Analytical Gaussian Process Cosmography: Unveiling Insights into Matter-Energy Density Parameter at Present

In this study, we introduce a novel analytical Gaussian Process (GP) cosmography methodology, leveraging the differentiable properties of GPs to derive key cosmological quantities analytically. Our approach combines cosmic chronometer (CC) Hubble parameter data with growth rate (f) observations to constrain the $\Omega_{\rm m0}$ parameter, offering insights into the underlying dynamics of the Universe. By formulating a consistency relation independent of specific cosmological models, we analyze under a flat FLRW metric and first-order Newtonian perturbation theory framework. Our analytical approach simplifies the process of Gaussian Process regression (GPR), providing a more efficient means of handling large datasets while offering deeper interpretability of results. We demonstrate the effectiveness of our methodology by deriving precise constraints on $\Omega_{\rm m0}h^2$, revealing $\Omega_{\rm m0}h^2=0.139\pm0.017$. Moreover, leveraging $H_0$ observations, we further constrain $\Omega_{\rm m0}$, uncovering an inverse correlation between mean $H_0$ and $\Omega_{\rm m0}$. Our investigation offers a proof of concept for analytical GP cosmography, highlighting the advantages of analytical methods in cosmological parameter estimation.