In

physical cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

, the

**cosmological constant** (usually denoted by the Greek capital letter

lambdaLambda is the 11th letter of the Greek alphabet. In the system of Greek numerals lambda has a value of 30. Lambda is related to the Phoenician letter Lamed . Letters in other alphabets that stemmed from lambda include the Roman L and the Cyrillic letter El...

: Λ) was proposed by

Albert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

as a modification of his original theory of

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

to achieve a stationary universe. Einstein abandoned the concept after the observation of the

Hubble redshiftHubble's law is the name for the astronomical observation in physical cosmology that: all objects observed in deep space are found to have a doppler shift observable relative velocity to Earth, and to each other; and that this doppler-shift-measured velocity, of various galaxies receding from...

indicated that the universe might not be stationary, as he had based his theory on the idea that the universe is unchanging. However, the discovery of

cosmic accelerationThe accelerating universe is the observation that the universe appears to be expanding at an increasing rate, which in formal terms means that the cosmic scale factor a has a positive second derivative, implying that the velocity at which a given galaxy is receding from us should be continually...

in 1998 has renewed interest in a cosmological constant.

## Equation

The cosmological constant Λ appears in

Einstein's modified field equationThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

in the form of

where

*R* and

*g*In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...

pertain to the structure of

spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

,

*T*The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

pertains to matter and energy (thought of as affecting that structure), and

*G*The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

and

*c*The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

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

vacuumIn everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

).

The cosmological constant has the same effect as an intrinsic

energy densityEnergy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

of the vacuum,

*ρ*_{vac} (and an associated

pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

). In this context it is commonly defined with a

proportionalityIn mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

factor of 8

: Λ = 8

*ρ*_{vac}, 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 energyIn physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...

and

cosmic inflationIn physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...

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

. In a flat universe

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 stateIn cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number \! w, equal to the ratio of its pressure \! p to its energy density \! \rho: \! w=p/\rho...

which is the ratio of pressure that dark energy puts on the Universe to the energy per unit volume.

## History

EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

included the cosmological constant as a term in his

field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

for

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

because he was dissatisfied that otherwise his equations did not allow, apparently, for a

static universeA static universe, also referred to as a "stationary" or "Einstein" universe, is a model in which space is neither expanding nor contracting. Albert Einstein proposed such a model as his preferred cosmology in 1917...

: 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 HubbleEdwin Powell Hubble was an American astronomer who profoundly changed the understanding of the universe by confirming the existence of galaxies other than the Milky Way - our own galaxy...

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

FriedmanAlexander Alexandrovich Friedman or Friedmann was a Russian and Soviet physicist and mathematician.- Life and work :...

. 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-Science:* Redshift, the increase of wavelength of detected electromagnetic radiation with respect to the original wavelength of the emission* Red shift, an informal term for a bathochromic shift...

, as the "biggest blunder" of his life.

In fact 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 energyVacuum energy is an underlying background energy that exists in space even when the space is devoid of matter . The concept of vacuum energy has been deduced from the concept of virtual particles, which is itself derived from the energy-time uncertainty principle...

, 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. 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 theoriesSince the 19th century, some physicists have attempted to develop a single theoretical framework that can account for the fundamental forces of nature – a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics...

, either introduced a cosmological constant on theoretical grounds or found that it arose naturally from the mathematics. For example, Sir

Arthur Stanley EddingtonSir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...

claimed that the cosmological constant version of the vacuum field equation expressed the "

epistemological" property that the universe is "self-

gaugingIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

", and

Erwin SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

's pure-

affineIn the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

theory using a simple

variational principleA variational principle in physics is an alternative method for determining the state or dynamics of a physical system, by identifying it as an extremum of a function or functional...

produced the field equation with a cosmological term.

## Positive value

From the observed galaxy distribution in 1992

PaalPaál György was a Hungarian astronomer and cosmologist.- Work :In the late 50-s Paál studied the quasar and galaxy cluster distributions. In 1970 from redshift quantization he came up with the idea that the Universe might has nontrivial topological structure.In 1992, G. Paal, et al. and A...

et al. was the first who suggested non zero cosmological constant. Two years later in another paper they suggested

.

Observations announced in 1998 of distance–redshift relation for Type Ia supernovae indicated that the expansion of the universe is accelerating. When combined with measurements of the

cosmic microwave background radiationIn cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

these implied a value of

, a result which has been supported and refined by more recent measurements. There are other possible causes of an

accelerating universeThe accelerating universe is the observation that the universe appears to be expanding at an increasing rate, which in formal terms means that the cosmic scale factor a has a positive second derivative, implying that the velocity at which a given galaxy is receding from us should be continually...

