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REVIEW 3 major objections 5 minor 58 references

The three-point correlation function of dark-matter halos carries most of the higher-order cosmological information, and the connected four-point function adds a further factor of about 1.4–1.5.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.5

2026-07-13 05:17 UTC pith:T4W5RNDV

load-bearing objection Solid first config-space N-point information ladder on Quijote; the ~1.4–1.5× connected-4PCF increment is the real claim and is better supported than the stress-test implies, while absolute Mν remains correctly flagged as unconverged. the 3 major comments →

arxiv 2607.09116 v1 pith:T4W5RNDV submitted 2026-07-10 astro-ph.CO

Climbing the N-point Ladder Part I: Information in the Higher-Order Configuration-Space Clustering of Dark Matter Halos

Sumi Kim , Cristiano G. Sabiu , Inkyu Park This is my paper
classification astro-ph.CO
keywords large-scale structureN-point correlation functionsthree-point correlation functionconnected four-point functionFisher forecastneutrino masshalo clusteringconfiguration space
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Late-time clustering of dark-matter halos is strongly non-Gaussian, so two-point statistics leave a large share of cosmological information on the table. This paper measures the configuration-space two-, three-, and connected four-point correlation functions of fixed-density Quijote halos at z=0, then forecasts how much each rung tightens the six standard parameters, including the summed neutrino mass, in real and redshift space. Treated as a ladder, the three-point function supplies most of the accessible higher-order gain—tightening every parameter and partially breaking the σ8–Mν degeneracy—while the connected four-point function adds a further ~1.4–1.5×. That rung-to-rung increment is stable against derivative noise and compression choices even though absolute errors remain limited by finite simulation ensembles and are flagged as preliminary. A tree-level model recovers a linear bias consistent with the two-point function, so the configuration-space ladder is presented as an independent route to the same physics already studied with Fourier poly-spectra.

Core claim

Climbing the configuration-space ladder 2PCF → +3PCF → +ζ_conn^(4) on fixed-number-density Quijote halos, the authors find that the three-point function supplies most of the accessible higher-order information—tightening all six cosmological parameters and most strongly σ8 and Mν, whose degeneracy it partially breaks—while the connected four-point function contributes a further, robust factor of roughly 1.4–1.5; absolute constraints are still limited by finite ensembles and are reported as preliminary.

What carries the argument

The N-point ladder (2PCF → +3PCF → +ζ_conn^(4)), with the connected four-point function isolated by subtracting the disconnected Gaussian products of two-point functions so that each rung’s incremental information can be counted cleanly.

Load-bearing premise

That the finite-difference neutrino-mass derivatives, built from at most a few hundred matched simulations, are clean enough for the absolute neutrino-mass error to be read even as a lower bound, despite the paper’s own tests showing the error still rising and changing strongly with the derivative scheme.

What would settle it

Recompute the same ladder Fisher matrices on a much larger derivative ensemble (or an independent simulation suite) and check whether the 2+3 → 2+3+ζ_conn^(4) information-gain ratio stays near 1.4–1.5 for every parameter, especially Mν; if the ratio collapses toward 1, the claimed four-point increment is a finite-sample artifact.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Most of the non-Gaussian clustering information available at this density and redshift is already captured by the three-point function; four-point measurements yield a real but modest further tightening.
  • Configuration-space N-point statistics can break the bias–amplitude and σ8–Mν degeneracies that limit the two-point function alone, in a manner consistent with Fourier bispectrum forecasts on the same simulations.
  • Because the relative rung-to-rung gains survive derivative-noise and compression tests, they are the quantities that can be trusted first when absolute errors remain unconverged.
  • Tree-level perturbation theory already describes the large-scale halo 3PCF well enough to recover a linear bias matching the 2PCF, so analytic models can anchor the lowest higher-order rung.

