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arxiv: 2512.07236 · v2 · submitted 2025-12-08 · 🌌 astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

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

Jiayi Li , Cheng Zhao
Authors on Pith no claims yet

Pith reviewed 2026-05-17 01:10 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords Betti curvespersistent homologylarge-scale structurecosmological parameterspower spectrumredshift-space distortionstopological statisticsmachine learning emulators
0
0 comments X

The pith

Betti curves from galaxy distributions tighten cosmological constraints when combined with the power spectrum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates Betti curves, derived from persistent homology, as a way to summarize the multiscale topology of large-scale structures including voids, filaments, and connected components. These topological features capture non-Gaussian information in galaxy distributions that conventional two-point statistics overlook. Using simulations, the authors train machine learning emulators to model how Betti curves respond to cosmological parameters and then perform Bayesian inference to recover unbiased values for parameters such as the scalar spectral index, fluctuation amplitude, and matter density. Accounting for redshift-space distortions boosts sensitivity to growth-related quantities. Most significantly, adding Betti curves to the power spectrum produces substantially narrower constraints on several key parameters than the power spectrum provides by itself.

Core claim

Betti curves derived from persistent homology quantify the birth and death of topological features such as voids and loops across scales in the galaxy distribution. Analysis of Quijote halo catalogs shows these curves respond to cosmological parameters. Machine-learning emulators enable Bayesian inference that recovers unbiased estimates of ns, sigma8, and Omega_m. Redshift-space distortions increase sensitivity to growth parameters. When Betti curves are analyzed jointly with the power spectrum, the resulting constraints on ns, sigma8, and w are significantly tighter than those obtained from the power spectrum alone.

What carries the argument

Betti curves from persistent homology, which count the appearance and disappearance of topological features like connected components, loops, and voids at multiple scales in the large-scale structure.

If this is right

  • Joint analysis with the power spectrum produces significantly tighter constraints on ns, sigma8, and w than the power spectrum alone.
  • Betti curves alone allow recovery of unbiased estimates for ns, sigma8, and Omega_m through Bayesian inference.
  • Validation on sub-box simulations confirms the results remain robust against cosmic variance.
  • Incorporating redshift-space distortions enhances sensitivity to parameters that govern structure growth.
  • The method offers an interpretable complement to two-point statistics for future galaxy survey data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The topological information could serve as an independent cross-check on parameters in cases where standard statistics show tensions between early- and late-universe measurements.
  • Extending the emulator framework to real observational catalogs from surveys would directly test whether the simulated performance translates to actual data.
  • Combining Betti curves with other higher-order statistics might yield even stronger limits on dark energy and modified gravity scenarios.
  • Tracking how Betti curves evolve with redshift could reveal new details about the growth history of cosmic web structures.

Load-bearing premise

Machine-learning emulators trained on simulated halo catalogs fully capture the dependence of Betti curves on cosmological parameters and redshift-space distortions without introducing bias or omitting effects present in real survey data.

What would settle it

Applying the same inference pipeline to actual galaxy survey catalogs and finding cosmological parameters that differ significantly from independent measurements such as those from the cosmic microwave background would falsify the claim of unbiased and improved constraints.

Figures

Figures reproduced from arXiv: 2512.07236 by Cheng Zhao, Jiayi Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Visualization of alpha filtration. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Normalized Betti curves for fiducial simulations in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Impact of RSD on Betti curves in fiducial cosmology. The solid lines stand for the average of Betti curves measured [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Betti curves in 0-, 1-, and 2-dimension in several cosmologies (top panel) and their SNR (bottom panel). [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Parameter derivatives of 0-, 1-, and 2-dimensional Betti curves relative to Ω [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Performance of emulators for 0-, 1-, and 2-dimensional Betti curves for serval test cosmologies. Top panel: The solid [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The RMSE for 0-, 1-, and 2-dimensional Betti curve [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The posterior distribution for cosmological parameters Ω [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Recovered versus true values for cosmological and nuisance parameters of test set without the inclusion of RSD. Red [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Parameter constraints under fiducial cosmology with (red contours) and without (blue contours) RSD from the [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Recovered versus true values for cosmological and nuisance parameters of test set with the inclusion of RSD. Red [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Constraints from Betti curves and power spectrum under fiducial cosmology without RSD. The gray, red, and blue [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Joint constraints from Betti curves and power spectrum under fiducial cosmology with RSD. Red and blue contours [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The comparison of Betti curves in fiducial boxes and sub-boxes. The upper panel plots the Betti curves not including [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The recovered parameter distributions for fiducial cosmological with or without RSD in sub-boxes. The blue (without [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
read the original 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 $n_{\mathrm{s}}$, $\sigma_8$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proposes Betti curves derived from persistent homology of halo distributions as a topological probe that captures non-Gaussian features of large-scale structure. Using the Quijote simulation suite, the authors construct Betti curves, train machine-learning emulators to model their cosmological dependence, perform Bayesian inference that recovers unbiased constraints on parameters including ns, σ8 and Ωm, examine the effects of redshift-space distortions, and report that the joint analysis of Betti curves with the power spectrum yields significantly tighter constraints on ns, σ8 and w than the power spectrum alone. Sub-box validation is used to assess robustness against cosmic variance.

