The Etherington's distance-duality equation is the relationship between the
luminosity distance of
standard candles and the
angular diameter distance
In astronomy, angular diameter distance is a distance defined in terms of an object's physical size, x, and its angular size, \theta, as viewed from Earth:
d_A= \frac
Cosmology dependence
The angular diameter distance depends on the assumed cos ...
.
The equation is as follows:
, where
is the redshift,
is the luminosity distance and
the angular-diameter distance.
History and derivations
When
Ivor Etherington
Ivor Malcolm Haddon Etherington FRSE (8 February 1908 -1 January 1994) was a mathematician who worked initially on general relativity, and later on genetics and introduced genetic algebras.
Life
He was born in Lewisham in London the son of Annie ...
introduced this equation in 1933, he mentioned that this equation was proposed by Tolman as a way to test a cosmological model. Ellis proposed a proof of this equation in the context of
Riemannian geometry.
A quote from Ellis: "The core of the reciprocity theorem is the fact that many geometric properties are invariant when the roles of the source and observer in astronomical observations are transposed". This statement is fundamental in the derivation of the reciprocity theorem.
Validation from astronomical observations
The Etherington's distance-duality equation has been validated from astronomical observations based on the X-ray surface brightness and the
Sunyaev–Zel'dovich effect of
galaxy clusters. The reciprocity theorem is considered to be true when photon number is conserved, gravity is described by a metric theory with photons traveling on unique null geodesics. Any violation of the distance duality would be attributed to exotic physics provided that astrophysical effects altering the cosmic distance measurements are well below the statistical errors. For instance, an incorrect modelling of the three-dimensional gas density profile in galaxy clusters may introduce systematic uncertainties in the determination of the cluster angular diameter distance from X-ray and/or SZ observations, thus altering the outcome of the distance-duality test. Similarly, unaccounted extinction from a diffuse dust component in the inter-galactic medium can affect the determination of luminosity distances and cause a violation of the distance-duality relation.
[{{cite journal , last=Corasaniti , first=P. S. , title=The impact of cosmic dust on supernova cosmology , journal=Monthly Notices of the Royal Astronomical Society , volume=372 , issue=1 , date=2006-10-11 , issn=0035-8711 , doi=10.1111/j.1365-2966.2006.10825.x, arxiv=astro-ph/0603833 , pages=191–198, bibcode=2006MNRAS.372..191C , doi-access=free]
See also
*
Distance measures (cosmology)
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 red ...
References
Physical quantities