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Cosmic Distance Duality Relation and the Shape of Galaxy Clusters

Observations in the cosmological domain are heavily dependent on the validity of the cosmic distance-duality (DD) relation, D_L(z) (1 + z)^{2}/D_{A}(z) = 1, an exact result required by the Etherington reciprocity theorem where D_L(z) and D_A(z) are, respectively, the luminosity and angular diameter distances. In the limit of very small redshifts D_A(z) = D_L(z) and this ratio is trivially satisfied. Measurements of Sunyaev-Zeldovich effect (SZE) and X-rays combined with the DD relation have been used to determine D_A(z)from galaxy clusters. This combination offers the possibility of testing the validity of the DD relation, as well as determining which physical processes occur in galaxy clusters via their shapes. We use WMAP (7 years) results by fixing the conventional LCDM model to verify the consistence between the validity of DD relation and different assumptions about galaxy cluster geometries usually adopted in the literature. We assume that $η$ is a function of the redshift parametrized by two different relations: η(z) = 1 + η_{0}z, and η(z)=1 + η_{0}z/(1+z), where η_0 is a constant parameter quantifying the possible departure from the strict validity of the DD relation. In order to determine the probability density function (PDF) of η_{0}, we consider the angular diameter distances from galaxy clusters recently studied by two different groups by assuming elliptical (isothermal) and spherical (non-isothermal) $β$ models. The strict validity of the DD relation will occur only if the maximum value of η_{0} PDF is centered on η_{0}=0. It was found that the elliptical $β$ model is in good agreement with the data, showing no violation of the DD relation (PDF peaked close to η_0=0 at 1-sigma), while the spherical (non-isothermal) one is only marginally compatible at 3-sigma.