Scale factor (Universe)
Encyclopedia
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. It is the (time-dependent) factor that relates the proper distance (which can change over time, unlike the comoving distance
which is constant) for a pair of objects moving with the Hubble flow in an expanding or contracting FLRW universe—the distance between a pair of galaxies, for example—at any arbitrary time
to their distance at some reference time, generally taken to be the present, by the formula:
where
is the proper distance at epoch 
, 
is the distance at the reference time 
(the present epoch), and 
is the scale factor. Thus, by definition, 
.
The scale factor could, in principle, have units of length or be dimensionless. Most commonly in modern usage, it is chosen to be dimensionless, with the current value equal to one:
, where 
is counted from the birth of the universe and 
is the present age of the universe
:
.
The evolution of the scale factor is a dynamical question, determined by the equations of general relativity
, which are presented in the case of a locally isotropic, locally homogeneous universe by the Friedmann equations
.
The Hubble parameter is defined:
where the dot represents a time derivative
. 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
.
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 which is used to model the expanding universe, if at the 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 given by the equation
.
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...
is a function of time which represents the relative expansion
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...
of the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
. It is sometimes called the Robertson-Walker scale factor. It is the (time-dependent) factor that relates the proper distance (which can change over time, unlike the comoving distance
Comoving distance
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
which is constant) for a pair of objects moving with the Hubble flow in an expanding or contracting FLRW universe—the distance between a pair of galaxies, for example—at any arbitrary time
to their distance at some reference time, generally taken to be the present, by the formula:where
is the proper distance at epoch , is the distance at the reference time (the present epoch), and is the scale factor. Thus, by definition, .The scale factor could, in principle, have units of length or be dimensionless. Most commonly in modern usage, it is chosen to be dimensionless, with the current value equal to one:
, where is counted from the birth of the universe and is the present age of the universeAge of the universe
The age of the universe is the time elapsed since the Big Bang posited by the most widely accepted scientific model of cosmology. The best current estimate of the age of the universe is 13.75 ± 0.13 billion years within the Lambda-CDM concordance model...
:
.The evolution of the scale factor is a dynamical question, determined by the 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 are presented in the case of a locally isotropic, locally homogeneous universe by the Friedmann equations
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...
.
The Hubble parameter is defined:
where the dot represents a time derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
. 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 lawHubble's law
Hubble'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...
.
Current evidence suggests that the expansion rate of the universe is accelerating
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...
, 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 which is used to model the expanding universe, if at the present time we receive light from a distant object with a redshift
Redshift
In physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...
of z, then the scale factor at the time the object originally emitted that light is given by the equation
.See also
- Friedmann equationsFriedmann equationsThe 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...
- Friedmann-Lemaître-Robertson-Walker metric
- RedshiftRedshiftIn physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...
- Cosmological principleCosmological PrincipleIn 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...
- Lambda-CDM modelLambda-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...
- Hubble's lawHubble's lawHubble'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...