Pith. sign in

REVIEW 3 major objections 5 minor 58 references

Weighting the cosmic density field by its local web shape, not just its overdensity, yields a consistent ~10% gain on key cosmological parameter constraints when combined with a density mark.

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 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 00:49 UTC pith:3OBTDZ2B

load-bearing objection Honest, systematic numerical study of morphology marks; the ~10% complementarity is plausible but unquantified leakage and missing error bars make the strong claims premature. the 3 major comments →

arxiv 2607.26021 v1 pith:3OBTDZ2B submitted 2026-07-28 astro-ph.CO

Weighted Webs: Morphology-Informed Marked Fields

Mikel Martin Barandiaran , Jessica A. Cowell , David Alonso , Javier Carr\'on Duque , Juan Garc\'ia-Bellido This is my paper
classification astro-ph.CO PACS 98.80.-k98.65.-r
keywords cosmic webmarked power spectrumtidal shearlocal fractal dimensionnon-Gaussian informationcosmological parameter constraintslarge-scale structure
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.

Marked power spectra recover non-Gaussian cosmological information by taking the two-point statistics of a density field weighted by a local environmental function. Almost all existing marks depend only on the smoothed local density. This paper asks whether marks that respond to the morphology of the cosmic web — the anisotropy of the local tidal field and the local fractal dimension of the matter distribution — carry information that density marks miss. Using N-body simulations and forecast-based error-improvement factors, it finds that morphology marks alone underperform the empirically tuned White mark (a density-based mark that upweights underdense regions), but combining any of them with the White mark consistently improves constraints on the matter density, the fluctuation amplitude, and the neutrino mass by about ten percent. The paper also argues that within the shear-based family the specific functional form of the mark hardly matters, so the information lives in the shear field itself rather than in the chosen nonlinear transformation.

Core claim

The central result is complementarity, not superiority. Density marks — above all the White mark — stay strongest alone (error-improvement factors ~2.2 for Ωm, ~1.9 for σ8 and Mν); shear-amplitude marks reach only ~1.3, with isotropy, filament, and Gaussian-shear variants within a few percent of that. Adding any shear-based mark to the White mark improves forecasted constraints by ~10% across all three parameters — Ωm rises from 2.24 to ~2.46, with comparable gains for σ8 and Mν. The local fractal dimension mark behaves similarly: individually the second-best single mark (~1.6 on Ωm), it still adds a consistent gain on top of the White mark. The paper reads the near-flatness across shear-mar

What carries the argument

The engine is the marked power spectrum: one takes the overdensity field, multiplies it by a local 'mark' function, and computes auto- and cross-power spectra of the marked field. The paper generalises the mark from a function of the smoothed density δR to a function of the tidal tensor's invariants (δR, s2_R, s3_R) — where s2 is the squared shear amplitude from the traceless Hessian of the gravitational potential — plus a Gaussian local fractal dimension that measures how enclosed mass scales with radius. The shear amplitude s = sqrt(s2) is a quadrupolar mode-coupling of the density field, so its marked spectra contain bispectrum- and trispectrum-like contributions (Appendix A), which is ho

Load-bearing premise

The claim that the gains are genuinely morphological rather than a leak of ordinary small-scale Gaussian variance rests on choosing smoothing scales R ≥ 5 Mpc/h, but the paper does not directly measure how much of the error-improvement factor would be reproduced by a Gaussian density field with the same power spectrum.

What would settle it

Run the same error-improvement-factor analysis on a Gaussianized version of the density field (randomized phases, identical power spectrum) with the same smoothing and covariance treatment. If the shear-marked and combined gains shrink to near unity or match the gains from simply extending the ordinary power spectrum to higher wavenumbers, the complementarity claim is refuted; if the gains persist, the morphological channel is real.

