Etherington's reciprocity theorem
The Etherington's distance-duality equation is the relationship between the luminosity distance of standard candles and the angular diameter distance.<ref name=Etherington1933>Etherington, I.M.H. (1933). "LX.On the definition of distance in general relativity". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Informa UK Limited. 15 (100): 761–773. doi:10.1080/14786443309462220. ISSN 1941-5982.</ref> The equation is as follows: <math>d_L=(1+z)^2 d_A</math>, where <math>z</math> is the redshift, <math>d_L</math> is the luminosity distance and <math>d_A</math> the angular-diameter distance.
History and derivations
When Ivor Etherington 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.<ref>G. F. R. Ellis, "Relativistic cosmology", Proceedings of the 47th International School of Physics "Enrico Fermi", edited by R. K. Sachs (Academic Press, New York and London), Vol. 15 (1971), pp. 104-182.</ref><ref name=Etherington1933 /><ref>Ellis, George F. R. (2006-10-24). "On the definition of distance in general relativity: I. M. H. Etherington (Philosophical Magazine ser. 7, vol. 15, 761 (1933))". General Relativity and Gravitation. Springer Science and Business Media LLC. 39 (7): 1047–1052. doi:10.1007/s10714-006-0355-5. ISSN 0001-7701. S2CID 121062656.</ref> 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.<ref>Uzan, Jean-Philippe; Aghanim, Nabila; Mellier, Yannick (2004-10-27). "Distance duality relation from x-ray and Sunyaev-Zel'dovich observationsof clusters". Physical Review D. American Physical Society (APS). 70 (8): 083533. arXiv:astro-ph/0405620. Bibcode:2004PhRvD..70h3533U. doi:10.1103/physrevd.70.083533. ISSN 1550-7998. S2CID 119430471.</ref><ref>De Bernardis, Francesco; Giusarma, Elena; Melchiorri, Alessandro (2006). "Constraints on Dark Energy and Distance Duality from Sunyaev-Zel'dovich Effect and Chandra X-Ray Measurements". International Journal of Modern Physics D. World Scientific Pub Co Pte Lt. 15 (5): 759–766. arXiv:gr-qc/0606029. doi:10.1142/s0218271806008486. ISSN 0218-2718. S2CID 119440320.</ref> 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.<ref>Bassett, Bruce A.; Kunz, Martin (2004-05-26). "Cosmic distance-duality as a probe of exotic physics and acceleration". Physical Review D. American Physical Society (APS). 69 (10): 101305(R). arXiv:astro-ph/0312443. Bibcode:2004PhRvD..69j1305B. doi:10.1103/physrevd.69.101305. ISSN 1550-7998. S2CID 580071.</ref> 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.<ref>Meng, Xiao-Lei; Zhang, Tong-Jie; Zhan, Hu; Wang, Xin (2012-01-04). "Morphology of Galaxy Clusters: A Cosmological Model-independent Test of the Cosmic Distance-Duality Relation". The Astrophysical Journal. IOP Publishing. 745 (1): 98. arXiv:1104.2833. Bibcode:2012ApJ...745...98M. doi:10.1088/0004-637x/745/1/98. ISSN 0004-637X.</ref> 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.<ref>Corasaniti, P. S. (2006-10-11). "The impact of cosmic dust on supernova cosmology". Monthly Notices of the Royal Astronomical Society. 372 (1): 191–198. arXiv:astro-ph/0603833. Bibcode:2006MNRAS.372..191C. doi:10.1111/j.1365-2966.2006.10825.x. ISSN 0035-8711.</ref>
See also
References
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