Pith. sign in

REVIEW 1 major objections 2 minor 2 cited by

The imprints of massive neutrinos on the three-point correlation function of large-scale structures

T0 review · 1 major / 2 minor · reviewed 2026-05-16 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read The three-point correlation function shows distinct massive-neutrino signals on elongated and right-angled triangles that differ from sigma_8 variations.

desk verdict Quijote 3PCF measurements show distinct signals on elongated and right-angled triangles that differ from sigma8 shifts, but resolution limits may affect the claimed degeneracy breaking. read the letter →

arxiv 2512.16992 v2 submitted 2025-12-18 astro-ph.CO

Andrea Labate , Massimo Guidi , Michele Moresco , Alfonso Veropalumbo This is my paper · ORCID
classification astro-ph.CO
keywords massiveneutrinosthree-pointcorrelationfunctionlarge-scalestructuresigma_8degeneracyhaloclusteringN-bodysimulations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper measures the isotropic connected three-point correlation function ζ and the reduced three-point correlation function Q in halo catalogs from N-body simulations with neutrino masses of 0.0, 0.1, 0.2, and 0.4 eV. It identifies the strongest effects in quasi-isosceles and squeezed triangles, with elongated triangles most sensitive in ζ and right-angled triangles providing complementary signal in Q. These patterns grow stronger at lower redshifts and look markedly different from the changes produced by varying sigma_8 instead of neutrino mass. The distinction suggests the three-point correlation function can break the well-known degeneracy between the two parameters, with potential application to stage-IV spectroscopic surveys.

What carries the argument

The isotropic connected three-point correlation function ζ and reduced three-point correlation function Q evaluated on different triangle shapes in simulated halo distributions.

What would settle it

If measurements from a stage-IV spectroscopic survey find that the three-point correlation function differences on elongated triangles in ζ and right-angled triangles in Q match those produced by sigma_8 variations alone, the claim that the 3PCF breaks the degeneracy would be falsified.

Watch

Extended reading notes

Core claim

Measurements of ζ and Q on halo catalogs from the Quijote simulations reveal that neutrino free-streaming produces its strongest signal in quasi-isosceles and squeezed triangles, increasing toward lower redshifts. Elongated triangles are most affected in ζ while right-angled triangles supply a complementary source in Q. Direct comparison with simulations that vary sigma_8 without neutrinos shows these signatures are significantly different, indicating that the three-point correlation function can break the M_ν–σ_8 degeneracy.

Load-bearing premise

The Quijote N-body simulations accurately capture the free-streaming effects of massive neutrinos on halo clustering without significant contamination from resolution limits or other modeling choices.

Editorial extensions

If this is right

  • Elongated triangles in ζ become the primary probe for isolating neutrino-mass effects in future galaxy surveys.
  • Right-angled triangles in Q supply an independent channel less entangled with sigma_8.
  • The neutrino signal strengthens at lower redshifts, favoring lower-redshift bins in survey analyses.
  • The triangle-shape framework can be applied directly to data from DESI, Euclid, 4MOST, and the Roman Space Telescope.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same shape-dependent analysis could be extended to other degenerate parameter pairs in large-scale structure studies.
  • Combining three-point correlation function measurements with two-point statistics may tighten neutrino-mass bounds beyond what either probe achieves separately.
  • Real survey data will ultimately test whether the simulated differences persist once observational systematics are included.
Share X Bluesky LinkedIn Reddit HN

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

1 major / 2 minor

Summary. The manuscript claims that measurements of the isotropic connected three-point correlation function ζ and reduced three-point correlation function Q from halo catalogs in the Quijote N-body suite (M_ν = 0.0, 0.1, 0.2, 0.4 eV) show the strongest neutrino free-streaming signals on quasi-isosceles, squeezed, and elongated triangles in ζ and right-angled triangles in Q. These signatures differ from those produced by σ8 variations, enabling the 3PCF to break the M_ν-σ8 degeneracy. A detectability framework is developed and applied to a stage-IV spectroscopic survey case, with signals strengthening toward lower redshifts.