Where Pith is reading between the lines

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

  • If the same ladder is applied to halo-occupation galaxy mocks with free bias parameters, the absolute gains will shrink, but the relative 3PCF and connected-4PCF increments should remain the most useful survey-facing numbers.
  • Extending the analysis to anisotropic multipoles, not only the monopole, is the natural next place to recover growth-rate information that the present redshift-space monopole still folds into the amplitude direction.
  • A joint configuration-plus-Fourier analysis could test whether residual information after the 3PCF is complementary across spaces rather than redundant.
  • The modest size of the connected-four-point increment already sets a practical cost–benefit ceiling for measuring a several-hundred-bin 4PCF data vector in Stage-IV catalogues.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper measures the configuration-space 2PCF, 3PCF, and connected 4PCF of Quijote FoF haloes at z=0 with fixed number density, using a GPU graph-database estimator on ~38,000 catalogues. It constructs Fisher forecasts for {Ω_m, Ω_b, h, n_s, σ_8, M_ν} in real space and the redshift-space monopole, treating the statistics as a ladder 2PCF → +3PCF → +ζ_conn^(4). The 3PCF supplies most of the higher-order information (tightening every parameter, especially σ_8 and M_ν, and partially breaking their degeneracy), with gains qualitatively consistent with the Fourier-space halo bispectrum on the same suite; the connected 4PCF adds a further ~1.4–1.5×. Absolute errors are flagged as limited by finite ensembles and derivative noise (Fisher’s mirage), while the rung-to-rung ratios are argued to be stable. The measured 3PCF is validated against a tree-level PT model, recovering a linear bias consistent with the 2PCF.

Significance. If the relative ladder gains hold, the work provides a clean, independent configuration-space route to the non-Gaussian information already explored in Fourier space with Quijote, with practical advantages for survey geometry and analytic contact. Strengths include the explicit connected/disconnected 4PCF split, the large GPU measurement campaign, Hartlap/Percival covariance corrections, three-axis LOS averaging, MOPED compression cross-checked against the direct Fisher, and an explicit Fisher’s-mirage diagnostic that tracks both absolute errors and the 2+3 → 2+3+4 ratio versus N_deriv. The careful separation of robust relative gains from preliminary absolute constraints is a methodological contribution in its own right for simulation-based higher-order forecasts.

major comments (3)
  1. [§7.5, §8.4, Table 1] §7.5, §8.4 and Table 1: The headline ~1.4–1.5× connected-4PCF increment is obtained with MOPED score compression (λ=0.1) as the primary estimator because the direct Nd=1484 covariance is poorly conditioned (Hartlap 0.70, Percival m1=1.42). Stability under λ∈[0.02,0.5] and agreement of the direct Fisher with the compressed one up to a uniform ~1.42 factor are necessary but incomplete. Please report the 2+3 → 2+3+ζ_conn^(4) gain ratio under the fully Percival-corrected direct Fisher for both rungs on the same footing (and, if feasible, under an alternative compression such as PCA/KL on the 4PCF block alone). Without that, it remains possible that part of the quoted increment is an artifact of how the noisy 632-bin 4PCF block is projected into score space.
  2. [§4.2, §7.5, §8.2, Table 1] §4.2 Eq. (4.2), §7.5 and §8.2: Absolute σ(M_ν) is scheme-dependent by ~2.3× (forward / three-point / four-point), still rising at N_deriv=500, and tighter for noisier schemes—the classic Fisher’s-mirage signature. Table 1 nonetheless quotes the four-point-scheme values (0.042 / 0.059 eV) in the same format as the other parameters. Either remove absolute M_ν from the main table (retaining only the robust gain ratios and the ≳0.1 eV lower bound in the text) or add a dedicated panel/table that shows all three schemes side-by-side so the reader cannot mistake the tabulated numbers for a forecast.
  3. [§7.1–7.4] §7.1–7.4 and comparison to Hahn et al. (2020): The large 3PCF-over-2PCF factors for σ_8 and M_ν (~9–14) are partly inflated by a weak, mirage-sensitive configuration-space 2PCF monopole baseline. The paper correctly cautions that ratios to this baseline are “indicative,” yet still presents them as tracking the Fourier bispectrum. Please add a short quantitative comparison that normalizes both analyses to a common, better-conditioned two-point baseline (e.g. the Fourier P(k) on the same catalogues, or the config-space 2PCF with multipoles) so the claimed consistency is not driven by the denominator.
minor comments (5)
  1. [§3.3, §8.3] §3.3 / §8.3: Binning (20/18/5 bins) is fixed and a full convergence sweep is deferred. A short appendix table with one coarser and one finer choice for the 3PCF and 4PCF would strengthen the claim that the ladder ordering is not binning-driven.
  2. [Fig. 2] Fig. 2 caption: Neutrino-mass response is omitted because of the separate derivative scheme; a companion panel (even noisy) would help the reader see where the M_ν sensitivity lives in configuration space, as done for the other parameters.
  3. [§5.2] §5.2: Reduced χ²/dof ≈ 4.6–5.7 is explained as sub-percent residuals on the mean of 5000 boxes; stating the absolute residual amplitude (e.g. median |d−t|/σ or fractional residual) would make the “physically meaningful validation is the bias agreement” argument more transparent.
  4. [§3.2, §9.4] Redshift-space analysis uses only the monopole. A sentence in §9.4 or §10 clarifying that the Kaiser anisotropy is not yet exploited (and that multipoles are left to future work) would prevent over-reading of the RSD columns in Table 1.
  5. [Contents, Abstract] Typographical: abstract and title use “N-point” / “N-point Ladder”; ensure consistent math-mode N throughout. Also “V alidation” in the contents has a stray space (§5 heading).