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

If this is right

  • Adding shear- or fractal-dimension marks to existing density-mark analyses yields a ~10% improvement on (Ωm, σ8, Mν) with no new estimators, so the method can be deployed on current and upcoming galaxy-survey data.
  • Because the mark's functional form has little effect, shear-mark analyses can use the simplest shear-amplitude mark, easing implementation and theoretical modeling.
  • The shear-marked auto- and cross-spectra receive leading contributions from bispectrum- and trispectrum-like terms, giving a target for perturbative models of morphology-marked statistics.
  • The local-fractal-dimension mark outperforms combining two density marks at different smoothing scales, indicating it captures environmental information not encoded in multiscale densities.
  • None of the morphology marks beats the empirically tuned White mark alone, so density weighting remains the primary probe; morphology is a complement, not a replacement.

Where Pith is reading between the lines

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

  • A direct test the paper does not perform: quantify what fraction of the shear-mark gain survives in a Gaussian realization with the same power spectrum. If most of the gain persists, the 'complementarity' would be a repackaging of two-point variance rather than a new non-Gaussian channel.
  • The near-indifference to mark shape suggests an interpretation the paper leaves implicit: the shear mark may function mainly as a monotone filter that reorders environments by tidal anisotropy; a hypothesis is that any mark strictly increasing in sR would reproduce the error-improvement factors, which is directly testable.
  • The cubic shear invariant s3 was tried and found weaker, but the paper leaves a systematic exploration to future work; a dedicated study of eigenvalue-based marks (e.g., ratios of eigenvalues) could map the geometry of the information gain.
  • If the shear field is the true carrier, then in galaxy surveys the mark could be constructed from reconstructed tidal fields, and the complementarity could extend to lensing or redshift-space data where density marks are systematics-limited.

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 proposes and tests marked power spectra whose marks depend on the local morphology of the cosmic web — specifically the tidal shear invariants, the local fractal dimension, isotropy, and filamentarity — rather than only on the smoothed density. Using Quijote N-body simulations, the authors build Fisher forecasts for (Ωm, σ8, Mν) from combinations of density, marked-density, and marked-auto power spectra, and report error improvement factors (EIFs) relative to the plain matter power spectrum. The headline results are that shear and fractal-dimension marks are individually weaker than the standard White density mark, but that combining them with the White mark improves constraints by roughly 10% (e.g. Ωm EIF from 2.24 to 2.46 for δ+Wh10+s10 in Table 2), and that the detailed functional form of shear-based marks has only a minor impact on the constraints.

Significance. If the central claim holds, this is a useful systematic step for marked-statistics cosmology: it broadens the mark design space beyond density-only weights and provides a practical, simulation-based ranking of morphology-sensitive marks. The paper is careful in several methodological respects: it uses an unbiased sample covariance from 2100 realisations with the Hartlap correction, checks Cholesky versus pseudo-inverse inversion, explicitly tests the Gaussian-covariance approximation, and frames honestly that tidal and LFD marks are deterministic functionals of δ and therefore reorganize rather than create information. The perturbative expressions in Appendix A are a helpful starting point for future analytic work. However, the central claim that the reported gains are non-Gaussian morphological information rather than leakage of small-scale Gaussian variance is not yet established, and the headline EIFs are presented without uncertainties, so the quantitative conclusions are not fully supported as they stand.