Significance. If the central results hold, the work would be significant for large-scale structure cosmology by identifying specific 3PCF triangle configurations that respond differently to neutrino free-streaming than to σ8, offering a practical complement to two-point statistics for neutrino mass constraints in surveys such as DESI, Euclid, and Roman. The use of the public Quijote suite with direct triangle counting across multiple masses and redshifts, plus the explicit detectability framework, provides a reproducible foundation for follow-up analyses.

major comments (1)
  1. The degeneracy-breaking claim (abstract) requires that differences in elongated triangles for ζ and right-angled triangles for Q between the M_ν runs and the σ8-varied runs arise purely from neutrino free-streaming. The Quijote setup deploys 512^3 CDM particles in 1 Gpc/h volumes with neutrinos as additional particles; for M_ν = 0.1–0.4 eV the free-streaming wavenumber lies near the resolution limit, where neutrino shot noise and force softening can alter small-scale halo clustering. No convergence tests of the 3PCF versus particle number or neutrino assignment scheme are referenced, so numerical artifacts would not cancel in the σ8 comparison and could mimic the reported distinct signatures.
minor comments (2)
  1. The abstract mentions both 'quasi-isosceles and squeezed triangles' and 'elongated triangles' as most affected; a short explicit statement of how these shape classes relate (e.g., via opening-angle or side-length ratios) would improve clarity.
  2. Details on covariance estimation, triangle binning, and error-bar construction for the detectability framework should be expanded in the methods section to allow full reproducibility of the stage-IV forecast.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We have carefully considered the major comment and provide our response below, along with plans for revision.

read point-by-point responses

  1. Referee: The degeneracy-breaking claim (abstract) requires that differences in elongated triangles for ζ and right-angled triangles for Q between the M_ν runs and the σ8-varied runs arise purely from neutrino free-streaming. The Quijote setup deploys 512^3 CDM particles in 1 Gpc/h volumes with neutrinos as additional particles; for M_ν = 0.1–0.4 eV the free-streaming wavenumber lies near the resolution limit, where neutrino shot noise and force softening can alter small-scale halo clustering. No convergence tests of the 3PCF versus particle number or neutrino assignment scheme are referenced, so numerical artifacts would not cancel in the σ8 comparison and could mimic the reported distinct signatures.

    Authors: We appreciate the referee pointing out the potential for numerical artifacts in the neutrino simulations. The Quijote suite uses 512^3 CDM particles and additional neutrino particles, and while the free-streaming scales for the considered M_ν values are indeed near the resolution limit, our measurements focus on triangle configurations where the signal is dominated by large-scale modes less affected by small-scale noise. The σ8-varied simulations share the same CDM resolution, facilitating a fair comparison of the shape-dependent responses. However, we agree that explicit convergence tests for the 3PCF are necessary to fully substantiate the claims. In the revised manuscript, we will add a dedicated section presenting convergence checks, including comparisons with available higher-resolution Quijote runs and assessments of neutrino particle loading effects, to confirm that the distinct signatures in elongated triangles for ζ and right-angled triangles for Q are physical and not artifacts. This will strengthen the degeneracy-breaking argument. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: results obtained via direct counting in N-body catalogs

full rationale

The paper reports measurements of the isotropic connected 3PCF ζ and reduced 3PCF Q extracted by direct enumeration of triangle configurations in halo catalogs drawn from the Quijote N-body suite. No analytic derivation, fitted functional form, or self-citation chain is invoked to obtain the reported signals on elongated or right-angled triangles; the differences between M_ν and σ_8 runs are presented as numerical outcomes of the simulation measurements themselves. The framework for detectability is likewise a post-processing quantification applied to those counts. Because the central claims rest on external simulation data rather than on any equation that reduces to its own inputs by construction, the derivation chain contains no circular steps.

Assumptions & free parameters 1 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the assumption that N-body simulations correctly model neutrino free-streaming and that the chosen triangle statistics capture the relevant physics without additional free parameters beyond the input neutrino masses.

free parameters (1)
  • M_nu simulation values
    Discrete neutrino mass sums (0.0, 0.1, 0.2, 0.4 eV) are chosen as input parameters for the simulation suite rather than derived.
assumptions (1)
  • domain assumption Quijote N-body simulations accurately reproduce the effects of neutrino free-streaming on halo clustering
    Invoked when interpreting differences in zeta and Q as direct imprints of M_nu

how reviews work

0 comments
Cite this review

Pith. "Pith review of The imprints of massive neutrinos on the three-point correlation function of large-scale structures." pith.science (2026). https://pith.science/paper/2512.16992