Circularity Check

0 steps flagged

No significant circularity: ladder gains are measured from independent Quijote ensembles; self-cites are to the authors' estimator code, not load-bearing for the Fisher ratios.

full rationale

The paper's central claims are empirical Fisher information ratios built from ~38k independent N-body measurements of the 2/3/connected-4PCF on Quijote halo catalogues (fixed n-bar selection). The connected 4PCF is isolated by the standard Wick subtraction of Eq. (2.2) so that the rung counts only incremental non-Gaussian information; this is definitional bookkeeping, not a circular derivation of a 'prediction'. The tree-level 3PCF validation (§5) fits (b1,b2) to the measured mean and recovers a linear bias consistent with the independent large-scale 2PCF; consistency is reported, not forced. Absolute errors are correctly flagged as preliminary (Fisher's mirage, scheme dependence of Mν derivatives, Hartlap/Percival factors). Self-citations ([24],[30]) document the GPU graph estimator used to obtain the measurements; they do not supply uniqueness theorems, ansätze, or uniqueness results that force the reported ~1.4–1.5× relative gain. Comparison to the independent Fourier bispectrum of Hahn et al. on the same suite further anchors the result externally. Minor residual risk is only ordinary self-citation of the measurement pipeline, which does not reduce any load-bearing claim by construction. Score 1 reflects that single non-load-bearing self-cite pattern; the derivation chain itself is self-contained.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central relative-gain claim rests on standard LSS assumptions (Gaussian Fisher likelihood, fixed-n_bar halo selection, tree-level PT validity on large scales) plus a handful of analysis choices (binning, shrinkage λ, derivative schemes) that the paper varies or flags. No new physical entities are postulated; the connected 4PCF is the standard cumulant.

