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

Constraints on the Hubble parameter, $H_0$, 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), $η= D_{L}(z)(1+z)^{-2}/D_{A}(z) = 1$, where $D_L$ and $D_A$ 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 $H^{*}_{0}=H_{0}/η^{2}$. We use 25 ADD of galaxy clusters to obtain simultaneous constraints on $H_{0}$ and possible violation of the DD relation in a flat $Λ$CDM model. Such a violation is parametrized by two functions: $η(z) = 1 + η_{0}z$ and $η(z) = 1 + η_{0}z/(1+z)$, where $η_0$ is a constant parameter quantifying possible departures from the strict validity. Finally, by marginalizing on the $η_{0}$ in both parameterizations, we obtain constraints on $H_0$ regardless of the validity of the DD relation. For the linear and non linear $η(z)$ functions, we obtain $H_{0}= 75^{+ 7}_{-7}$ km/s/Mpc and $H_{0}= 75^{+ 10}_{-7}$ km/s/Mpc, respectively (without systematic erros). Our results support recent $H_{0}$ measurements by using X-ray and SZE observations of galaxy clusters which have taken the distance duality as valid.