@misc{pith2026251216992,
  author       = {Pith},
  title        = {Pith review of: The imprints of massive neutrinos on the three-point correlation function of large-scale structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2512.16992}},
  note         = {Machine review of arXiv:2512.16992}
}
abstract

Free-streaming of cosmic neutrinos affects the distribution and growth of cosmic structures on small scales. This enables the sum of neutrino masses $M_\nu$ to be constrained from clustering studies. We investigate the possibility of disentangling massive neutrino cosmologies with the three-point correlation function (3PCF) for the first time. We measured the isotropic connected 3PCF $\zeta$ and the reduced 3PCF $Q$ of halo catalogs from the Quijote suite of $N$-body simulations, considering $M_\nu =0.0, 0.1, 0.2,$ and $0.4 \, \mathrm{eV}$ in different redshift bins. We developed a framework to quantify the detectability of massive neutrinos for different triangle configurations and shapes, and applied it to a case compatible with a stage-IV spectroscopic survey. We also compared our results with the analysis of simulations without neutrinos, but with different $\sigma_8$ values, to test whether the 3PCF can break the well-known degeneracy between the two parameters. We found that as a result of free-streaming, the strongest signal is found for quasi-isosceles and squeezed triangles; this signal increases for decreasing redshifts. Among these configurations, elongated triangles, tracing the filamentary structure of the cosmic web, are the most affected by massive neutrinos, with a 3PCF signal increasing with $M_\nu$. A complementary source of signal comes from right-angled triangles in $Q$. Importantly, we found that the signatures of a $\sigma_8$ variation appear to be significantly different on elongated triangles in $\zeta$ and right-angled triangles in $Q$, suggesting that the 3PCF can be used to effectively break the $M_\nu - \sigma_8$ degeneracy. These results open the possibility to use the 3PCF as a powerful complementary tool for constraining neutrino masses in current and future spectroscopic surveys such as DESI, Euclid, 4MOST, and the Nancy Grace Roman Space Telescope.

Figures

Figures reproduced from arXiv: 2512.16992 by the authors.

Figure 1
Figure 1. The values of the parameter ˜χ 2 (s12, s13) defined in Eq. 10, for the single-scale connected 3PCF, obtained for Mν = 0.4 eV. Each panel corresponds to a different redshift: from left to right, z = 0, 1, and 2. The lines overplotted on the left panel are taken as representative of regions of enhanced signal, and identify isosceles triangles (black dashed line with η = 0, where the signal is enhanced only on scales ≲… view at source ↗
Figure 2
Figure 2. Single-scale connected 3PCF for the triangle configurations selected in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Detectability matrices of massive neutrinos. Each matrix element shows the scale scross (Eq. 13) below which, considering all triangles with scale larger than scross, we get a significant detection of the signal from massive neutrinos in the connected 3PCF. The matrices in the upper and lower rows show the values for a 1σ and 3σ statistical significance (Eq. 12), respectively, computed for a volume of 10 h −3 Gpc3 .… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The detectability of the halo connected 3PCF (upper panels) and reduced 3PCF (lower panels) as a function of the triangle shape. Here, the results are reported at z = 0 and for Mν = 0.4 eV. The triangle shapes are determined by the side ratios s12/s23 and s13/s23, with…
Figure 5
Figure 5. Figure 5: Comparison between the detectability of a variation in Mν and σ8 from the halo 3PCF as a function of triangle shape. We show the results for the simulations at z = 0 with Mν = 0.1 eV and fiducial σ8 = 0.834 (red-scale colormaps), and with Mν = 0 eV and σ8 = 0.849 (uppe…

Discussion (0). Continue with ORCID to comment.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. First full-shape joint analysis of the two- and three-point correlation functions on real data: $\Lambda$CDM cosmological constraints from BOSS DR12

    astro-ph.CO 2026-06 unverdicted novelty 8.0 of 10

    First joint 2PCF+3PCF full-shape analysis on BOSS DR12 real data improves σ(h) by ~29%, σ(ω_cdm) by ~10%, and σ(A_s) by ~24% over 2PCF alone via extra BAO information in 3PCF triangles.