free parameters (4)
  • fixed comoving number density n_bar = 1.5e-4 h^3 Mpc^{-3}
    Set by hand to 1.5×10^{-4} h^3 Mpc^{-3} (150 000 most massive halos) to remove abundance response; changes the effective bias and therefore the absolute Fisher errors.
  • MOPED shrinkage regularization λ = 0.1
    Chosen as λ=0.1 after scanning [0.02,0.5]; the connected-4PCF gain varies <10 % across the range, but the absolute compressed Fisher depends on it.
  • 3PCF fit scale cut r_fit_min = 40 h^{-1} Mpc
    Triangles with shortest side <40 h^{-1} Mpc excluded from the tree-level bias fit; affects the recovered (b1,b2) and the claim that the measurement is physical.
  • pair-separation binning (20/18/5 bins) = 20, 18, 5 bins
    Fixed measurement binning out to 150/100/65 h^{-1} Mpc; absolute gains depend on it and a full convergence sweep is deferred.
axioms (4)
  • domain assumption Gaussian likelihood with parameter-independent covariance for the Fisher matrix (Eq. 4.1)
    Standard for this class of forecast; untested by simulation-based inference on the high-dimensional non-Gaussian data vector.
  • domain assumption Tree-level Eulerian bias model with local-Lagrangian tidal bias bs2 = −4/7(b1−1) is adequate for the 3PCF on scales >40 h^{-1} Mpc
    Used in §5 to validate the measurement; reduced χ^{2} >1 is attributed to sub-percent residuals rather than model failure.
  • standard math Connected four-point function obtained by subtracting the disconnected Wick products of the measured 2PCF (Eq. 2.2) isolates genuinely new information
    Standard cumulant decomposition; essential for the ladder accounting.
  • domain assumption One-sided higher-order finite-difference formula (Eq. 4.2) correctly estimates ∂μ/∂Mν given the Zel’dovich initial conditions of the massive-neutrino runs
    Adopted from Hahn et al. 2020; the paper shows the absolute Mν error remains scheme-dependent.

pith-pipeline@v1.1.0-grok45 · 35632 in / 3378 out tokens · 40313 ms · 2026-07-13T05:17:12.258425+00:00 · methodology

0 comments
read the original abstract

The two-point correlation function completely describes a Gaussian random field, but nonlinear gravitational growth, halo bias, and redshift-space distortions drive the late-time halo field strongly non-Gaussian, moving a substantial part of the cosmological information into higher-order correlations. We quantify the information content of the configuration-space two-, three-, and connected four-point correlation functions of Quijote dark-matter haloes at $z=0$ and fixed number density. We build Fisher forecasts for $\{\Omega_m, \Omega_b, h, n_s, \sigma_8, M_\nu\}$ in real and redshift space from ${\sim}38{,}000$ GPU-accelerated $N$-point measurements. Treating the statistics as a ladder, $\mathrm{2PCF} \rightarrow +\mathrm{3PCF} \rightarrow +\zeta^{(4)}_{\mathrm{conn}}$, we report the information gained at each rung. The 3PCF supplies most of the accessible higher-order information: it tightens every parameter, most strongly $\sigma_8$ and $M_\nu$, whose degeneracy it partially breaks, with per-parameter gains consistent with those of the Fourier-space halo bispectrum on the same simulations. The connected 4PCF adds a further $\sim1.4$--$1.5\times$. This rung-to-rung increment is stable against derivative-sample noise and compression regularization, whereas the absolute constraints remain limited by the finite simulation ensembles and are reported as preliminary. We validate the measured 3PCF against a tree-level perturbation-theory model, recovering a linear bias consistent with the 2PCF. The configuration-space ladder thus offers an independent and complementary route to the higher-order information probed by the Fourier-space poly-spectra.

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Reference graph

Works this paper leans on

58 extracted references · 22 linked inside Pith

  1. [1]

    P. J. E. Peebles,The large-scale structure of the universe. Princeton University Press, 1980

    1980

  2. [2]

    J. N. Fry,The Galaxy correlation hierarchy in perturbation theory,ApJ279(Apr., 1984) 499–510. – 22 –

    1984

  3. [3]

    Bernardeau, S

    F. Bernardeau, S. Colombi, E. Gazta˜ naga, and R. Scoccimarro,Large-scale structure of the Universe and cosmological perturbation theory,Phys. Rep.367(Sept., 2002) 1–248, [astro-ph/0112551]

    Pith/arXiv arXiv 2002

  4. [4]

    Aghamousa, J

    DESI Collaboration, A. Aghamousa, J. Aguilar, S. Ahlen, S. Alam, L. E. Allen, C. Allende Prieto, J. Annis, S. Bailey, C. Balland, O. Ballester, C. Baltay, L. Beaufore, C. Bebek, T. C. Beers, E. F. Bell, J. L. Bernal, R. Besuner, F. Beutler, C. Blake, H. Bleuler, M. Blomqvist, R. Blum, A. S. Bolton, C. Briceno, D. Brooks, J. R. Brownstein, E. Buckley-Geer,...

