Enhancing Cosmological Constraints by Two-dimensional $\beta$-cosmic-web Weighted Angular Correlation Functions

In this study, we investigate the potential of mark-weighted angular correlation functions (MACFs), which integrate $\beta$-cosmic-web classification with angular correlation function analysis to improve cosmological constraints. Using SDSS DR12 CMASS-NGC galaxies and mock catalogs with $\Omega_m$ varying from 0.25 to 0.40, we assess the discriminative power of different statistics via the average improvement in chi-squared, $\Delta \overline{\chi^2}$, across six redshift bins. This metric quantifies how effectively each statistic distinguishes between different cosmological models. Incorporating cosmic-web weights leads to substantial improvements. Using statistics weighted by the mean neighbor distance ($\bar{D}_{\rm nei}$) increases $\Delta \overline{\chi^2}$ by approximately 40%-130%, while applying inverse mean neighbor distance weighting ($1/\bar{D}_{\rm nei}$) yields even larger gains, boosting $\Delta \overline{\chi^2}$ by a factor of 2-3 compared to traditional unweighted angular statistics. These enhancements are consistent with previous 3D clustering results, demonstrating the superior sensitivity of the $\beta$-weighted approaches. Our method, based on thin redshift slices, is particularly suited for slitless surveys (e.g., Euclid, CSST) where redshift uncertainties limit 3D analyses. This study also offers a framework for applying marked statistics to 2D angular clustering.