Tracing the Evolution of ${\mathrm{\Omega }}_m(z)$ over the Last 10 Billion Years with Non-parametric Methods

We investigate the redshift evolution of the matter density parameter, ${\mathrm{\Omega }}_m(z)$, 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 ${\mathrm{\Omega }}_m(z)$ 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 $\mathrm{\Lambda }$CDM model. We obtain ${\mathrm{\Omega }}_{m0}=0.296\pm 0.044$ from the 44-cluster sample, and ${\mathrm{\Omega }}_{m0}=0.271\pm 0.016$, $0.253\pm 0.017$, and $0.210\pm 0.013$ for the 103-cluster compilation, depending on the assumed mass calibration. While ${\mathrm{\Omega }}_m(z)$ follows the expected redshift behaviour, the inferred value of ${\mathrm{\Omega }}_{m0}$ shows a strong dependence on the cluster mass calibration. Within this framework, mass bias emerges as the dominant source of uncertainty, exceeding statistical errors.