    Pith/arXiv arXiv 2016

  5. [5]

    Laureijs, J

    R. Laureijs, J. Amiaux, S. Arduini, J. L. Augu` eres, J. Brinchmann, R. Cole, M. Cropper, C. Dabin, L. Duvet, A. Ealet, B. Garilli, P. Gondoin, L. Guzzo, J. Hoar, H. Hoekstra, R. Holmes, T. Kitching, T. Maciaszek, Y. Mellier, F. Pasian, W. Percival, J. Rhodes, G. Saavedra Criado, M. Sauvage, R. Scaramella, L. Valenziano, S. Warren, R. Bender, F. Castander...

    Pith/arXiv arXiv 2011

  6. [6]

    Dor´ e, J

    O. Dor´ e, J. Bock, M. Ashby, P. Capak, A. Cooray, R. de Putter, T. Eifler, N. Flagey, Y. Gong, S. Habib, K. Heitmann, C. Hirata, W.-S. Jeong, R. Katti, P. Korngut, E. Krause, D.-H. Lee, D. Masters, P. Mauskopf, G. Melnick, B. Mennesson, H. Nguyen, K. ¨Oberg, A. Pullen, A. Raccanelli, R. Smith, Y.-S. Song, V. Tolls, S. Unwin, T. Venumadhav, M. Viero, M. W...

  7. [7]

    arXiv:1412.4872, [arXiv:1412.4872]

    Pith/arXiv arXiv

  8. [8]

    Lesgourgues and S

    J. Lesgourgues and S. Pastor,Massive neutrinos and cosmology,Phys. Rep.429(July, 2006) 307–379, [astro-ph/0603494]

    Pith/arXiv arXiv 2006

  9. [9]

    Sefusatti, M

    E. Sefusatti, M. Crocce, S. Pueblas, and R. Scoccimarro,Cosmology and the bispectrum, Phys. Rev. D74(July, 2006) 023522, [astro-ph/0604505]

    Pith/arXiv arXiv 2006

  10. [10]

    Gil-Mar´ ın, W

    H. Gil-Mar´ ın, W. J. Percival, L. Verde, J. R. Brownstein, C.-H. Chuang, F.-S. Kitaura, S. A. Rodr´ ıguez-Torres, and M. D. Olmstead,The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement from the power spectrum and bispectrum of the DR12 BOSS galaxies,MNRAS465(Feb., 2017) 1757–1788, [arXiv:1606.0043]

    arXiv 2017

  11. [11]

    C. G. Sabiu, D. F. Mota, C. Llinares, and C. Park,Probing scalar tensor theories for gravity in redshift space,A&A592(July, 2016) A38, [arXiv:1603.0575]

    arXiv 2016

  12. [12]

    Villaescusa-Navarro, C

    F. Villaescusa-Navarro, C. Hahn, E. Massara, A. Banerjee, A. M. Delgado, D. K. Ramanah, T. Charnock, E. Giusarma, Y. Li, E. Allys, A. Brochard, C. Uhlemann, C.-T. Chiang, S. He, A. Pisani, A. Obuljen, Y. Feng, E. Castorina, G. Contardo, C. D. Kreisch, A. Nicola, J. Alsing, R. Scoccimarro, L. Verde, M. Viel, S. Ho, S. Mallat, B. Wandelt, and D. N. Spergel,...

    arXiv 2020

  13. [13]

    C. Hahn, F. Villaescusa-Navarro, E. Castorina, and R. Scoccimarro,Constraining M ν with the bispectrum. Part I. Breaking parameter degeneracies,JCAP2020(Mar., 2020) 040, [arXiv:1909.1110]

    arXiv 2020

  14. [14]

    Hahn and F

    C. Hahn and F. Villaescusa-Navarro,Constraining M ν with the bispectrum. Part II. The information content of the galaxy bispectrum monopole,JCAP2021(Apr., 2021) 029, [arXiv:2012.0220]

    arXiv 2021

  15. [15]