major comments (3)
  1. [§3.4 and Tables 1–2] This is a specific technical concern.
  2. [Tables 1–3, §4.2] The headline EIFs are quoted to two decimal places with no uncertainties. The derivatives are estimated from only 50 realisations per cosmology, and the covariance from 2100 realisations; the differences that drive the paper’s main conclusions are small — e.g. Table 2 shows a ∼10% EIF improvement for White+shear versus White alone, and the different shear-mark families differ at the few-percent level. Without bootstrap/jackknife errors on the EIFs, the statements that the improvements are ‘consistent’ and that the functional form has ‘only a minor impact’ are not yet supported. This is particularly important because §4.3 calls the White+GLFD improvement ‘statistically significant’, but no significance test or error bar is provided.
  3. [§4.3, Table 3] The comparison between the GLFD mark and the two-scale density mark (δ+δ10+δ15) is used to argue that the GLFD’s information is not merely multiscale density information. However, δ15 is only one possible second smoothing scale, and the Fisher information is invariant under affine mark transformations, so the comparison is not a strong discriminator. More importantly, the GLFD is a deterministic functional of δ and the claim that it carries ‘complementary’ information to the White mark is subject to the same unresolved Gaussian-leakage concern as the shear marks. The conclusion that the GLFD ‘retains some complementary cosmological information’ should be conditional on the outcome of the control test requested above.
minor comments (5)
  1. [Table 2 header] The White mark entry is labelled ‘White Mark (R=10, p=2, b=0.25)’; the last parameter is δ_s, not b. Please fix the notation to match Eq. (2.22).
  2. [§2.1 and §2.2] The notation δ_R and s_R is used for both the smoothed fields and the mark values derived from them; in places such as Figure 5 the distinction between the underlying field and the mark function is implicit. A short glossary or explicit statement would improve readability.
  3. [§4.1, Fig. 11] The statement that increasing kmax ‘always increases the total amount of information available’ is true only for the specific quantities shown; consider rephrasing to avoid implying a general monotonicity theorem for Fisher information when the covariance itself depends on kmax.
  4. [§A.1] The perturbative expansion of s_R around its mean is labelled heuristic and the paper correctly does not use it for the main results. Still, the expression in Eq. (A.8) drops a constant term that contributes at k=0; this should be noted explicitly so future users do not mistakenly include a k=0 mode.
  5. [References] Refs. [6] and [45] are the same paper (Forero-Romero et al. 2009) and are duplicated; likewise refs. [4] and [53] are duplicated in spirit. Please consolidate.

Circularity Check

0 steps flagged

No significant circularity: the EIFs are measured from Quijote simulations, not derived from the marks' definitions; the main caveat is an untested Gaussian-variance leakage assumption, which is a physical confound, not a circular reduction.

full rationale

The paper's central results are numerical Fisher-matrix forecasts. The mark fields are deterministic functionals of the density field (acknowledged in Eq. 2.21 and the Conclusions), and the Error Improvement Factors are computed from Quijote N-body power-spectrum covariances, not from the definitions of the marks. No parameter is fitted to the EIFs; White-mark hyperparameters are taken from the literature, and the shear/LFD mark parameters are scanned rather than optimized against the target. The perturbative expansion in Appendix A is explicitly heuristic and is not used to predict the EIFs. The only notable self-citations are Refs. [35] and [36] (Cowell et al., with overlapping authors) used to justify the affine-invariance interpretation and the claim that R>=5 h^-1 Mpc avoids Gaussian-variance leakage. That smoothing-scale rationale is load-bearing for the interpretation of the gains as non-Gaussian morphological information, yet it is not a circular reduction: it invokes an external result and is testable by extending kmax or quantifying the Gaussian-variance fraction, which the paper does not do. The absence of such a control test is a correctness/robustness concern, not evidence that the EIFs are constructed from their own outputs. I therefore find no step that reduces, by the paper's own equations or by self-citation, to its own inputs.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The central claim depends on standard cosmological perturbation theory, the equivalence principle, and the Fisher-formalism statistical assumptions. The mark hyperparameters are analysis choices, not fitted physical constants. No new physical entities are introduced.

