Distance measures (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. 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 common 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–Lemaître–Robertson–Walker solution is used to describe the universe.
There are a few different definitions of "distance" in cosmology which all coincide for sufficiently small redshifts. The expressions for these distances are most practical when written as functions of redshift , since redshift is always the observable. They can easily be written as functions of scale factor , cosmic time or conformal time as well by performing a simple transformation of variables. By defining the dimensionless Hubble parameter and the Hubble distance , the relation between the different distances becomes apparent.
Here, is the total radiation energy density, is the total matter density, is the dark energy density, represents the curvature, is the Hubble parameter today and is the speed of light. The Hubble parameter at a given redshift is then .
To compute the distance to an object from its redshift, we must integrate the above equation. Although for some limited choices of parameters (e.g. matter-only: ) the comoving distance integral defined below has a closed analytic form, in general—and specifically for the parameters of our Universe—we can only find a solution numerically. Cosmologists commonly use the following measures for distances from the observer to an object at redshift along the line of sight:
Where here is the Hubble distance, defined as the speed of light times the Hubble time; .
Transverse comoving distance:
Angular diameter distance:
Note that the comoving distance is recovered from the transverse comoving distance by taking the limit , such that the two distance measures are equivalent in a flat universe.
Age of the universe is , and the time elapsed since redshift until now is
Peebles (1993) calls the transverse comoving distance the "angular size distance", which is not to be mistaken for the angular diameter distance. Even though it is not a matter of nomenclature, the comoving distance is equivalent to the proper motion distance, which is defined as the ratio of the transverse velocity and its proper motion in radians per time. Occasionally, the symbols or are used to denote both the comoving and the angular diameter distance. Sometimes, the light-travel distance is also called the "lookback distance".
The comoving distance between fundamental observers, i.e. observers that are both moving with the Hubble flow, does not change with time, as comoving distance accounts for the expansion of the universe. Comoving distance is obtained by integrating the proper distances of nearby fundamental observers along the line of sight (LOS), where the proper distance is what a measurement at constant cosmic time would yield.
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to measure distances between objects; the comoving distance is the proper distance at the present time.
Proper distance roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe. Comoving distance factors out the expansion of the universe, which gives a distance that does not change in time due to the expansion of space (though this may change due to other, local factors, such as the motion of a galaxy within a cluster); the comoving distance is the proper distance at the present time.
Transverse comoving distance
Two comoving objects at constant redshift that are separated by an angle on the sky are said to have the distance , where the transverse comoving distance is defined appropriately.
Angular diameter distance
An object of size at redshift that appears to have angular size has the angular diameter distance of . This is commonly used to observe so called standard rulers, for example in the context of baryon acoustic oscillations.
If the intrinsic luminosity of a distant object is known, we can calculate its luminosity distance by measuring the flux and determine , which turns out to be equivalent to the expression above for . This quantity is important for measurements of standard candles like type Ia supernovae, which were first used to discover the acceleration of the expansion of the universe.
This distance is the time (in years) that it took light to reach the observer from the object multiplied by the speed of light. For instance, the radius of the observable universe in this distance measure becomes the age of the universe multiplied by the speed of light (1 light year/year) i.e. 13.8 billion light years. Also see misconceptions about the size of the visible universe.
Etherington's distance duality
The Etherington's distance-duality equation  is the relationship between the luminosity distance of standard candles and the angular-diameter distance. It is expressed as follows:
- Big Bang
- Comoving distance
- Friedmann equations
- Physical cosmology
- Cosmic distance ladder
- Friedmann-Lemaître-Robertson-Walker metric
- Subatomic scale
- David W. Hogg (2000). "Distance measures in cosmology". arXiv:astro-ph/9905116v4.
- Peebles, P. J. E. (1993). Principles of Physical Cosmology. Princeton University Press. pp. 310–320. Bibcode:1993ppc..book.....P. ISBN 978-0-691-01933-8.
- I.M.H. Etherington, “LX. On the Definition of Distance in General Relativity”, Philosophical Magazine, Vol. 15, S. 7 (1933), pp. 761-773.
- Scott Dodelson, Modern Cosmology. Academic Press (2003).
- '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.