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

TL;DR

This work introduces Betti curves, topological summaries from persistent homology, as a powerful cosmological probe of large-scale structure. By constructing an alpha filtration from halo catalogs in the QUIJOTE simulations and normalizing carefully, the authors extract Betti curves that encode multiscale topology beyond standard two-point statistics. They develop Gaussian-process emulators optimized by AutoML to predict Betti curves as functions of cosmological parameters, and perform Bayesian inference to constrain $n_{\mathrm{s}}$, $\sigma_8$, $\Omega_{\mathrm{m}}$, and $w$, with RSD boosting sensitivity. The results show Betti curves, especially when combined with the power spectrum, provide significantly tighter and complementary constraints, and sub-box tests confirm robustness to cosmic variance; this motivates applying Betti curves to upcoming Stage-V galaxy surveys. The framework directly links topology to cosmology in an interpretable way and outlines paths to address observational systematics and extensions to non-standard gravity models.

Abstract

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 $n_{\mathrm{s}}$, $σ_8$, and $Ω_{\mathrm{m}}$, 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 $n_{\mathrm{s}}$, $σ_8$, 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.

Figures (15)

• Figure 1: Visualization of alpha filtration.
• Figure 2: Normalized Betti curves for fiducial simulations in three dimensions. We rescale the amplitude of the Betti curves here for better visualization. The solid dark lines represent the mean Betti curves in three dimensions, while the shaded regions are the 1-$\sigma$ scatter of Betti curves inferred from 500 realizations.
• Figure 3: Impact of RSD on Betti curves in fiducial cosmology. The solid lines stand for the average of Betti curves measured from 500 realizations in fiducial cosmology with RSD (orange lines) and without RSD (blue lines). The error bars stand for the standard deviation. From the left to the right, they are $\hat{\beta}_0(\hat{\alpha})$, $\hat{\beta}_1(\hat{\alpha})$, and $\hat{\beta}_2(\hat{\alpha})$ respectively.
• Figure 4: Betti curves in 0-, 1-, and 2-dimension in several cosmologies (top panel) and their SNR (bottom panel).
• Figure 5: Parameter derivatives of 0-, 1-, and 2-dimensional Betti curves relative to $\Omega_\text{m}, \Omega_\text{b}, h, n_{\text{s}}, \sigma_{\text{8}},w,M_{\nu}$ around fiducial cosmology.