De Sitter universe
Encyclopedia
A de Sitter universe is a cosmological
solution to Einstein
's field equations of General Relativity
which is named after Willem de Sitter
. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the cosmological constant
, thought to correspond to dark energy
in our universe or the inflaton field in the early universe. According to the models of inflation
and current observations of the accelerating universe
, the concordance models of physical cosmology are converging on a consistent model where our universe was best described as a de Sitter universe at about a time
seconds after the fiducial Big Bang
singularity
, and far into the future
.
which sets the expansion rate,
. A larger cosmological constant leads to a larger expansion rate:
,
where the constants of proportionality depend on conventions. The cosmological constant is
.
It is common to describe a patch of this solution as an expanding universe of the FLRW
form where the scale factor is given by
,
where the constant
is the Hubble expansion rate and 
is time. As in all FLRW spaces, 
, the scale factor
, describes the expansion of physical spatial distances
.
Unique to universes described by the FLRW metric, a de Sitter universe has a Hubble Law which is not only consistent through all space, but also through all time (since the deceleration parameter is equal to
), thus satisfying the perfect cosmological principle
that assumes isotropy and homogeneity throughout space and time. As a class of models with different values of the Hubble constant, the static universe
that Einstein developed, and for which he invented the cosmological constant, can be considered a special case of the de Sitter universe where the expansion is finely tuned
to just cancel out the collapse associated with the positive curvature associated with a non-zero matter density. There are ways to cast de Sitter space with static coordinates (see de Sitter space
), so unlike other FLRW models, de Sitter space can be thought of as a static solution to Einstein's equations even though the geodesic
s followed by observers necessarily diverge in the normal way expected from the expansion of physical spatial dimensions. As a model for the universe, de Sitter's solution was not considered viable for the observed universe
until models for inflation
and dark energy
were developed. Before then, it was assumed that the Big Bang
implied only an acceptance of the weaker cosmological principle
which holds isotropy true only for spatial extents but not temporal extents.
a few billion years ago, our universe is probably approaching a de Sitter universe in the infinite future. If the current acceleration of our universe is due to a cosmological constant then as the universe continues to expand all of the matter and radiation will be diluted. Eventually there will be almost nothing left but the cosmological constant, and our universe will have become a de Sitter universe.
. At this point those two observers will no longer be able to make contact. Therefore any observer in a de Sitter universe would see event horizon
s beyond which that observer can never see nor learn any information. If our universe is approaching a de Sitter universe then eventually we will not be able to observe any galaxies
other than our own Milky Way
(and any others in the gravitationally bound Local Group
, assuming they were to somehow survive to that time without merging).
. Many inflationary models are approximately de Sitter space and can be modelled by giving the Hubble parameter a mild time dependence. For simplicity, some calculations involving inflation in the early universe can be performed in de Sitter space rather than a more realistic inflationary universe. By using the de Sitter universe instead, where the expansion is truly exponential, there are many simplifications.
Physical cosmology
Physical 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...
solution to Einstein
Albert Einstein
Albert 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...
's field equations of General Relativity
General relativity
General 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...
which is named after Willem de Sitter
Willem de Sitter
Willem de Sitter was a Dutch mathematician, physicist and astronomer.-Life and work:Born in Sneek, De Sitter studied mathematics at the University of Groningen and then joined the Groningen astronomical laboratory. He worked at the Cape Observatory in South Africa...
. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
, thought to correspond to dark energy
Dark energy
In 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...
in our universe or the inflaton field in the early universe. According to the models of inflation
Cosmic inflation
In 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...
and current observations of the accelerating universe
Accelerating universe
The 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...
, the concordance models of physical cosmology are converging on a consistent model where our universe was best described as a de Sitter universe at about a time
seconds after the fiducial Big BangBig Bang
The 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...
singularity
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
, and far into the future
Ultimate fate of the universe
The ultimate fate of the universe is a topic in physical cosmology. Many possible fates are predicted by rival scientific theories, including futures of both finite and infinite duration....
.
Mathematical Expression
A de Sitter universe has no ordinary matter content but with a positive cosmological constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
which sets the expansion rate,
. A larger cosmological constant leads to a larger expansion rate:,where the constants of proportionality depend on conventions. The cosmological constant is
.It is common to describe a patch of this solution as an expanding universe of the FLRW
Friedmann-Robertson-Walker metric
The Friedmann–Lemaître–Robertson–Walker metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic expanding or contracting universe that may be simply connected or multiply connected...
form where the scale factor is given by
,where the constant
is the Hubble expansion rate and is time. As in all FLRW spaces, , the scale factorScale factor (Universe)
The scale factor or cosmic scale factor parameter of the Friedmann equations is a function of time which represents the relative expansion of the universe. It is sometimes called the Robertson-Walker scale factor...
