Counting voids and filaments: Betti Curves as a Powerful Probe for Cosmology

Topological analysis of galaxy distributions has gathered increasing attention in cosmology, as they are able to capture non-Gaussian features of large-scale structures (LSS) that are overlooked by conventional two-point clustering statistics. We utilize Betti curves, a summary statistic derived from persistent homology, to characterize the multiscale topological features of the LSS, including connected components, loops, and voids, as a complementary cosmological probe. Using halo catalogs from the \textsc{Quijote} suite, we construct Betti curves, assess their sensitivity to cosmological parameters, and train automated machine learning based emulators to model their dependence on cosmological parameters. Our Bayesian inference recovers unbiased estimation of cosmological parameters, notably $n_{\mathrm{s}}$, $\sigma_8$, and $\Omega_{\mathrm{m}}$, while validation on sub-box simulations confirms robustness against cosmic variance. We further investigate the impact of redshift-space distortions (RSD) on Betti curves and demonstrate that including RSD enhances sensitivity to growth-related parameters. By jointly analyzing Betti curves and the power spectrum, we achieve significantly tightened constraints than using power spectrum alone on parameters such as $n_{\mathrm{s}}$, $\sigma_8$, and $w$. These findings highlight Betti curves -- especially when combined with traditional two-point statistics -- as a promising, interpretable tool for future galaxy survey analyses.