Distance measures (cosmology)

Physical cosmology

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Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. The age of the Universe is the time elapsed between the theory of the Big Bang and the present day This timeline of the Big Bang describes the events according to the Scientific theory of the Big Bang, using the cosmological time parameter of Comoving coordinates The ultimate fate of the universe is a topic in Physical cosmology. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its Distance is a numerical description of how far apart objects are The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy 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). Luminosity has different meanings in several different fields of science A quasar (contraction of QUASi-stellAR radio source) is an extremely powerful and distant Active galactic nucleus. In Physics and Astronomy, redshift occurs when Electromagnetic radiation – usually Visible light – emitted or reflected by A galaxy is a massive gravitationally bound system consisting of Stars an Interstellar medium of gas and dust, and Dark matter In standard cosmology, ' comoving' distance and ' proper distance' are two closely related distance measures used by cosmologists to define distances between 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-Robertson-Walker solution is used to describe the universe. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

1 Types of Distance Measures
2 Comparison of Distance Measures
3 See also
4 References
5 External links

Types of Distance Measures

Angular diameter distance or proper motion distance. The angular diameter distance is a distance measure used in Astronomy. 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. Luminosity distance DL is defined in terms of the relationship between the Absolute magnitude M and Apparent magnitude m
Comoving distance. In standard cosmology, ' comoving' distance and ' proper distance' are two closely related distance measures used by cosmologists to define distances between The distance between two points measured along a path defined at the present cosmological time. This timeline of the Big Bang describes the events according to the Scientific theory of the Big Bang, using the cosmological time parameter of Comoving coordinates
Cosmological proper distance. In standard cosmology, ' comoving' distance and ' proper distance' are two closely related distance measures used by cosmologists to define distances between 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 distance. In relativistic Physics, proper Length is an invariant quantity which is the rod Distance between Spacelike
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 light. 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. In Big Bang Cosmology, the observable universe is the region of space bounded by a Sphere, centered on the observer that is small enough that
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. Hubble's law is the statement in Physical cosmology that the Redshift in light coming from distant galaxies is proportional to their distance Hubble's law is the statement in Physical cosmology that the Redshift in light coming from distant galaxies is proportional to their distance In Physics and Astronomy, redshift occurs when Electromagnetic radiation – usually Visible light – emitted or reflected by "

Comparison of Distance Measures

A comparison of cosmological distance measures, from redshift zero to redshift of 0. 5. The background cosmology is Hubble parameter 72 km/s/Mpc, Omega_lambda = 0. 732, Omega_matter = 0. 266, Omega_radiation = 0. 266/3454, and Omega_k chosen so that the sum of Omega parameters is one.

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:

d_a = d_pm / (1 + z) = d_L /(1 + z)²

where z is the redshift. Luminosity distance DL is defined in terms of the relationship between the Absolute magnitude M and Apparent magnitude m The angular diameter distance is a distance measure used in Astronomy.

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,

$d_{\mbox{pm}} = R_C \sinh {\chi \over R_C}$ ,

while for positive curvature,

$d_{\mbox{pm}} = R_C \sin {\chi \over R_C}$ ,

where $R C$ is the (absolute value of the) radius of curvature. In Mathematics, the absolute value (or modulus) 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

$d_p \equiv \chi(z) = {c \over H_0} \int^{a'=1}_{a'=1/(1+z)} {\mbox{d}a \over a \sqrt{ \Omega_m /a - (\Omega_m + \Omega_\Lambda -1) + \Omega_\Lambda a^{-(1+3w)} } },$

A comparison of cosmological distance measures, from redshift zero to redshift of 10,000, corresponding to the epoch of matter/radiation equality. In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification The background cosmology is Hubble parameter 72 km/s/Mpc, Omega_lambda = 0. 732, Omega_matter = 0. 266, Omega_radiation = 0. 266/3454, and Omega_k chosen so that the sum of Omega parameters is one.

where c is the speed of light and H₀ is the Hubble constant. Hubble's law is the statement in Physical cosmology 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.

References

P. The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. In standard cosmology, ' comoving' distance and ' proper distance' are two closely related distance measures used by cosmologists to define distances between The Friedmann equations are a set of Equations in cosmology that govern the expansion of space in homogeneous and isotropic models Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its The cosmic distance ladder (also known as the Extragalactic Distance Scale) is the succession of methods by which astronomers determine the Distances to celestial J. E. Peebles, Principles of Physical Cosmology. Princeton University Press (1993)
Scott Dodelson, Modern Cosmology. Academic Press (2003).

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.

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Contents

Types of Distance Measures

Comparison of Distance Measures

See also

References

External links