Constraining the Hubble Constant with a Simulated Full Covariance Matrix Using Neural Networks

The Hubble parameter, $H(z)$, plays a crucial role in understanding the expansion history of the universe and constraining the Hubble constant, $\mathrm{H}_0$. The Cosmic Chronometers (CC) method provides an independent approach to measuring $H(z)$, but existing studies either neglect off-diagonal elements in the covariance matrix or use an incomplete covariance matrix, limiting the accuracy of $\mathrm{H}_0$ constraints. To address this, we use a Positive-Definite Covariance Network (PD-CovNet) to simulate the full $33 \times 33$ covariance matrix based on a previously published $15 \times 15$ covariance matrix. Hyperparameters are chosen via leave-one-z-out validation, and performance is benchmarked against a Gaussian-process (GP) baseline. Under identical five-fold cross-validation over redshift groups, we prove that PD-CovNet is a reliable generator of the full covariance compared to the GP baseline. Using this full PD-CovNet-simulated covariance alongside three comparators with different covariance specifications, we constrain $\mathrm{H}_0$ with two independent methods (EMCEE and GP). Across all covariance specifications and both constraint methods, standardized differences and two-sided p-values show no statistically meaningful shift in the central value of the constrained $\mathrm{H}_0$. However, the precision of the constrained $\mathrm{H}_0$ depends on both covariance and method: EMCEE is uniformly more precise than GP once covariance is modeled; within a fixed method, incorporating more covariance reduces precision; and PD-CovNet hyperparameters have a modest effect on uncertainty. These results indicate the importance of accurate covariance modeling in CC-based $\mathrm{H}_0$ constraints.