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. Physical cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... Universe is a word derived from the Old French univers, which in turn comes from the Latin roots unus (one) and versus (a form of vertere, to turn). Based on observations of the observable universe, physicists attempt to describe the whole of space-time, including all matter and energy and... Luminosity has different meanings in several different fields of science. ... Artists impression of quasar GB1508 A quasar (contraction of QUASi-stellAR radio source) is an astronomical source of electromagnetic energy, including radio waves and visible light. ... Redshift of spectral lines in the optical spectrum of a supercluster of distant galaxies (right), as compared with that of the Sun (left). ... NGC 4414, a typical spiral galaxy in the constellation Coma Berenices, is about 17,000 parsecs in diameter and approximately 20 million parsecs distant. ... WMAP image of the CMB anisotropy,Cosmic microwave background radiation(June 2003) The cosmic microwave background radiation (CMB) is a form of electromagnetic radiation that fills the whole of the universe. ... The comoving distance or conformal distance of two objects in the universe is the distance divided by a time-varying scale factor representing the expansion of the universe. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

In accord with our present understanding of cosmology, these measures are calculated within the context of general relativity, where the Friedmann-Robertson-Walker solution is used to describe the universe. General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ...

Types of Distance Measures

• Angular diameter distance or Proper motion distance
• Luminosity distance
• Comoving distance
• Lookback time (this is how long ago light left an object of given redshift)
• Naive Hubble's law, taking z = H₀d/c, with H₀ today's Hubble constant, z the redshift of the object, c the speed of light, and d the "distance."

The angular diameter distance is a distance measure used in astronomy. ... The angular diameter distance is a distance measure used in astronomy. ... Luminosity distance DL is defined in terms of the relationship between the absolute magnitude M and apparent magnitude m of an astronomical object: which gives: Calculating the luminosity distance of an object correctly from its real distance is quite complex, but there are a number of useful webpages for performing... The comoving distance or conformal distance of two objects in the universe is the distance divided by a time-varying scale factor representing the expansion of the universe. ... Hubbles law is the statement in physical cosmology that the redshift in light coming from distant galaxies is proportional to their distance. ... Hubbles law is the statement in astronomy that the redshift in light coming from distant galaxies is proportional to their distance. ... Redshift of spectral lines in the optical spectrum of a supercluster of distant galaxies (right), as compared with that of the Sun (left). ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation...

Comparison of Distance Measures

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• 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 [1])
  • 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: Luminosity distance DL is defined in terms of the relationship between the absolute magnitude M and apparent magnitude m of an astronomical object: which gives: Calculating the luminosity distance of an object correctly from its real distance is quite complex, but there are a number of useful webpages for performing... The angular diameter distance is a distance measure used in astronomy. ... The angular diameter distance is a distance measure used in astronomy. ...

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. d_pm = χ.

For negative curvature,

,

while for positive curvature,

,

where R_C is the (absolute value of the) radius of curvature. In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

A practical formula for numerically integrating d_p to a redshift z for arbitrary values of the matter density parameter Ω_m, the cosmological constant Ω_Λ, and the quintessence parameter w is The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... The word quintessence is used in different fields: In physics, quintessence is a hypothetical form of energy postulated to exist as a possible explanation of observations of an accelerating universe. ...

where c is the speed of light and H₀ is the Hubble constant. A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation... Hubbles law is the statement in astronomy that the redshift in light coming from distant galaxies is proportional to their distance. ...

By using sin and sinh functions, proper motion distance d_pm can be obtained from d_p.

See also

• Big Bang
• Comoving distance
• Friedmann equations
• Physical Cosmology

According to the Big Bang model, the universe emerged from an extremely dense and hot state. ... The comoving distance or conformal distance of two objects in the universe is the distance divided by a time-varying scale factor representing the expansion of the universe. ... The Friedmann equations relate various cosmological parameters within the context of general relativity. ... Physical cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ...

References

• P. J. E. Peebles, Principles of Physical Cosmology. Princeton University Press (1993)
• Scott Dodelson, Modern Cosmology. Academic Press (2003).