    Massara, F

    E. Massara, F. Villaescusa-Navarro, S. Ho, N. Dalal, and D. N. Spergel,Using the Marked Power Spectrum to Detect the Signature of Neutrinos in Large-Scale Structure,Phys. Rev. Lett. 126(Jan., 2021) 011301, [arXiv:2001.1102]

    arXiv 2021

  16. [16]

    Valogiannis and C

    G. Valogiannis and C. Dvorkin,Towards an optimal estimation of cosmological parameters with the wavelet scattering transform,Phys. Rev. D105(May, 2022) 103534, [arXiv:2108.0782]

    arXiv 2022

  17. [17]

    Banerjee and T

    A. Banerjee and T. Abel,Nearest neighbour distributions: New statistical measures for cosmological clustering,MNRAS500(Jan., 2021) 5479–5499, [arXiv:2007.1334]

    arXiv 2021

  18. [18]

    Villaescusa-Navarro, A

    F. Villaescusa-Navarro, A. Banerjee, N. Dalal, E. Castorina, R. Scoccimarro, R. Angulo, and D. N. Spergel,The Imprint of Neutrinos on Clustering in Redshift Space,ApJ861(July, 2018) 53, [arXiv:1708.0115]. – 24 –

    arXiv 2018

  19. [19]

    A. E. Bayer, F. Villaescusa-Navarro, E. Massara, J. Liu, D. N. Spergel, L. Verde, B. D. Wandelt, M. Viel, and S. Ho,Detecting Neutrino Mass by Combining Matter Clustering, Halos, and Voids,ApJ919(Sept., 2021) 24, [arXiv:2102.0504]

    arXiv 2021

  20. [20]

    Szapudi and A

    I. Szapudi and A. S. Szalay,A New Class of Estimators for the N-Point Correlations,ApJL 494(Feb., 1998) L41–L44, [astro-ph/9704241]

    Pith/arXiv arXiv 1998

  21. [21]

    Slepian and D

    Z. Slepian and D. J. Eisenstein,Computing the three-point correlation function of galaxies in O(Nˆ2) time,MNRAS454(Dec., 2015) 4142–4158, [arXiv:1506.0204]

    arXiv 2015

  22. [22]

    Slepian and D

    Z. Slepian and D. J. Eisenstein,A practical computational method for the anisotropic redshift-space three-point correlation function,MNRAS478(Aug., 2018) 1468–1483, [arXiv:1709.1015]

    arXiv 2018

  23. [23]

    O. H. E. Philcox, Z. Slepian, J. Hou, C. Warner, R. N. Cahn, and D. J. Eisenstein,ENCORE: an O (N g2) estimator for galaxy N-point correlation functions,MNRAS509(Jan., 2022) 2457–2481, [arXiv:2105.0872]

    arXiv 2022

  24. [24]

    O. H. E. Philcox and D. J. Eisenstein,Computing the small-scale galaxy power spectrum and bispectrum in configuration space,MNRAS492(Feb., 2020) 1214–1242, [arXiv:1912.0101]

    arXiv 2020

  25. [25]

    C. G. Sabiu, B. Hoyle, J. Kim, and X.-D. Li,Graph Database Solution for Higher-order Spatial Statistics in the Era of Big Data,ApJS242(June, 2019) 29, [arXiv:1901.0029]

    arXiv 2019

  26. [26]

    Gazta˜ naga and R

    E. Gazta˜ naga and R. Scoccimarro,The three-point function in large-scale structure: redshift distortions and galaxy bias,MNRAS361(Aug., 2005) 824–836, [astro-ph/0501637]

    Pith/arXiv arXiv 2005

  27. [27]

    Slepian, D

    Z. Slepian, D. J. Eisenstein, J. R. Brownstein, C.-H. Chuang, H. Gil-Mar´ ın, S. Ho, F.-S. Kitaura, W. J. Percival, A. J. Ross, G. Rossi, H.-J. Seo, A. Slosar, and M. Vargas-Maga˜ na, Detection of baryon acoustic oscillation features in the large-scale three-point correlation function of SDSS BOSS DR12 CMASS galaxies,MNRAS469(Aug., 2017) 1738–1751, [arXiv...