  2. Cosmological constraints from the DESI DR1 Bispectrum Full-Shape and DR2 BAO

    astro-ph.CO 2026-06 unverdicted novelty 4.0 of 10

    DESI DR1 bispectrum plus DR2 BAO data raise σ8 and S8 by ~1.1-1.2σ while tightening uncertainties, shift DESI-only w0waCDM toward ΛCDM, produce a 2.8σ deviation from ΛCDM when combined with CMB, and yield a neutrino m...

Reference graph

Works this paper leans on

107 extracted references · 107 canonical work pages · cited by 2 Pith papers

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

    work page

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

    work page

  3. [3]

    2025, , 112, 083515

    Abdul Karim , M., Aguilar , J., Ahlen , S., et al. 2025, , 112, 083515

    work page 2025

  4. [4]

    G., Aguilar , J., Ahlen , S., et al

    Adame , A. G., Aguilar , J., Ahlen , S., et al. 2025, , 2025, 028

    work page 2025

  5. [5]

    2006, arXiv e-prints, astro

    Albrecht , A., Bernstein , G., Cahn , R., et al. 2006, arXiv e-prints, astro

    work page 2006

  6. [6]

    M., Bond , J

    Bardeen , J. M., Bond , J. R., Kaiser , N., & Szalay , A. S. 1986, , 304, 15

    work page 1986

  7. [7]

    2002, , 367, 1

    Bernardeau , F., Colombi , S., Gazta \ n aga , E., & Scoccimarro , R. 2002, , 367, 1

    work page 2002

  8. [8]

    R., Efstathiou , G., & Silk , J

    Bond , J. R., Efstathiou , G., & Silk , J. 1980, , 45, 1980

    work page 1980

Show all 107 references
  1. [9]

    2008, , 2008, 020

    Brandbyge , J., Hannestad , S., Haugb lle , T., & Thomsen , B. 2008, , 2008, 020

    2008

  2. [10]

    Brandbyge , J., Hannestad , S., Haugb lle , T., & Wong , Y. Y. Y. 2010, , 2010, 014

    2010

  3. [11]

    2015, , 2015, 043

    Castorina , E., Carbone , C., Bel , J., Sefusatti , E., & Dolag , K. 2015, , 2015, 043

    2015

  4. [12]

    K., Villaescusa-Navarro , F., & Viel , M

    Castorina , E., Sefusatti , E., Sheth , R. K., Villaescusa-Navarro , F., & Viel , M. 2014, , 2014, 049

    2014

  5. [13]

    & Matarrese , S

    Celoria , M. & Matarrese , S. 2018, arXiv e-prints, arXiv:1812.08197

    2018 arXiv

  6. [14]

    & Kaiser , N

    Cole , S. & Kaiser , N. 1989, , 237, 1127

    1989

  7. [15]

    2011, European Physical Journal C, 71, 1554

    Cowan , G., Cranmer , K., Gross , E., & Vitells , O. 2011, European Physical Journal C, 71, 1554

    2011

  8. [16]

    S., & White , S

    Davis , M., Efstathiou , G., Frenk , C. S., & White , S. D. M. 1985, , 292, 371

    1985

  9. [17]

    S., & Novaes , C

    de Carvalho , E., Bernui , A., Xavier , H. S., & Novaes , C. P. 2020, , 492, 4469

    2020

  10. [18]

    S., Agertz , O., Berbel , A

    de Jong , R. S., Agertz , O., Berbel , A. A., et al. 2019, The Messenger, 175, 3

    2019

  11. [19]

    2016, arXiv e-prints, arXiv:1611.00036

    DESI Collaboration: Aghamousa , A., Aguilar , J., Ahlen , S., et al. 2016, arXiv e-prints, arXiv:1611.00036

    2016 arXiv

  12. [20]

    2018, , 733, 1

    Desjacques , V., Jeong , D., & Schmidt , F. 2018, , 733, 1

    2018

  13. [21]

    2019, , 51, 341

    Dore , O., Hirata , C., Wang , Y., et al. 2019, , 51, 341

    2019

  14. [22]

    E., et al

    Elbers , W., Aviles , A., Noriega , H. E., et al. 2025, , 112, 083513

    2025

  15. [23]

    2025, arXiv e-prints, arXiv:2506.22257

    Euclid Collaboration: Guidi , M., Veropalumbo , A., Pugno , A., et al. 2025, arXiv e-prints, arXiv:2506.22257

    2025

  16. [24]

