Unveiling the Hubble Constant through Galaxy Cluster Gas Mass Fractions

In this work, we obtain Hubble constant ($H_0$) estimates by using two galaxy cluster gas mass fraction measurement samples, Type Ia supernovae luminosity distances, and the validity of the cosmic distance duality relation. Notably, the angular diameter distance (ADD) to each galaxy cluster in the samples is determined by combining its gas mass fraction measurement with galaxy clustering observations, more precisely, the $\Omega_b/\Omega_m$ ratio. Such a combination results in a $H_0$ estimate that is independent of a specific cosmological framework. In one of the samples, the gas fraction measurements were calculated in spherical shells at radii near $r_{\rm 2500}$ (44 data points), while in the other (103 data points) the measurements were calculated within $ r_{\rm 500}$. We find $H_0=72.7^{+6.3}_{-5.6}$ km/s/Mpc at 68\% CL for the joint analysis of these data sets. We also investigate the impact on the $H_0$ determination by exploring the precision and number of gas mass fraction data by performing a data Monte Carlo simulation. Our simulations show that future measurements could achieve a precision of up to 5\% for $H_0$.