free parameters (7)
  • White mark exponent p = 2
    Strength of the density weighting in the White mark; adopted from literature [26,31]; not optimized here. Affects baseline EIFs.
  • White mark density scale δ_s = 0.25
    Characteristic density above which the White mark suppresses weight; adopted from [26,31].
  • Smoothing scale R = 10 h^-1 Mpc (fiducial); also 5,15,20
    Hyperparameter controlling environmental scale; EIFs vary with R (Table 1). Chosen as compromise in §3.4.
  • Isotropy mark parameters (b,p) = b=5,p=2; variants b=1,p=1; b=1,p=2; b=5,p=1
    Chosen by hand in §2.3.1; performance varies only at percent level (Table 2).
  • Filament mark density suppression δ* = 0.5 and 1.0
    Controls density scale for Gaussian suppression; chosen by hand; EIF varies by ~5% between values.
  • Gaussian shear mark (s0,s*) = (1,0.5), (1,1), (2,0.5), (2,1)
    Target shear and width; chosen to cover plausible ranges; EIFs differ by ~10% between s0=2 and s0=1.
  • Maximum wavenumber k_max = 0.3 h/Mpc (fiducial); 0.5 in part
    Truncation of data vector; EIFs depend on it (Fig. 11).
axioms (6)
  • standard math The density field relates to the tidal tensor via the rescaled Poisson equation ∇²Φ = δ (Eq. 2.3).
    Standard Newtonian Poisson equation in comoving coordinates with cosmological factors absorbed; used to construct shear invariants.
  • domain assumption Gravitational collapse is governed only by the tidal tensor ∂i∂jΦ (equivalence principle).
    Invoked at the start of §2.1 to justify tidal-based morphological descriptors; standard gravitational physics.
  • domain assumption The Fisher formalism with a multivariate Gaussian likelihood is a valid approximation for the marked power-spectrum data vector.
    Stated in §3.2; the paper uses it as a 'standard first approximation' and notes exact values may change in a realistic setting.
  • standard math The sample covariance from 2,100 realizations, after Hartlap correction, is an unbiased estimator of the true covariance.
    Standard result used in §3.3; convergence checked by increasing number of realizations.
  • domain assumption Quijote simulations correctly reproduce the late-time matter distribution at z=0 over the scales and smoothing radii used.
    The entire EIF calculation is built on N-body outputs; the paper cites [55] but does not validate against independent simulations.
  • domain assumption Smoothing at R≥5 h^-1 Mpc ensures EIF gains are dominated by non-Gaussian mode coupling rather than leakage of small-scale Gaussian variance.
    Stated in §3.4 following [35]; not directly quantified in this paper. This is the paper's weakest load-bearing statistical/physical premise.

pith-pipeline@v1.3.0-alltime-deepseek · 24797 in / 16644 out tokens · 148568 ms · 2026-08-01T00:49:33.110120+00:00 · methodology

0 comments
read the original abstract

The morphology of the cosmic web formed by the late-time matter distribution encodes cosmological information beyond that contained in standard two-point statistics. Marked power spectra provide a computationally efficient framework to access this higher-order information, by studying the two-point statistics of the density field ``marked'' (i.e. weighted) by a function of its local environmental density. In this work we explore the potential of marks that are sensitive to the morphology of this local environment, rather than simply its density. We study a broad range of such marks, considering mark functions based on the smoothed density, tidal shear amplitude, local degree of isotropy and filamentarity, Gaussian transformations of these quantities and a local fractal dimension estimator. We quantify the merit of different marks in terms of their constraints on key cosmological parameters, including the matter abundance $\Omega_m$, the amplitude of fluctuations $\sigma_8$, and the mass of neutrinos $M_\nu$. We have found that density-dependent marks continue to provide the largest improvements over the standard power spectrum, while morphology-based marks yield more modest improvements on their own. Nevertheless, combining density- and morphology-based marks consistently enhances cosmological constraints beyond what either class achieves separately, demonstrating that they probe complementary aspects of the underlying matter distribution. These results provide a systematic assessment of morphology-based marked statistics and explore which geometric properties of the cosmic web contribute most effectively to cosmological parameter inference. They also establish a physically motivated framework for future investigations of optimal marks and their perturbative connection to higher-order correlation functions.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 45 linked inside Pith

  1. [1]

    Zeldovich,Gravitational instability: An Approximate theory for large density perturbations,Astron

    Y.B. Zeldovich,Gravitational instability: An Approximate theory for large density perturbations,Astron. Astrophys.5(1970) 84

    1970

  2. [2]