    A., et al

    Euclid Collaboration: Mellier , Y., Abdurro'uf , Acevedo Barroso , J. A., et al. 2025, , 697, A1

    2025

  17. [25]

    2022, , 662, A112

    Euclid Collaboration: Scaramella , R., Amiaux , J., Mellier , Y., et al. 2022, , 662, A112

    2022

  18. [26]

    2020, , 497, 2699

    Fang , X., Eifler , T., & Krause , E. 2020, , 497, 2699

    2020

  19. [27]

    2024, arXiv e-prints, arXiv:2408.03036

    Farina , A., Veropalumbo , A., Branchini , E., & Guidi , M. 2024, arXiv e-prints, arXiv:2408.03036

    2024

  20. [28]

    Fisher , K. B. 1995, , 448, 494

    1995

  21. [29]

    Frieman , J. A. & Gaztanaga , E. 1994, , 425, 392

    1994

  22. [30]

    Fry , J. N. 1984, , 279, 499

    1984

  23. [31]

    Fry , J. N. 1994, , 73, 215

    1994

  24. [32]

    Fry , J. N. & Gaztanaga , E. 1993, , 413, 447

    1993

  25. [33]

    1998, , 81, 1562

    Fukuda , Y., Hayakawa , T., Ichihara , E., et al. 1998, , 81, 1562

    1998

  26. [34]

    2009, , 399, 801

    Gazta \ n aga , E., Cabr \'e , A., Castander , F., Crocce , M., & Fosalba , P. 2009, , 399, 801

    2009

  27. [35]

    M., & Croton , D

    Gazta \ n aga , E., Norberg , P., Baugh , C. M., & Croton , D. J. 2005, , 364, 620

    2005

  28. [36]

    N., S \'a nchez , A

    Grieb , J. N., S \'a nchez , A. G., Salazar-Albornoz , S., et al. 2017, , 467, 2085

    2017

  29. [37]

    Groth , E. J. & Peebles , P. J. E. 1977, , 217, 385

    1977

  30. [38]

    2023, , 2023, 066

    Guidi , M., Veropalumbo , A., Branchini , E., Eggemeier , A., & Carbone , C. 2023, , 2023, 066

    2023

  31. [39]

    & Villaescusa-Navarro , F

    Hahn , C. & Villaescusa-Navarro , F. 2021, , 2021, 029

    2021

  32. [40]

    2020, , 2020, 040

    Hahn , C., Villaescusa-Navarro , F., Castorina , E., & Scoccimarro , R. 2020, , 2020, 040

    2020

  33. [41]

    Hamilton , A. J. S. 1992, , 385, L5

    1992

  34. [42]

    2007, , 464, 399

    Hartlap , J., Simon , P., & Schneider , P. 2007, , 464, 399

    2007

  35. [43]

    R., Colombi , S., & Juszkiewicz , R

    Hivon , E., Bouchet , F. R., Colombi , S., & Juszkiewicz , R. 1995, , 298, 643

    1995

  36. [44]

    J., & Tegmark , M

    Hu , W., Eisenstein , D. J., & Tegmark , M. 1998, , 80, 5255

    1998

  37. [45]

    M., Simonovi \'c , M., & Zaldarriaga , M

    Ivanov , M. M., Simonovi \'c , M., & Zaldarriaga , M. 2020, , 101, 083504

    2020

  38. [46]

    Jing , Y. P. & B \"o rner , G. 2004, , 607, 140

    2004

  39. [47]

    P., Borner , G., & Valdarnini , R

    Jing , Y. P., Borner , G., & Valdarnini , R. 1995, , 277, 630

    1995

  40. [48]

    1984, , 284, L9

    Kaiser , N. 1984, , 284, L9

    1984

  41. [49]

    1987, , 227, 1

    Kaiser , N. 1987, , 227, 1

    1987

  42. [50]

    & Slepian , Z

    Kamalinejad , F. & Slepian , Z. 2025 a , arXiv e-prints, arXiv:2508.06759

    2025

  43. [51]

    & Slepian , Z

    Kamalinejad , F. & Slepian , Z. 2025 b , , 112, 083501

    2025

  44. [52]

    Landy , S. D. & Szalay , A. S. 1993, , 412, 64

    1993

  45. [53]

