Constraints on the Hubble Parameter from galaxy clusters and the Validity of the Cosmic Distance Duality Relation

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Constraints on the Hubble parameter, H₀, via X-ray surface brightness and Sunyaev-Zel'dovich effect (SZE) observations of the galaxy clusters depend on the validity of the cosmic distance duality relation (DD relation), η= Dₗ(z)(1+z)⁻²/Dₐ(z) = 1, where Dₗ and Dₐ are the luminosity distance and angular diameter distance (ADD), respectively. In this work, we argue that if the DD relation does not hold the X-ray plus SZE technique furnishes a H*₀=H₀/η². We use 25 ADD of galaxy clusters to obtain simultaneous constraints on H₀ and possible violation of the DD relation in a flat ΛCDM model. Such a violation is parametrized by two functions: η(z) = 1 + η₀z and η(z) = 1 + η₀z/(1+z), where η₀ is a constant parameter quantifying possible departures from the strict validity. Finally, by marginalizing on the η₀ in both parameterizations, we obtain constraints on H₀ regardless of the validity of the DD relation. For the linear and non linear η(z) functions, we obtain H₀= 75⁺ ⁷₋₇ km/s/Mpc and H₀= 75⁺ ¹⁰₋₇ km/s/Mpc, respectively (without systematic erros). Our results support recent H₀ measurements by using X-ray and SZE observations of galaxy clusters which have taken the distance duality as valid.