    Gunn and J.R

    J.E. Gunn and J.R. Gott, III,On the Infall of Matter Into Clusters of Galaxies and Some Effects on Their Evolution, ApJ176(1972) 1

    1972

  3. [3]

    J.R. Bond, L. Kofman and D. Pogosyan,How filaments of galaxies are woven into the cosmic web, Nature380(1996) 603 [astro-ph/9512141]

    Pith/arXiv arXiv 1996

  4. [4]

    Springel, C.S

    V. Springel, C.S. Frenk and S.D.M. White,The large-scale structure of the Universe, Nature 440(2006) 1137 [astro-ph/0604561]

    Pith/arXiv arXiv 2006

  5. [5]

    van de Weygaert and W

    R. van de Weygaert and W. Schaap,The Cosmic Web: Geometric Analysis, inData Analysis in Cosmology, V.J. Mart ´ ınez, E. Saar, E. Mart ´ ınez-Gonz´ alez and M.-J. Pons-Border ´ ıa, eds., vol. 665, pp. 291–413 (2009), DOI

    2009

  6. [7]

    Cautun, R

    M. Cautun, R. van de Weygaert, B.J.T. Jones and C.S. Frenk,Evolution of the cosmic web, MNRAS441(2014) 2923 [1401.7866]

    Pith/arXiv arXiv 2014

  7. [8]

    Libeskind, R

    N.I. Libeskind, R. van de Weygaert, M. Cautun, B. Falck, E. Tempel, T. Abel et al.,Tracing the cosmic web, MNRAS473(2018) 1195 [1705.03021]

    Pith/arXiv arXiv 2018

  8. [9]

    Scoccimarro, S

    R. Scoccimarro, S. Colombi, J.N. Fry, J.A. Frieman, E. Hivon and A. Melott,Nonlinear Evolution of the Bispectrum of Cosmological Perturbations, ApJ496(1998) 586 [astro-ph/9704075]

    Pith/arXiv arXiv 1998

  9. [10]

    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(2002) 1 [astro-ph/0112551]

    Pith/arXiv arXiv 2002

  10. [11]

    Mecke, T

    K.R. Mecke, T. Buchert and H. Wagner,Robust morphological measures for large-scale structure in the Universe,A&A288(1994) 697 [astro-ph/9312028]

    Pith/arXiv arXiv 1994

  11. [12]

    Schmalzing and T

    J. Schmalzing and T. Buchert,Beyond Genus Statistics: A Unifying Approach to the Morphology of Cosmic Structure,ApJl482(1997) L1 [astro-ph/9702130]

    Pith/arXiv arXiv 1997

  12. [13]

    Wilding, K

    G. Wilding, K. Nevenzeel, R. van de Weygaert, G. Vegter, P. Pranav, B.J.T. Jones et al., Persistent homology of the cosmic web - I. Hierarchical topology inΛCDM cosmologies, MNRAS507(2021) 2968 [2011.12851]

    Pith/arXiv arXiv 2021

  13. [14]

    Pranav, H

    P. Pranav, H. Edelsbrunner, R. van de Weygaert, G. Vegter, M. Kerber, B.J.T. Jones et al., The topology of the cosmic web in terms of persistent Betti numbers, MNRAS465(2017) 4281 [1608.04519]

    Pith/arXiv arXiv 2017

  14. [15]

    Cheng, Y.-S

    S. Cheng, Y.-S. Ting, B. M’enard and J. Bruna,A new approach to observational cosmology using the scattering transform,MNRAS(2020)

    2020

  15. [16]

    Valogiannis and C

    G. Valogiannis and C. Dvorkin,Towards an optimal estimation of cosmological parameters with the wavelet scattering transform,Phys. Rev. D105(2022) 103534 [2108.07821]

    Pith/arXiv arXiv 2022

  16. [17]

    Pisani, E

    A. Pisani, E. Massara, D.N. Spergel, D. Alonso, T. Baker, Y.-C. Cai et al.,Cosmic voids: a novel probe to shed light on our Universe,BAAS51(2019) 40 [1903.05161]

