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), $\eta=
D_{L}(z)(1+z)^{-2}/D_{A}(z) = 1$, 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 $H^{*}_{0}=H_{0}/\eta^{2}$. We use 25 ADD of galaxy clusters to
obtain simultaneous constraints on $H_{0}$ and possible violation of the DD
relation in a flat $\Lambda$CDM model. Such a violation is parametrized by two
functions: $\eta(z) = 1 + \eta_{0}z$ and $\eta(z) = 1 + \eta_{0}z/(1+z)$, where
$\eta_0$ is a constant parameter quantifying possible departures from the
strict validity. Finally, by marginalizing on the $\eta_{0}$ in both
parameterizations, we obtain constraints on $H_0$ regardless of the validity of
the DD relation. For the linear and non linear $\eta(z)$ 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.