, such as

quintessenceIn physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. It has been proposed by some physicists to be a fifth fundamental force...

, but the cosmological constant is in most respects the simplest solution. Thus, the current standard model of cosmology, the

Lambda-CDM modelΛCDM or Lambda-CDM is an abbreviation for Lambda-Cold Dark Matter, which is also known as the cold dark matter model with dark energy...

, 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} GeV

^{4}, 10

^{−29} g/cm

^{3},.

As was only recently seen, by works of

't HooftGerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...

,

SusskindLeonard Susskind is the Felix Bloch Professor of Theoretical Physics at Stanford University. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology...

and others, a positive cosmological constant has surprising consequences, such as a finite maximum

entropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

of the observable universe (see the

holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...

).

### Quantum field theory

A major outstanding

problemThis is a list of some of the major unsolved problems in physics. Some of these problems are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result...

is that most

quantum field theoriesQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

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.

Such arguments are usually based on

dimensional analysisIn physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

and

effective field theoryIn physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...

. If the universe is described by an effective local quantum field theory down to the

Planck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...

, then we would expect a cosmological constant of the order of

. 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!".

Some

supersymmetricIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

theories require a cosmological constant that is exactly zero, which further complicates things. This is the

**cosmological constant problem**, the worst problem of

fine-tuningIn theoretical physics, fine-tuning refers to circumstances when the parameters of a model must be adjusted very precisely in order to agree with observations. Theories requiring fine-tuning are regarded as problematic in the absence of a known mechanism to explain why the parameters happen to...

in

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

: there is no known natural way to derive the tiny cosmological constant used in

cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

from

particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

.

### Anthropic principle

One possible explanation for the small but non-zero value was noted by

Steven WeinbergSteven Weinberg is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles....

in 1987 following the

anthropic principleIn astrophysics and cosmology, the anthropic principle is the philosophical argument that observations of the physical Universe must be compatible with the conscious life that observes it. Some proponents of the argument reason that it explains why the Universe has the age and the fundamental...

. 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. Using this argument, Weinberg predicted that the cosmological constant would have a value of less than a hundred times the currently accepted value. In 1992 Weinberg refined this prediction of the cosmological constant to 5 to 10 times the matter density.

This argument depends on a lack of a variation of the distribution (spatial or otherwise) in the vacuum energy density, as would be expected if the 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

inflatonThe inflaton is the generic name of the hypothetical and hitherto unidentified scalar field that may be responsible for the hypothetical inflation in the very early universe...

(also see

quintessenceIn physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. It has been proposed by some physicists to be a fifth fundamental force...

). Another theoretical approach that deals with the issue is that of

multiverseThe multiverse is the hypothetical set of multiple possible universes that together comprise all of reality.Multiverse may also refer to:-In fiction:* Multiverse , the fictional multiverse used by DC Comics...

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 fallacyThe inverse gambler's fallacy, named by philosopher Ian Hacking, is a formal fallacy of Bayesian inference which is similar to the better known gambler's fallacy. It is the fallacy of concluding, on the basis of an unlikely outcome of a random process, that the process is likely to have occurred...

.

In 1995 Weinberg's argument was refined by

Alexander VilenkinAlexander Vilenkin is Professor of Physics and Director of the Institute of Cosmology at Tufts University. A theoretical physicist who has been working in the field of cosmology for 25 years, Vilenkin has written over 150 papers and is responsible for introducing the ideas of eternal inflation and...

to predict a value for the cosmological constant that was only ten times the matter density, i.e. about three times the current value since determined.

### Cyclic model

More recent work has suggested the problem may be indirect evidence of a

cyclic universeA cyclic model is any of several cosmological models in which the universe follows infinite, self-sustaining cycles. For example, the oscillating universe theory briefly considered by Albert Einstein in 1930 theorized a universe following an eternal series of oscillations, each beginning with a...

possibly as allowed by

string theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

. With every cycle of the universe (

Big BangThe Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

then eventually a

Big CrunchIn physical cosmology, the Big Crunch is one possible scenario for the ultimate fate of the universe, in which the metric expansion of space eventually reverses and the universe recollapses, ultimately ending as a black hole singularity.- Overview :...

) taking about a trillion (10

^{12}) 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.”

## 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.

## Further reading

- Michael, E., University of Colorado, Department of Astrophysical and Planetary Sciences, "The Cosmological Constant",
*http://super.colorado.edu/~michaele/Lambda/lambda.html*
- Ferguson, Kitty (1991).
*Stephen Hawking: Quest For A Theory of Everything*, Franklin Watts. ISBN 0-553-29895-X.

## External links