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 $\eta$ is a function of the redshift
parametrized by two different relations: \eta(z) = 1 + \eta_{0}z, and \eta(z)=1
+ \eta_{0}z/(1+z), where \eta_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 \eta_{0}, we consider the
angular diameter distances from galaxy clusters recently studied by two
different groups by assuming elliptical (isothermal) and spherical
(non-isothermal) $\beta$ models. The strict validity of the DD relation will
occur only if the maximum value of \eta_{0} PDF is centered on \eta_{0}=0. It
was found that the elliptical $\beta$ model is in good agreement with the data,
showing no violation of the DD relation (PDF peaked close to \eta_0=0 at
1-sigma), while the spherical (non-isothermal) one is only marginally
compatible at 3-sigma.
