Tracing the Evolution of $\Omega_m(z)$ over the Last 10 Billion Years with Non-parametric Methods

We investigate the redshift evolution of the matter density parameter, $\Omega_m(z)$, using galaxy cluster gas mass fraction measurements combined with cosmic chronometer $H(z)$ data and type Ia supernova luminosity distances. Our approach employs Gaussian Process Regression to reconstruct $\Omega_m(z)$ in a non-parametric way, remaining only weakly dependent on a specific background cosmology. The reconstructed evolution is consistent with the standard $\rho_m \propto (1+z)^3$ scaling predicted by the $\Lambda$CDM model. We obtain $\Omega_{m0}=0.296 \pm 0.044$ from the 44-cluster sample, and $\Omega_{m0}=0.271 \pm 0.016$, $0.253 \pm 0.017$, and $0.210 \pm 0.013$ for the 103-cluster compilation, depending on the assumed mass calibration. While $\Omega_m(z)$ follows the expected redshift behaviour, the inferred value of $\Omega_{m0}$ shows a strong dependence on the cluster mass calibration. Within this framework, mass bias emerges as the dominant source of uncertainty, exceeding statistical errors.