    Pith/arXiv arXiv 2019

  17. [18]

    Davies, M

    C.T. Davies, M. Cautun, B. Giblin, B. Li, J. Harnois-D’eraps and Y.-C. Cai,Constraining cosmology with weak lensing voids,MNRAS(2020)

    2020

  18. [19]

    Fluri, T

    J. Fluri, T. Kacprzak, R. Sgier, A. Refregier and A. Amara,Weak lensing peak statistics in the era of large scale cosmological surveys, J. Cosmology Astropart. Phys.2018(2018) 051 [1803.08461]. – 31 –

    Pith/arXiv arXiv 2018

  19. [20]

    Stoyan,On Correlations of Marked Point Processes,Mathematische Nachrichten116(1984) 197

    D. Stoyan,On Correlations of Marked Point Processes,Mathematische Nachrichten116(1984) 197

    1984

  20. [21]

    Beisbart and M

    C. Beisbart and M. Kerscher,Luminosity- and Morphology-dependent Clustering of Galaxies, ApJ545(2000) 6 [astro-ph/0003358]

    Pith/arXiv arXiv 2000

  21. [22]

    Skibba, R.K

    R. Skibba, R.K. Sheth, A.J. Connolly and R. Scranton,The luminosity-weighted or ‘marked’ correlation function,MNRAS369(2006) 68 [astro-ph/0512463]

    Pith/arXiv arXiv 2006

  22. [23]

    Skibba, S.P

    R.A. Skibba, S.P. Bamford, R.C. Nichol, C.J. Lintott, D. Andreescu, E.M. Edmondson et al., Galaxy Zoo: disentangling the environmental dependence of morphology and colour, MNRAS 399(2009) 966 [0811.3970]

    Pith/arXiv arXiv 2009

  23. [24]

    K. Joy, U. Sureshkumar, A. Narayanan, S. Bellstedt, A. Durkalec, A. Pollo et al., Environmental dependence of galaxy properties in the southern GAMA regions,arXiv e-prints (2025) arXiv:2505.10429 [2505.10429]

    arXiv 2025

  24. [25]

    Sureshkumar, A

    U. Sureshkumar, A. Durkalec, A. Pollo, M. Bilicki, J. Loveday, D.J. Farrow et al.,Galaxy and Mass Assembly (GAMA). Tracing galaxy environment using the marked correlation function, A&A653(2021) A35 [2102.04177]

    Pith/arXiv arXiv 2021

  25. [26]

    White,A marked correlation function for constraining modified gravity models, J

    M. White,A marked correlation function for constraining modified gravity models, J. Cosmology Astropart. Phys.2016(2016) 057 [1609.08632]

    Pith/arXiv arXiv 2016

  26. [27]

    Armijo, Y.-C

    J. Armijo, Y.-C. Cai, N. Padilla, B. Li and J.A. Peacock,Testing modified gravity using a marked correlation function,MNRAS478(2018) 3627 [1801.08975]

    Pith/arXiv arXiv 2018

  27. [28]

    K¨ archer, J

    M. K¨ archer, J. Bel and S. de la Torre,Towards an optimal marked correlation function analysis for the detection of modified gravity, A&A694(2025) A253 [2406.02504]

    Pith/arXiv arXiv 2025

  28. [29]

    Valogiannis and R

    G. Valogiannis and R. Bean,Beyondδ: Tailoring marked statistics to reveal modified gravity, Phys. Rev. D97(2018) 023535 [1708.05652]

    Pith/arXiv arXiv 2018

  29. [30]

    Hamaus, M.-C

    N. Hamaus, M.-C. Cousinou, A. Pisani, M. Aubert, S. Escoffier and J. Weller,Multipole analysis of redshift-space distortions around cosmic voids, J. Cosmology Astropart. Phys.2017 (2017) 014 [1705.05328]

    Pith/arXiv arXiv 2017

  30. [31]