    2011, arXiv e-prints, arXiv:1110.3193

    Laureijs , R., Amiaux , J., Arduini , S., et al. 2011, arXiv e-prints, arXiv:1110.3193

    2011 arXiv

  46. [54]

    & Pastor , S

    Lesgourgues , J. & Pastor , S. 2006, , 429, 307

    2006

  47. [55]

    I., Brinchmann , J., et al

    Mainieri , V., Anderson , R. I., Brinchmann , J., et al. 2024, arXiv e-prints, arXiv:2403.05398

    2024

  48. [56]

    A., Blake , C., Poole , G

    Mar \' n , F. A., Blake , C., Poole , G. B., et al. 2013, , 432, 2654

    2013

  49. [57]

    2011, , 418, 346

    Marulli , F., Carbone , C., Viel , M., Moscardini , L., & Cimatti , A. 2011, , 418, 346

    2011

  50. [58]

    K., Connolly , A

    McBride , C. K., Connolly , A. J., Gardner , J. P., et al. 2011, , 739, 85

    2011

  51. [59]

    D., Green , D., Flauger , R., et al

    Meerburg , P. D., Green , D., Flauger , R., et al. 2019, , 51, 107

    2019

  52. [60]

    Mo , H. J. & White , S. D. M. 1996, , 282, 347

    1996

  53. [61]

    2014, , 443, 2874

    Moresco , M., Marulli , F., Baldi , M., Moscardini , L., & Cimatti , A. 2014, , 443, 2874

    2014

  54. [62]

    2021, , 919, 144

    Moresco , M., Veropalumbo , A., Marulli , F., Moscardini , L., & Cimatti , A. 2021, , 919, 144

    2021

  55. [63]

    2023, , 2023, 025

    Moretti , C., Tsedrik , M., Carrilho , P., & Pourtsidou , A. 2023, , 2023, 025

    2023

  56. [64]

    2025, , 2025, 045

    Nadal-Matosas , A., Gil-Mar \' n , H., & Verde , L. 2025, , 2025, 045

    2025

  57. [65]

    2024, , 110, 030001

    Navas , S., Amsler , C., Gutsche , T., et al. 2024, , 110, 030001

    2024

  58. [66]

    2021, , 2021, 038

    Oddo , A., Rizzo , F., Sefusatti , E., Porciani , C., & Monaco , P. 2021, , 2021, 038

    2021

  59. [67]

    2022, , 2022, 066

    Pardede , K., Rizzo , F., Biagetti , M., et al. 2022, , 2022, 066

    2022

  60. [68]

    2021, , 2021, 009

    Parimbelli , G., Anselmi , S., Viel , M., et al. 2021, , 2021, 009

    2021

  61. [69]

    Peebles , P. J. E. 1973, , 185, 413

    1973

  62. [70]

    Peebles , P. J. E. 1980, The Large-Scale Structure of the Universe (Princeton University Press)

    1980

  63. [71]

    Peebles , P. J. E. & Groth , E. J. 1975, , 196, 1

    1975

  64. [72]

    2015, , 2015, 001

    Peloso , M., Pietroni , M., Viel , M., & Villaescusa-Navarro , F. 2015, , 2015, 001

    2015

  65. [73]

    Philcox , O. H. E. 2021, , 104, 123529

    2021

  66. [74]

    2020, , 641, A6

    Planck Collaboration , Aghanim , N., Akrami , Y., et al. 2020, , 641, A6

    2020

  67. [75]

    2025, , 2025, 075

    Pugno , A., Eggemeier , A., Porciani , C., & Kuruvilla , J. 2025, , 2025, 075

    2025

  68. [76]

    2018, , 2018, 003

    Ruggeri , R., Castorina , E., Carbone , C., & Sefusatti , E. 2018, , 2018, 003

    2018

  69. [77]

    G., Montesano , F., Kazin , E

    S \'a nchez , A. G., Montesano , F., Kazin , E. A., et al. 2014, , 440, 2692

    2014

  70. [78]

    G., Scoccimarro , R., Crocce , M., et al

    S \'a nchez , A. G., Scoccimarro , R., Crocce , M., et al. 2017, , 464, 1640

    2017

  71. [79]

    2004, , 70, 083007

    Scoccimarro , R. 2004, , 70, 083007

    2004

  72. [80]

