter Λ 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 (Einstein static universe).
• In 1922, the Russian physicist Alexander Friedmann mathematically shows that Einstein's equations (whatever Λ) 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 1932 with the Dutch physicist and astronomer Willem de Sitter, a model of a continuously expanding Universe with zero cosmological constant (Einstein–de Sitter spacetime).
• In 1998, two teams of astrophysicists, one led by
  Standard cosmological model).

  For this work, Perlmutter, Schmidt, and Riess
  jointly received the Nobel Prize in physics in 2011.

The cosmological constant Λ appears in the Einstein field equations in the form

${\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu },}$

where

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 moved to the right-hand side of the equation using Λ = κρ_vac. It is common to quote values of energy density directly, though still using the name "cosmological constant". The dimension of Λ is generally understood as length⁻².

Using the values known in 2018 and Planck units for Ω_Λ = 0.6889±0.0056 and the

${\displaystyle {\begin{aligned}\Lambda =3\,\left({\frac {\,H_{0}\,}{c}}\right)^{2}\Omega _{\Lambda }&=1.1056\times 10^{-52}\ {\text{m}}^{-2}\\&=2.888\times 10^{-122}\,l_{\text{P}}^{-2}\end{aligned}}}$

where ${\displaystyle l_{\text{P}}}$ is the

Ω_Λ (Omega sub Lambda)

Instead of the cosmological constant itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and the

Another ratio that is used by scientists is the equation of state, usually denoted w, which is the ratio of pressure that dark energy puts on the universe to the energy per unit volume.^[20] This ratio is w = −1 for the cosmological constant used in the Einstein equations; alternative time-varying forms of vacuum energy such as quintessence generally use a different value. The value w = −1.028±0.032, measured by the Planck Collaboration (2018)^[16] is consistent with −1, assuming w does not change over cosmic time.

Positive value

Observations announced in 1998 of distance–redshift relation for

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 Holographic principle).^[30]

Predictions

Quantum field theory

A major outstanding

Such arguments are usually based on

Planck scale

, then we would expect a cosmological constant of the order of ${\displaystyle M_{\rm {pl}}^{2}}$ (${\displaystyle 1}$ in reduced Planck units). As noted above, the measured cosmological constant is smaller than this by a factor of ~10¹²⁰. This discrepancy has been called "the worst theoretical prediction in the history of physics".

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.

No vacuum in the string theory landscape is known to support a metastable, positive cosmological constant, and in 2018 a group of four physicists advanced a controversial conjecture which would imply that no such universe exists.^[32]

Anthropic principle

One possible explanation for the small but non-zero value was noted by Steven Weinberg in 1987 following the anthropic principle.^[33] 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. Specifically, if the vacuum energy is negative and its absolute value is substantially larger than it appears to be in the observed universe (say, a factor of 10 larger), holding all other variables (e.g. matter density) constant, that would mean that the universe is closed; furthermore, its lifetime would be shorter than the age of our universe, possibly too short for intelligent life to form. On the other hand, a universe with a large positive cosmological constant would expand too fast, preventing galaxy formation. According to Weinberg, domains where the vacuum energy is compatible with life would be comparatively rare. Using this argument, Weinberg predicted that the cosmological constant would have a value of less than a hundred times the currently accepted value.^[34] In 1992, Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.^[35]

This argument depends on the vacuum energy density being constant throughout spacetime, as would be expected if dark energy were the cosmological constant. 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). Another theoretical approach that deals with the issue is that of multiverse theories, which predict a large number of "parallel" universes with different laws of physics and/or values of fundamental constants. Again, the anthropic principle states that we can only live in one of the universes that is compatible with some form of intelligent life. Critics claim that these theories, when used as an explanation for fine-tuning, commit the inverse gambler's fallacy.

In 1995, Weinberg's argument was refined by Alexander Vilenkin to predict a value for the cosmological constant that was only ten times the matter density,^[36] i.e. about three times the current value since determined.

Failure to detect dark energy

An attempt to directly observe and relate quanta or fields like the chameleon particle or the symmetron theory to dark energy, in a laboratory setting, failed to detect a new force.^[37] Inferring the presence of dark energy through its interaction with baryons in the cosmic microwave background has also led to a negative result,^[38] although the current analyses have been derived only at the linear perturbation regime. It is also possible that the difficulty in detecting dark energy is due to the fact that the cosmological constant describes an existing, known interaction (e.g. electromagnetic field).^[39]

See also

References

Footnotes