    Massara, F

    E. Massara, F. Villaescusa-Navarro, C. Hahn, M.M. Abidi, M. Eickenberg, S. Ho et al., Cosmological Information in the Marked Power Spectrum of the Galaxy Field,ApJ951(2023) 70 [2206.01709]

    Pith/arXiv arXiv 2023

  31. [32]

    Massara, C

    E. Massara, C. Hahn, M. Eickenberg, S. Ho, J. Hou, P. Lemos et al.,Cosmological constraints using simulation-based inference of galaxy clustering with marked power spectra, Phys. Rev. D 112(2025) 083507 [2404.04228]

    Pith/arXiv arXiv 2025

  32. [33]

    Satpathy, R

    S. Satpathy, R. A C Croft, S. Ho and B. Li,Measurement of marked correlation functions in SDSS-III Baryon Oscillation Spectroscopic Survey using LOWZ galaxies in Data Release 12, MNRAS484(2019) 2148 [1901.01447]

    Pith/arXiv arXiv 2019

  33. [34]

    Armijo, C.M

    J. Armijo, C.M. Baugh, P. Norberg and N.D. Padilla,A new test of gravity - I. Introduction to the method,MNRAS529(2024) 2866 [2304.06218]

    Pith/arXiv arXiv 2024

  34. [35]

    Cowell, J

    J.A. Cowell, J. Armijo, L. Thiele, G.A. Marques, C.P. Novaes, D. Grand´ on et al.,First constraints from marked angular power spectra with Subaru Hyper Suprime-Cam Survey First-Year Data, MNRAS546(2026) stag033 [2507.12315]

    Pith/arXiv arXiv 2026

  35. [36]

    Cowell, D

    J.A. Cowell, D. Alonso and J. Liu,Optimizing marked power spectra for cosmology, MNRAS 535(2024) 3129 [2409.05695]

    Pith/arXiv arXiv 2024

  36. [37]

    Philcox, E

    O.H.E. Philcox, E. Massara and D.N. Spergel,What does the marked power spectrum measure? Insights from perturbation theory, Phys. Rev. D102(2020) 043516 [2006.10055]. – 32 –

    Pith/arXiv arXiv 2020

  37. [38]

    Philcox, A

    O.H.E. Philcox, A. Aviles and E. Massara,Modeling the marked spectrum of matter and biased tracers in real- and redshift-space,J. Cosmology Astropart. Phys.2021(2021) 038 [2010.05914]

    Pith/arXiv arXiv 2021

  38. [39]

    Marinucci, G

    M. Marinucci, G. Jung, M. Liguori, A. Ravenni, F. Spezzati, A. Andrews et al.,The constraining power of the marked power spectrum: an analytical study,J. Cosmology Astropart. Phys2025(2025) 036 [2411.14377]

    Pith/arXiv arXiv 2025

  39. [40]

    Ebina and M

    H. Ebina and M. White,An analytically tractable marked power spectrum, J. Cosmology Astropart. Phys.2025(2025) 150 [2409.17133]

    Pith/arXiv arXiv 2025

  40. [41]

    Ebina, M

    H. Ebina, M. White and E. Chaussidon,The Marked Power Spectrum as a Practical Bispectrum Measure for Galaxy Redshift Surveys,arXiv e-prints(2026) arXiv:2603.03581 [2603.03581]

    arXiv 2026

  41. [42]

    Marinucci, M

    M. Marinucci, M. Liguori, F. Semenzato and B.D. Wandelt,Optimal marked statistics from the Effective Field Theory of Large Scale Structure,arXiv e-prints(2026) arXiv:2606.14332 [2606.14332]

    arXiv 2026

  42. [43]

    Alonso, E

    D. Alonso, E. Eardley and J.A. Peacock,Halo abundances within the cosmic web, MNRAS447 (2015) 2683 [1406.4159]

    Pith/arXiv arXiv 2015

  43. [44]

    S. Alam, Y. Zu, J.A. Peacock and R. Mandelbaum,Cosmic web dependence of galaxy clustering and quenching in SDSS, MNRAS483(2019) 4501 [1801.04878]