    arXiv 2017

  28. [28]

    O. H. E. Philcox,Probing parity violation with the four-point correlation function of BOSS galaxies,Phys. Rev. D106(Sept., 2022) 063501, [arXiv:2206.0422]

    arXiv 2022

  29. [29]

    J. Hou, Z. Slepian, and R. N. Cahn,Measurement of parity-odd modes in the large-scale 4-point correlation function of Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey twelfth data release CMASS and LOWZ galaxies,MNRAS522(May, 2023) 5701–5739, [arXiv:2206.0362]

    arXiv 2023

  30. [30]

    R. N. Cahn, Z. Slepian, and J. Hou,Test for Cosmological Parity Violation Using the 3D Distribution of Galaxies,Phys. Rev. Lett.130(May, 2023) 201002, [arXiv:2110.1200]

    arXiv 2023

  31. [31]

    C. G. Sabiu,Fast graph-based higher-order clustering statistics on the gpu,ApJS submitted (2026) [arXiv:2607.0660]

    arXiv 2026

  32. [32]

    Hartlap, P

    J. Hartlap, P. Simon, and P. Schneider,Why your model parameter confidences might be too optimistic. Unbiased estimation of the inverse covariance matrix,A&A464(Mar., 2007) 399–404, [astro-ph/0608064]

    Pith/arXiv arXiv 2007

  33. [33]

    Dodelson and M

    S. Dodelson and M. D. Schneider,The effect of covariance estimator error on cosmological parameter constraints,Phys. Rev. D88(Sept., 2013) 063537, [arXiv:1304.2593]

    Pith/arXiv arXiv 2013

  34. [34]

    W. J. Percival, A. J. Ross, A. G. S´ anchez, L. Samushia, A. Burden, R. Crittenden, A. J. Cuesta, M. V. Magana, M. Manera, F. Beutler, C.-H. Chuang, D. J. Eisenstein, S. Ho, C. K. McBride, F. Montesano, N. Padmanabhan, B. Reid, S. Saito, D. P. Schneider, H.-J. Seo, R. Tojeiro, and B. A. Weaver,The clustering of Galaxies in the SDSS-III Baryon Oscillation ...

    Pith/arXiv arXiv 2014

  35. [35]

    W. R. Coulton and B. D. Wandelt,How to estimate Fisher information matrices from simulations,arXiv e-prints(May, 2023) arXiv:2305.08994, [arXiv:2305.0899]. – 25 –

    Pith/arXiv arXiv 2023

  36. [36]

    M. H. Goroff, B. Grinstein, S.-J. Rey, and M. B. Wise,Coupling of modes of cosmological mass density fluctuations,ApJ311(Dec., 1986) 6–14

    1986

  37. [37]

    McDonald and A

    P. McDonald and A. Roy,Clustering of dark matter tracers: generalizing bias for the coming era of precision LSS,JCAP2009(Aug., 2009) 020, [arXiv:0902.0991]

    Pith/arXiv arXiv 2009

  38. [38]

    Desjacques, D

    V. Desjacques, D. Jeong, and F. Schmidt,Large-scale galaxy bias,Phys. Rep.733(Feb., 2018) 1–193, [arXiv:1611.0978]

    arXiv 2018

  39. [39]

    Baldauf, M

    T. Baldauf, M. Mirbabayi, M. Simonovi´ c, and M. Zaldarriaga,Equivalence principle and the baryon acoustic peak,Phys. Rev. D92(Aug., 2015) 043514, [arXiv:1504.0436]

    arXiv 2015

  40. [40]

    Slepian and D

    Z. Slepian and D. J. Eisenstein,Accelerating the two-point and three-point galaxy correlation functions using Fourier transforms,MNRAS455(Jan., 2016) L31–L35, [arXiv:1506.0474]

    arXiv 2016

  41. [41]

    Barriga and E

    J. Barriga and E. Gazta˜ naga,The three-point function in large-scale structure - I. The weakly non-linear regime in N-body simulations,MNRAS333(June, 2002) 443–453, [astro-ph/0112278]

