Test of the cosmic distance duality relation for arbitrary spatial curvature

Jin Qin, Fulvio Melia, Tong Jie Zhang

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5 Scopus citations


The cosmic distance duality relation (CDDR), η(z) = (1 + z)2dA(z)/dL(z) = 1, is one of the most fundamental and crucial formulae in cosmology. This relation couples the luminosity and angular diameter distances, two of the most often used measures of structure in the Universe. We here propose a new model-independent method to test this relation, using strong gravitational lensing (SGL) and the high-redshift quasar Hubble diagram reconstructed with a Bézier parametric fit. We carry out this test without pre-assuming a zero spatial curvature, adopting instead the value ωK = 0.001 ± 0.002 optimized by Planck in order to improve the reliability of our result. We parametrize the CDDR using η(z) = 1 + η0z, 1 + η1z + η2z2, and 1 + η3z/(1 + z), and consider both the SIS and non-SIS lens models for the strong lensing. Our best-fitting results are: η 0=-0.021+0.068-0.048, η 1=-0.404 +0.123-0.090, η 2=0.106 +0.028-0.034, and η 3=-0.507+0.193-0.133 for the SIS model, and η 0=-0.109+0.044-0.031 for the non-SIS model. The measured η(z), based on the Planck parameter ωK, is essentially consistent with the value (=1) expected if the CDDR were fully respected. For the sake of comparison, we also carry out the test for other values of ωK, but find that deviations of spatial flatness beyond the Planck optimization are in even greater tension with the CDDR. Future measurements of SGL may improve the statistics and alter this result but, as of now, we conclude that the CDDR favours a flat Universe.

Original languageEnglish (US)
Pages (from-to)3500-3509
Number of pages10
JournalMonthly Notices of the Royal Astronomical Society
Issue number3
StatePublished - Apr 1 2021


  • Cosmological parameters
  • Cosmology: observations
  • Cosmology: theory

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science


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