    Scoccimarro , R., Couchman , H. M. P., & Frieman , J. A. 1999, , 517, 531

    1999

  73. [81]

    2006, , 74, 023522

    Sefusatti , E., Crocce , M., Pueblas , S., & Scoccimarro , R. 2006, , 74, 023522

    2006

  74. [82]

    G., Pezzotta , A., et al

    Semenaite , A., S \'a nchez , A. G., Pezzotta , A., et al. 2023, , 521, 5013

    2023

  75. [83]

    Sheth , R. K. & Tormen , G. 1999, , 308, 119

    1999

  76. [84]

    & Eisenstein , D

    Slepian , Z. & Eisenstein , D. J. 2015 a , , 454, 4142

    2015

  77. [85]

    & Eisenstein , D

    Slepian , Z. & Eisenstein , D. J. 2015 b , , 448, 9

    2015

  78. [86]

    & Eisenstein , D

    Slepian , Z. & Eisenstein , D. J. 2017, , 469, 2059

    2017

  79. [87]

    & Eisenstein , D

    Slepian , Z. & Eisenstein , D. J. 2018, , 478, 1468

    2018

  80. [88]

    J., Beutler , F., et al

    Slepian , Z., Eisenstein , D. J., Beutler , F., et al. 2017 a , , 468, 1070

    2017

  81. [89]

    J., Blazek , J

    Slepian , Z., Eisenstein , D. J., Blazek , J. A., et al. 2018, , 474, 2109

    2018

  82. [90]

    J., Brownstein , J

    Slepian , Z., Eisenstein , D. J., Brownstein , J. R., et al. 2017 b , , 469, 1738

    2017

  83. [91]

    Springel , V., White , S. D. M., Jenkins , A., et al. 2005, , 435, 629

    2005

  84. [92]

    S., Saito , S., Beutler , F., & Seo , H.-J

    Sugiyama , N. S., Saito , S., Beutler , F., & Seo , H.-J. 2021, , 501, 2862

    2021

  85. [93]

    2004, , 605, L89

    Szapudi , I. 2004, , 605, L89

    2004

  86. [94]

    & Szalay , A

    Szapudi , I. & Szalay , A. S. 1998, , 494, L41

    1998

  87. [95]

    2014, Progress of Theoretical and Experimental Physics, 2014, 06B105

    Takahashi , T. 2014, Progress of Theoretical and Experimental Physics, 2014, 06B105

    2014

  88. [96]

    2010, , 82, 063522

    Taruya , A., Nishimichi , T., & Saito , S. 2010, , 82, 063522

    2010

  89. [97]

    2021, , 2021, 035

    Umeh , O. 2021, , 2021, 035

    2021

  90. [98]

    F., & Kamionkowski , M

    Verde , L., Wang , L., Heavens , A. F., & Kamionkowski , M. 2000, , 313, 141

    2000

  91. [99]

    2025, , 112, 043545

    Verdiani , F., Bellini , E., Moretti , C., et al. 2025, , 112, 043545

    2025

  92. [100]

    2022, , 2022, 033

    Veropalumbo , A., Binetti , A., Branchini , E., et al. 2022, , 2022, 033

    2022

  93. [101]

    2021, , 507, 1184

    Veropalumbo , A., S \'a ez Casares , I., Branchini , E., et al. 2021, , 507, 1184

    2021

  94. [102]

    G., & Springel , V

    Viel , M., Haehnelt , M. G., & Springel , V. 2010, , 2010, 015

    2010

  95. [103]

    2023, , 2023, 010

    Vielzeuf , P., Calabrese , M., Carbone , C., Fabbian , G., & Baccigalupi , C. 2023, , 2023, 010

    2023

  96. [104]

    2018, , 861, 53

    Villaescusa-Navarro , F., Banerjee , A., Dalal , N., et al. 2018, , 861, 53

    2018

  97. [105]

    2020, , 250, 2

    Villaescusa-Navarro , F., Hahn , C., Massara , E., et al. 2020, , 250, 2

    2020

  98. [106]

    2014, , 2014, 011

    Villaescusa-Navarro , F., Marulli , F., Viel , M., et al. 2014, , 2014, 011

    2014

  99. [107]

    Zel'dovich , Y. B. 1970, , 5, 84

    1970

Pith tools

Reviewed May 16, 2026 · model on record in the stance chip above.