    Pith/arXiv arXiv 2002

  42. [42]

    Lewis, A

    A. Lewis, A. Challinor, and A. Lasenby,Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models,ApJ538(Aug.,

  43. [43]

    473–476, [astro-ph/9911177]

    Pith/arXiv arXiv

  44. [44]

    Hamann, S

    J. Hamann, S. Hannestad, J. Lesgourgues, C. Rampf, and Y. Y. Y. Wong,Cosmological parameters from large scale structure - geometric versus shape information,JCAP2010(July,

  45. [45]

    022, [arXiv:1003.3999]

    Pith/arXiv arXiv

  46. [46]

    D. J. Eisenstein and W. Hu,Baryonic Features in the Matter Transfer Function,ApJ496 (Mar., 1998) 605–614, [astro-ph/9709112]

    Pith/arXiv arXiv 1998

  47. [47]

    D. J. Eisenstein, H.-J. Seo, and M. White,On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter,ApJ664(Aug., 2007) 660–674, [astro-ph/0604361]

    Pith/arXiv arXiv 2007

  48. [48]

    D. Blas, M. Garny, M. M. Ivanov, and S. Sibiryakov,Time-sliced perturbation theory II: baryon acoustic oscillations and infrared resummation,JCAP2016(July, 2016) 028, [arXiv:1605.0214]

    arXiv 2016

  49. [49]

    Kaiser,Clustering in real space and in redshift space,MNRAS227(July, 1987) 1–21

    N. Kaiser,Clustering in real space and in redshift space,MNRAS227(July, 1987) 1–21

    1987

  50. [50]

    Scoccimarro, H

    R. Scoccimarro, H. M. P. Couchman, and J. A. Frieman,The Bispectrum as a Signature of Gravitational Instability in Redshift Space,ApJ517(June, 1999) 531–540, [astro-ph/9808305]

    Pith/arXiv arXiv 1999

  51. [51]

    Wilson and R

    C. Wilson and R. Bean,Implications of noisy numerical derivatives for simulation-based cosmological inference and Fisher forecasts,Phys. Rev. D111(May, 2025) 103532, [arXiv:2406.0606]

    arXiv 2025

  52. [52]

    A. F. Heavens, R. Jimenez, and O. Lahav,Massive lossless data compression and multiple parameter estimation from galaxy spectra,MNRAS317(Oct., 2000) 965–972, [astro-ph/9911102]

    Pith/arXiv arXiv 2000

  53. [53]

    Alsing and B

    J. Alsing and B. Wandelt,Generalized massive optimal data compression,MNRAS476(May,

  54. [54]

    L60–L64, [arXiv:1712.0001]

    arXiv

  55. [55]

    Ledoit and M

    O. Ledoit and M. Wolf,A well-conditioned estimator for large-dimensional covariance matrices,Journal of Multivariate Analysis88(2004), no. 2 365–411

    2004

  56. [56]

    A. C. Pope and I. Szapudi,Shrinkage estimation of the power spectrum covariance matrix, MNRAS389(Sept., 2008) 766–774, [arXiv:0711.2509]

    Pith/arXiv arXiv 2008

  57. [57]

    Forero-S´ anchez, S

    D. Forero-S´ anchez, S. Novell Masot, H. Gil-Mar´ ın, L. Verde, J. Aguilar, S. Ahlen, D. Bianchi, A. Brodzeller, D. Brooks, F. J. Castander, S. Cole, A. de la Macorra, J. Della Costa, B. Dey, P. Doel, S. Ferraro, A. Font-Ribera, J. E. Forero-Romero, S. G. A. Gontcho, G. Gutierrez, C. Hahn, H. K. Herrera-Alcantar, K. Honscheid, D. Huterer, M. Ishak, D. Kir...

    Pith/arXiv arXiv 2026

  58. [58]

    Labate, M

    A. Labate, M. Guidi, M. Moresco, and A. Veropalumbo,The imprints of massive neutrinos on the three-point correlation function of large-scale structures,A&A708(Apr., 2026) A210, [arXiv:2512.1699]. – 27 –

    arXiv 2026