The
relative expansion of the
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. ...
is parametrized by a
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
scale factor
. Also known as the cosmic scale factor or sometimes the Robertson Walker scale factor,
this is a key parameter of the
Friedmann equations.
In the early stages of the
Big Bang, most of the energy was in the form of radiation, and that radiation was the dominant influence on the expansion of the universe. Later, with cooling from the expansion the roles of matter and radiation changed and the universe entered a matter-dominated era. Recent results suggest that we have already entered an era dominated by
dark energy
In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
, but examination of the roles of matter and radiation are most important for understanding the early universe.
Using the dimensionless scale factor to characterize the expansion of the universe, the effective energy densities of radiation and matter scale differently. This leads to a radiation-dominated era in the very early universe but a transition to a matter-dominated era at a later time and, since about 4 billion years ago, a subsequent dark-energy-dominated era.
Detail
Some insight into the expansion can be obtained from a Newtonian expansion model which leads to a simplified version of the Friedmann equation. It relates the proper distance (which can change over time, unlike the
comoving distance
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a spec ...
which is constant and set to today's distance) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting
FLRW universe at any arbitrary time
to their distance at some reference time
. The formula for this is:
:
where
is the proper distance at epoch
,
is the distance at the reference time
, usually also referred to as comoving distance, and
is the scale factor. Thus, by definition,
and
.
The scale factor is dimensionless, with
counted from the birth of the universe and
set to the present
age of the universe
In physical cosmology, the age of the universe is the time elapsed since the Big Bang. Astronomers have derived two different measurements of the age of the universe:
a measurement based on direct observations of an early state of the universe, ...
:
giving the current value of
as
or
.
The evolution of the scale factor is a dynamical question, determined by the equations of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, which are presented in the case of a locally isotropic, locally homogeneous universe by the
Friedmann equations
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann ...
.
The
Hubble parameter
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
is defined as:
:
where the dot represents a time
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
. The Hubble parameter varies with time, not with space, with the Hubble constant
being its current value.
From the previous equation
one can see that
, and also that
, so combining these gives
, and substituting the above definition of the Hubble parameter gives
which is just
Hubble's law
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
.
Current evidence suggests that
the expansion rate of the universe is accelerating, which means that the second derivative of the scale factor
is positive, or equivalently that the first derivative
is increasing over time. This also implies that any given galaxy recedes from us with increasing speed over time, i.e. for that galaxy
is increasing with time. In contrast, the Hubble parameter seems to be decreasing with time, meaning that if we were to look at some fixed distance d and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.
According to the
Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
which is used to model the expanding universe, if at present time we receive light from a distant object with a
redshift of ''z'', then the scale factor at the time the object originally emitted that light is
.
Chronology
Radiation-dominated era
After
Inflation
In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reduct ...
, and until about 47,000 years
after the Big Bang, the dynamics of the
early universe
The chronology of the universe describes the history and future of the universe according to Big Bang cosmology.
Research published in 2015 estimates the earliest stages of the universe's existence as taking place 13.8 billion years ago, wit ...
were set by
radiation (referring generally to the constituents of the universe which moved
relativistically, principally
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s and
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s).
For a radiation-dominated universe the evolution of the scale factor in the
Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
is obtained solving the
Friedmann equations
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann ...
:
:
Matter-dominated era
Between about 47,000 years and 9.8 billion years
after the Big Bang, the energy density of matter exceeded both the energy density of radiation and the vacuum energy density.
When the
early universe
The chronology of the universe describes the history and future of the universe according to Big Bang cosmology.
Research published in 2015 estimates the earliest stages of the universe's existence as taking place 13.8 billion years ago, wit ...
was about 47,000 years old (redshift 3600),
mass–energy density surpassed the
radiation energy, although the universe remained
optically thick to radiation until the universe was about 378,000 years old (redshift 1100). This second moment in time (close to the time of
recombination), at which the photons which compose the
cosmic microwave background radiation
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all space ...
were last scattered, is often mistaken as marking the end of the radiation era.
For a matter-dominated universe the evolution of the scale factor in the
Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
is easily obtained solving the
Friedmann equations
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann ...
:
:
Dark-energy-dominated era
In
physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
, the dark-energy-dominated era is proposed as the last of the three phases of the known universe, the other two being the
matter-dominated era
The relative expansion of the universe is parametrized by a dimensionless scale factor a . Also known as the cosmic scale factor or sometimes the Robertson Walker scale factor, this is a key parameter of the Friedmann equations.
In the early st ...
and the
radiation-dominated era
The relative expansion of the universe is parametrized by a dimensionless scale factor a . Also known as the cosmic scale factor or sometimes the Robertson Walker scale factor, this is a key parameter of the Friedmann equations.
In the early s ...
. The dark-energy-dominated era began after the matter-dominated era, i.e. when the Universe was about 9.8 billion years old. In the era of
cosmic inflation, the Hubble parameter is also thought to be constant, so the expansion law of the dark-energy-dominated era also holds for the inflationary prequel of the big bang.
The
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
is given the symbol Λ, and, considered as a source term in the Einstein field equation, can be viewed as equivalent to a "mass" of empty space, or
dark energy
In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
. Since this increases with the volume of the universe, the expansion pressure is effectively constant, independent of the scale of the universe, while the other terms decrease with time. Thus, as the density of other forms of matter – dust and radiation – drops to very low concentrations, the cosmological constant (or "dark energy") term will eventually dominate the energy density of the Universe. Recent measurements of the change in Hubble constant with time, based on observations of distant
supernovae, show this acceleration in expansion rate,
The Nobel Prize in Physics 2011
Retrieved 18 May 2017. indicating the presence of such dark energy.
For a dark-energy-dominated universe, the evolution of the scale factor in the Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
is easily obtained solving the Friedmann equations
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann ...
:
:
Here, the coefficient in the exponential, the Hubble constant
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving ...
, is
:
This exponential dependence on time makes the spacetime geometry identical to the de Sitter universe
A de Sitter universe is a cosmological solution to the Einstein field equations of general relativity, named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominat ...
, and only holds for a positive sign of the cosmological constant, which is the case according to the currently accepted value of the cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
, Λ, that is approximately
The current density of the observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these ob ...
is of the order of and the age of the universe is of the order of 13.8 billion years, or . The Hubble constant, , is (The Hubble time is 13.79 billion years).
See also
*Cosmological principle
In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...
*Lambda-CDM model
The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda (Greek Λ) associated with ...
* Redshift
Notes
References
*
*
External links
Relation of the scale factor with the cosmological constant and the Hubble constant
{{DEFAULTSORT:Scale Factor (Cosmology)
Physical cosmology