    Pith/arXiv arXiv 2019

  44. [45]

    Forero-Romero, Y

    J.E. Forero-Romero, Y. Hoffman, S. Gottl¨ ober, A. Klypin and G. Yepes,A dynamical classification of the cosmic web, MNRAS396(2009) 1815 [0809.4135]

    Pith/arXiv arXiv 2009

  45. [46]

    Ay¸ coberry, A

    E. Ay¸ coberry, A. Barthelemy and S. Codis,A theoretical view of the T-web statistical description of the cosmic web, A&A686(2024) A276 [2310.03548]

    Pith/arXiv arXiv 2024

  46. [47]

    Armijo and L

    J. Armijo and L. Da Costa,Marked statistics across the cosmic web: environmental dependent clustering in modified gravity simulations, MNRAS550(2026) stag1217 [2601.05934]

    arXiv 2026

  47. [48]

    Doroshkevich,Spatial structure of perturbations and origin of galactic rotation in fluctuation theory,Astrophysics6(1970) 320

    A.G. Doroshkevich,Spatial structure of perturbations and origin of galactic rotation in fluctuation theory,Astrophysics6(1970) 320

    1970

  48. [49]

    van de Weygaert and E

    R. van de Weygaert and E. Bertschinger,Peak and gravity constraints in Gaussian primordial density fields: An application of the Hoffman-Ribak method, MNRAS281(1996) 84 [astro-ph/9507024]

    Pith/arXiv arXiv 1996

  49. [50]

    Bardeen, J.R

    J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay,The Statistics of Peaks of Gaussian Random Fields, ApJ304(1986) 15

    1986

  50. [51]

    Desjacques,Environmental dependence in the ellipsoidal collapse model, MNRAS388 (2008) 638 [0707.4670]

    V. Desjacques,Environmental dependence in the ellipsoidal collapse model, MNRAS388 (2008) 638 [0707.4670]

    Pith/arXiv arXiv 2008

  51. [52]

    McDonald and A

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

    Pith/arXiv arXiv 2009

  52. [53]

    Desjacques, D

    V. Desjacques, D. Jeong and F. Schmidt,Large-scale galaxy bias, Phys. Rep.733(2018) 1 [1611.09787]

    Pith/arXiv arXiv 2018

  53. [54]

    O. Hahn, C. Porciani, C.M. Carollo and A. Dekel,Properties of dark matter haloes in clusters, filaments, sheets and voids, MNRAS375(2007) 489 [astro-ph/0610280]

    Pith/arXiv arXiv 2007

  54. [55]

    Villaescusa-Navarro, C

    F. Villaescusa-Navarro, C. Hahn, E. Massara, A. Banerjee, A.M. Delgado, D.K. Ramanah et al.,The Quijote Simulations, ApJS250(2020) 2 [1909.05273]

    Pith/arXiv arXiv 2020

  55. [56]

    Pylians: Python libraries for the analysis of numerical simulations

    F. Villaescusa-Navarro, “Pylians: Python libraries for the analysis of numerical simulations.” Astrophysics Source Code Library, record ascl:1811.008, November, 2018

    2018

  56. [57]

    C.F. Park, E. Allys, F. Villaescusa-Navarro and D. Finkbeiner,Quantification of High-dimensional Non-Gaussianities and Its Implication to Fisher Analysis in Cosmology, ApJ 946(2023) 107 [2204.05435]. – 33 –

    Pith/arXiv arXiv 2023

  57. [58]

    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(2007) 399 [astro-ph/0608064]

    Pith/arXiv arXiv 2007

  58. [59]

    Lucie-Smith, H.V

    L. Lucie-Smith, H.V. Peiris, A. Pontzen, B. Nord and J. Thiyagalingam,Deep learning insights into cosmological structure formation, Phys. Rev. D109(2024) 063524 [2011.10577]. – 34 –

    Pith/arXiv arXiv 2024