Distance measures (cosmology)
Encyclopedia
Distance measures are used in physical cosmology
to give a natural notion of the distance
between two objects or events in the universe
. They are often used to tie some observable quantity (such as the luminosity
of a distant quasar
, the redshift
of a distant galaxy
, or the angular size of the acoustic peaks in the CMB
power spectrum) to another quantity that is not directly observable, but is more convenient for calculations (such as the comoving coordinates of the quasar, galaxy, etc). The distance measures discussed here all reduce to the naïve notion of Euclidean
distance at low redshift.
In accord with our present understanding of cosmology, these measures are calculated within the context of general relativity
, where the Friedmann–Lemaitre–Robertson–Walker solution is used to describe the universe.
The latter three are related by:
where z is the redshift.
If and only if the curvature is zero, then proper motion distance and comoving distance are identical, i.e. .
For negative curvature,
while for positive curvature,
where is the (absolute value
of the) radius of curvature.
A practical formula for numerically integrating to a redshift for arbitrary values of the matter density parameter , the cosmological constant
density parameter , and the equation of state parameter is
where c is the speed of light
and H_{0} is the Hubble constant.
By using sin and sinh functions, proper motion distance d_{pm} can be obtained from d_{p}.
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...
to give a natural notion of the distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
between two objects or events in 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...
. They are often used to tie some observable quantity (such as the luminosity
Luminosity
Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...
of a distant quasar
Quasar
A quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...
, the 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 a distant galaxy
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...
, or the angular size of the acoustic peaks in the CMB
CMB
CMB can mean:*The IATA airport code for Bandaranaike International Airport, Colombo – Sri Lanka's only international airport*C.M.B., the debut album of American R&B and pop group Color Me Badd...
power spectrum) to another quantity that is not directly observable, but is more convenient for calculations (such as the comoving coordinates of the quasar, galaxy, etc). The distance measures discussed here all reduce to the naïve notion of Euclidean
Euclidean
Euclidean relates to Euclid , a town or others. It may refer to:Geometry...
distance at low redshift.
In accord with our present understanding of cosmology, these measures are calculated within the context 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...
, where the Friedmann–Lemaitre–Robertson–Walker solution is used to describe the universe.
Types of distance measures
- Angular diameter distance is a good indication (especially in a flat universe) of how near an astronomical object was to us when it emitted the light that we now see.
- Luminosity distance.
- Comoving distanceComoving distanceIn standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
. The distance between two points measured along a path defined at the present cosmological time. - Cosmological proper distanceComoving distanceIn standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
. The distance between two points measured along a path defined at a constant cosmological time. The cosmological proper distance should not be confused with the more general proper length or proper distanceProper lengthIn relativistic physics, proper length is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime....
. - Light travel time or lookback time. This is how long ago light left an object of given redshift.
- Light travel distance (LTD). The light travel time times the speed of lightSpeed of lightThe 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...
. For values above 2 billion light years, this value does not equal the comoving distance or the angular diameter distance anymore, because of the expansion of the universe. Also see misconceptions about the size of the visible universe. - Naive 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...
, taking z = H_{0}d/c, with H_{0} today's Hubble constant, z the 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...
of the object, c the speed of lightSpeed of lightThe 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...
, and d the "distance."
Comparison of distance measures
- light-travel distance – simply the speed of light times the cosmological time interval, i.e. integral of c dt, while the comoving distance is the integral of c dt/a(t).
- d_{L} luminosity distance
- d_{pm} proper motion distance
- called the angular size distance by Peebles 1993, but should not be confused with angular diameter distance http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?bibcode=1993ppc..book.....P&db_key=AST&high=3ece3bb64809032)
- sometimes called the coordinate distance
- sometimes d_{pm} is called the angular diameter distance
- d_{a} angular diameter distance
The latter three are related by:
- d_{a} = d_{pm}/(1 + z) = d_{L}/(1 + z)^{2}
where z is the redshift.
If and only if the curvature is zero, then proper motion distance and comoving distance are identical, i.e. .
For negative curvature,
while for positive curvature,
where is the (absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
of the) radius of curvature.
A practical formula for numerically integrating to a redshift for arbitrary values of the matter density parameter , 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...
density parameter , and the equation of state parameter is
where c is the speed of light
Speed 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...
and H_{0} is the Hubble constant.
By using sin and sinh functions, proper motion distance d_{pm} can be obtained from d_{p}.
See also
- Big BangBig 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...
- Comoving distanceComoving distanceIn standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
- 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...
- Physical cosmologyPhysical 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...
- Cosmic distance ladderCosmic distance ladderThe cosmic distance ladder is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement of an astronomical object is possible only for those objects that are "close enough" to Earth...
- Friedmann–Lemaître–Robertson–Walker metric
External links
- 'The Distance Scale of the Universe' compares different cosmological distance measures.
- 'Distance measures in cosmology' explains in detail how to calculate the different distance measures as a function of world model and redshift.
- iCosmos: Cosmology Calculator (With Graph Generation ) calculates the different distance measures as a function of cosmological model and redshift, and generates plots for the model from redshift 0 to 20.