, describes the expansion of physical spatial distances
Metric expansion of space
The metric expansion of space is the increase of distance between distant parts of the universe with time. It is an intrinsic expansion—that is, it is defined by the relative separation of parts of the universe and not by motion "outward" into preexisting space...
.
Unique to universes described by the FLRW metric, a de Sitter universe has a Hubble Law which is not only consistent through all space, but also through all time (since the deceleration parameter is equal to
), thus satisfying the perfect cosmological principlePerfect Cosmological Principle
The Perfect Cosmological Principle states that the Universe is homogenous and isotropic in space and time. In this view the universe looks the same everywhere , the same as it always has and always will...
that assumes isotropy and homogeneity throughout space and time. As a class of models with different values of the Hubble constant, the static universe
Static universe
A 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...
that Einstein developed, and for which he invented the cosmological constant, can be considered a special case of the de Sitter universe where the expansion is finely tuned
Fine Tuning
Fine Tuning was the name of XM Satellite Radio's eclectic music channel. The program director for Fine Tuning was Ben Smith and the tag line was, "The World's Most Interesting Music"....
to just cancel out the collapse associated with the positive curvature associated with a non-zero matter density. There are ways to cast de Sitter space with static coordinates (see de Sitter space
De Sitter space
In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space , denoted dS_n, is the Lorentzian manifold analog of an n-sphere ; it is maximally symmetric, has constant positive curvature,...
), so unlike other FLRW models, de Sitter space can be thought of as a static solution to Einstein's equations even though the geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s followed by observers necessarily diverge in the normal way expected from the expansion of physical spatial dimensions. As a model for the universe, de Sitter's solution was not considered viable for the observed universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...
until models for inflation
Cosmic inflation
In 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...
and dark energy
Dark energy
In 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...
were developed. Before then, it was assumed that the Big Bang
Big Bang
The 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...
implied only an acceptance of the weaker cosmological principle
Cosmological Principle
In modern physical cosmology, the cosmological principle is the working assumption that observers on Earth do not occupy an unusual or privileged location within the universe as a whole, judged as observers of the physical phenomena produced by uniform and universal laws of physics...
which holds isotropy true only for spatial extents but not temporal extents.
Potential for Universe
Because our Universe entered the Dark Energy Dominated EraDark Energy Dominated Era
The dark-energy-dominated era refers to one of the three phases of the known universe, the other two being the matter-dominated era and the radiation-dominated era. The dark-energy-dominated era began after the matter-dominated era, i.e. when the Universe was about 5 billion years old...
a few billion years ago, our universe is probably approaching a de Sitter universe in the infinite future. If the current acceleration of our universe is due to a cosmological constant then as the universe continues to expand all of the matter and radiation will be diluted. Eventually there will be almost nothing left but the cosmological constant, and our universe will have become a de Sitter universe.
Relative Expansion
The exponential expansion of the scale factor means that the physical distance between any two non-accelerating observers will eventually be growing faster than the speed of lightSpeed of light
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...
. At this point those two observers will no longer be able to make contact. Therefore any observer in a de Sitter universe would see event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
s beyond which that observer can never see nor learn any information. If our universe is approaching a de Sitter universe then eventually we will not be able to observe any galaxies
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
other than our own Milky Way
Milky Way
The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky...
(and any others in the gravitationally bound Local Group
Local Group
The Local Group is the group of galaxies that includes Earth's galaxy, the Milky Way. The group comprises more than 30 galaxies , with its gravitational center located somewhere between the Milky Way and the Andromeda Galaxy...
, assuming they were to somehow survive to that time without merging).
Modelling cosmic inflation
Another application of de Sitter space is in the early universe during cosmic inflationCosmic inflation
In 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...
. Many inflationary models are approximately de Sitter space and can be modelled by giving the Hubble parameter a mild time dependence. For simplicity, some calculations involving inflation in the early universe can be performed in de Sitter space rather than a more realistic inflationary universe. By using the de Sitter universe instead, where the expansion is truly exponential, there are many simplifications.
See also
- Cosmic inflationCosmic 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...
- De Sitter spaceDe Sitter spaceIn mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space , denoted dS_n, is the Lorentzian manifold analog of an n-sphere ; it is maximally symmetric, has constant positive curvature,...
for more mathematical properties - Deceleration parameter
- Causal patchCausal patchAfter Leonard Susskind proposed the black hole complementarity conjecture for black holes in quantum gravity, he realized it would also apply to a de Sitter universe with a positive cosmological constant with the cosmological horizon in place of the event horizon. The region within the horizon is...