Significance. If the emulators prove accurate and the joint constraints are free of systematic bias, the work would establish Betti curves as a practical, interpretable complement to two-point statistics, potentially improving parameter constraints from future galaxy surveys. The use of an external simulation suite and standard Bayesian methods is a strength, as is the explicit sub-box validation for cosmic variance; however, the significance of the tightening claim rests on unshown quantitative checks of emulator fidelity under RSD.

major comments (2)
  1. [Emulator training and validation sections] The central claim of significantly tightened joint constraints on ns, σ8 and w requires that the ML emulators accurately reproduce the full cosmological and RSD dependence of Betti curves without introducing bias that would be absorbed into the joint posterior. The manuscript provides no quantitative emulator validation metrics (e.g., residual distributions, parameter bias from held-out simulations, or comparison of emulator-induced covariance to statistical errors) across the full parameter volume, which is load-bearing for the reported improvement over the power spectrum alone.
  2. [RSD impact and joint-analysis results] The abstract states that including RSD enhances sensitivity to growth parameters and that the joint analysis tightens constraints, yet no demonstration is given that emulator residuals remain sub-dominant to statistical errors or that the joint covariance matrix is correctly estimated when Betti curves and the power spectrum are combined. Any unmodeled RSD systematic in the emulator would directly affect the claimed tightening.
minor comments (3)
  1. Clarify the precise definition and normalization of the Betti curves (e.g., which homology dimensions are retained and how the curves are binned or smoothed) so that readers can reproduce the topological summary statistic.
  2. Specify the machine-learning architecture, training hyperparameters, and any regularization used for the emulators; also report the training/validation split sizes and any cross-validation procedure.
  3. Add a brief discussion of how the Betti-curve computation choices (e.g., persistence threshold or filtration parameter) propagate into the final cosmological posteriors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of our results on Betti curves as a cosmological probe. We address each major comment below and outline the revisions we will implement to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Emulator training and validation sections] The central claim of significantly tightened joint constraints on ns, σ8 and w requires that the ML emulators accurately reproduce the full cosmological and RSD dependence of Betti curves without introducing bias that would be absorbed into the joint posterior. The manuscript provides no quantitative emulator validation metrics (e.g., residual distributions, parameter bias from held-out simulations, or comparison of emulator-induced covariance to statistical errors) across the full parameter volume, which is load-bearing for the reported improvement over the power spectrum alone.

    Authors: We appreciate the referee's emphasis on rigorous quantitative validation of the emulators. The manuscript demonstrates unbiased cosmological parameter recovery via Bayesian inference and includes sub-box tests for cosmic variance, which indirectly support emulator fidelity. However, we agree that explicit metrics such as residual distributions, bias assessments on held-out simulations, and comparisons of emulator covariance to statistical errors across the full parameter space were not presented in sufficient detail. In the revised manuscript, we will add these quantitative checks, including residual plots and validation results, to directly substantiate the joint constraint improvements. revision: yes

  2. Referee: [RSD impact and joint-analysis results] The abstract states that including RSD enhances sensitivity to growth parameters and that the joint analysis tightens constraints, yet no demonstration is given that emulator residuals remain sub-dominant to statistical errors or that the joint covariance matrix is correctly estimated when Betti curves and the power spectrum are combined. Any unmodeled RSD systematic in the emulator would directly affect the claimed tightening.

    Authors: We acknowledge that the manuscript would benefit from explicit demonstrations that emulator residuals under RSD remain sub-dominant and that the joint covariance is properly estimated. The current analysis shows enhanced sensitivity to growth parameters with RSD included and reports tighter joint constraints, supported by the overall unbiased inference. In the revision, we will add targeted validation of emulator residuals for RSD cases, confirm they are sub-dominant to statistical errors, and detail the construction and validation of the joint covariance matrix between Betti curves and the power spectrum. These additions will reinforce the robustness of the reported tightening. revision: yes

Circularity Check

0 steps flagged

No circularity: standard emulator-based inference on external simulations

full rationale

The paper constructs Betti curves from the external Quijote halo catalogs, trains ML emulators to capture their cosmological and RSD dependence, performs Bayesian inference that recovers unbiased parameter estimates, and validates on sub-box splits for cosmic variance. The joint analysis with the power spectrum uses complementary statistics from the same simulation suite. No equation reduces the reported tightening of constraints on ns, σ8 or w to a fitted parameter by construction, nor does any self-citation or ansatz serve as the load-bearing justification. This is self-contained against external benchmarks and follows conventional simulation-calibrated inference practices.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that persistent-homology summaries of halo catalogs are sensitive to cosmology in a manner not fully captured by the power spectrum, and that emulators trained on a finite set of simulations generalize to the parameter space of interest.

free parameters (1)
  • emulator hyperparameters
    Architecture depth, training-set size, and regularization choices in the machine-learning emulators are tuned to match the Quijote runs.
axioms (2)
  • domain assumption Standard flat wCDM cosmology with the usual six parameters plus w
    The parameter recovery targets ns, sigma8, Om, and w, which presupposes the background model used to generate the Quijote suite.
  • domain assumption Halo catalogs faithfully trace the underlying dark-matter topology at the scales probed
    Betti curves are computed on halos rather than galaxies or particles; this mapping is taken as given.

pith-pipeline@v0.9.0 · 5535 in / 1479 out tokens · 79945 ms · 2026-05-17T01:10:14.555563+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    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

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Reference graph

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