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Context. The angular diameter distances toward galaxy clusters can be determined with measurements of Sunyaev-Zel’dovich effect and X-ray surface brightness combined with the validity of the distance-duality relation, D L(z)(1 + z)2/D A(z) = 1, where D L(z) and D A(z) are, respectively, the luminosity and angular diameter distances. This combination enables us to probe galaxy cluster physics or even to test the validity of the distance-duality relation itself.Aims. We explore these possibilities based on two different, but complementary approaches. Firstly, in order to constrain the possible galaxy cluster morphologies, the validity of the distance-duality relation (DD relation) is assumed in the ΛCDM framework (WMAP7). Secondly, by adopting a cosmological-model-independent test, we directly confront the angular diameters from galaxy clusters with two supernovae Ia (SNe Ia) subsamples (carefully chosen to coincide with the cluster positions). The influence of the different SNe Ia light-curve fitters in the previous analysis are also discussed.Methods. We assumed 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 considered the angular diameter distances from galaxy clusters recently studied by two different groups by assuming elliptical and spherical isothermal β models and spherical non-isothermal β model. The strict validity of the DD relation will occur only if the maximum value of η 0 PDF is centered on η 0 = 0.Results. For both approaches we find that the elliptical β model agrees with the distance-duality relation, whereas the non-isothermal spherical description is, in the best scenario, only marginally compatible. We find that the two-light curve fitters (SALT2 and MLCS2K2) present a statistically significant conflict, and a joint analysis involving the different approaches suggests that clusters are endowed with an elliptical geometry as previously assumed. Conclusions. The statistical analysis presented here provides new evidence that the true geometry of clusters is elliptical. In principle, it is remarkable that a local property such as the geometry of galaxy clusters might be constrained by a global argument like the one provided by the cosmological distance-duality relation.

In this work we examine a relativistic model for the observed inhomogeneities of the large-scale structure where we make the hypothesis that this structure can be described as being a self-similar fractal system. The old Charlier concept of hierarchical clustering is identified with a fractal distribution, and the problems raised by the use of fractal ideas in a relativistic model are discussed, as well as their relations to the Copernican and cosmological principles. Voids, clusters, and superclusters of galaxies are assumed to be part of a smoothed-out fractal structure described by a Tolman solution. The basic concepts of the Newtonian model presented by Pietronero are reinterpreted and applied to this inhomogeneous curved spacetime. This fractal system is also assumed to have a crossover to homogeneity which leads to a "Swiss cheese "-type model, composed by an interior Tolman metric and an exterior dust Friedmann solution. The Darmois junction conditions between the two spacetimes are calculated, and we also obtain for the interior region the observational relations necessary to compare the model with observations. The differential equations of the interior spacetime are set up, and we devise a numerical strategy for finding particular Tolman solutions representing a fractal behavior along the past light cone.