The Hubble constant from galaxy cluster scaling-relation and SNe Ia observations: a consistency test

In this paper, we propose a self-consistent test for a Hubble constant estimate using galaxy cluster and type Ia supernovae (SNe Ia) observations. The approach consists, in a first step, of obtaining the observational value of the galaxy cluster scaling-relation $Y_{SZE}D_{A}^{2}/C_{XSZ}Y_X = C $ by combining the X-Ray and SZ observations of galaxy clusters at low redshifts ($z < 0.1$) from the first {\it Planck mission} all-sky data set ($0.044 \leq z \leq 0.444$), along with SNe Ia observations and making use of the cosmic distance duality relation validity. Then, by considering a flat $\Lambda$CDM model for $D_A$, the constant $C$ from the first step and the Planck prior on $\Omega_M$ parameter, we obtain $H_0$ by using the galaxy cluster data with $z>0.1$. As a result, we obtain $H_0 = 73.014^{+7.435}_{-6.688}$ km/s/Mpc, which represents $9.7\%$ accuracy measurement on the Hubble constant.. We also compare our method with that one where the $C$ parameter is obtained from hydrodynamical simulations of massive galaxy clusters.