Cosmological perturbation theory

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, as he had based his theory on the idea that the universe is unchanging.[1] 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 of

R_μν -1/2 R g_μν + Λg_μν = 8π GT_μν / c⁴

where R and g pertain to the structure of spacetime, T pertains to matter and energy (thought of as affecting that structure), and G and c are conversion factors that 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).

The cosmological constant has the same effect as an intrinsic energy density of the vacuum, ρvac (and an associated pressure). In this context it is commonly defined with a proportionality factor of 8π: Λ = 8πρvac, where unit conventions of general relativity are used (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 itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and the critical density of the universe. This ratio is usually denoted Ω_Λ. In a flat universe Ω_Λ corresponds to the fraction of the energy density of the Universe due to 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.[2]

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.[1] 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. Einstein later referred to his failure to predict the expansion of the universe from theory, before it was proven by observation of the cosmological red shift, as the "biggest blunder" of his life.

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.

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.[1] The explanation of this small but positive value is an outstanding theoretical challenge (see the section 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 announced in 1998 of distance–redshift relation for Type Ia supernovae[3][4] indicated that the expansion of the universe is accelerating. When combined with measurements of the cosmic microwave background radiation these implied a value of Ω_Λ approx 0.7,[5] 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 simplest 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 10⁻⁵² m⁻², in metric units. Multiplied by other constants that appear in the equations, it is often expressed as 10⁻³⁵ s⁻², 10⁻⁴⁷ GeV⁴, 10⁻²⁹ g/cm³,[6] or even 10⁻¹²⁰ in reduced Planck units.

Cosmological constant problem
See also: Vacuum catastrophe
Unsolved problems in physics Why can't the zero-point energy of the vacuum be interpreted as a cosmological constant? What causes the discrepancies? Question mark2.svg

A major outstanding problem is that most quantum field theories predict a huge value for the quantum vacuum. A common assumption is that the quantum vacuum is equivalent to the cosmological constant. Although no theory exists that supports this assumption, arguments can be made in its favor. [7]

Such arguments are usually based on dimensional analysis and effective field theory. If the universe is described by an effective local quantum field theory down to the Planck scale, then we would expect a cosmological constant of the order of M_pl⁴. As noted above, the measured cosmological constant is smaller than this by a factor of 10⁻¹²⁰. This discrepancy has been called "the worst theoretical prediction in the history of physics!".[8]

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.[9] 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 the 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 are compatible with life would be comparatively rare. This argument depends crucially on the reality of a distribution (spatial or otherwise) 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). Another theoretical approach that deals with the issue is that of multiverse theories - predicting a large number of "parallel" universes, possibly 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.

As was only recently seen, by works of 't Hooft, Susskind[10] 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."[11] 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.”[12]

de Sitter relativity
Main article: de Sitter relativity

In de Sitter relativity (which is an all-energy-scale applicable example of doubly special relativity), special relativity is modified so that the symmetry group is a de Sitter rather than Poincaré group. It is thought de Sitter relativity will be more accurate than special relativity at high energies. The de Sitter group naturally incorporates an invariant length–parameter and results in a residual spacetime curvature even in the absence of matter or energy. This corresponds to a special relativity with a built-in cosmological constant and a correspondingly modified de Sitter general relativity.