pith. machine review for the scientific record. sign in

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

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

Andrea Labate , Massimo Guidi , Michele Moresco , Alfonso Veropalumbo This is my paper

Pith reviewed 2026-05-16 20:57 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords massive neutrinosthree-point correlation functionlarge-scale structuresigma_8 degeneracyhalo clusteringN-body simulations
0
0 comments X

The pith

The three-point correlation function shows distinct massive-neutrino signals on elongated and right-angled triangles that differ from sigma_8 variations.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

Figures

Figures reproduced from arXiv: 2512.16992 by Alfonso Veropalumbo, Andrea Labate, Massimo Guidi, Michele Moresco.

Figure 1
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 o… view at source ↗
Figure 2
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. 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… view at source ↗
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 s12 ≤ s13 ≤ s23. In each panel, each pixel represents a given triangle shape, where the color shows the absolute value of the detectability… view at source ↗
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 (upper blue-scale colormap) and σ8 = 0.819 (lower blue-scale colormap). The top and bottom pairs of plots refer to the connected and reduced 3PCF… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

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

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.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 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.
axioms (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

pith-pipeline@v0.9.0 · 5692 in / 1366 out tokens · 25919 ms · 2026-05-16T20:57:42.604611+00:00 · methodology

discussion (0)

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

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.

Reference graph

Works this paper leans on

107 extracted references · 107 canonical work pages · 3 internal anchors

  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

  9. [9]

    2008, , 2008, 020

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

    work page 2008

  10. [10]

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

    work page 2010

  11. [11]

    2015, , 2015, 043

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

    work page 2015

  12. [12]

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

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

    work page 2014

  13. [13]

    Primordial Non-Gaussianity

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

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  14. [14]

    & Kaiser , N

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

    work page 1989

  15. [15]

    2011, European Physical Journal C, 71, 1554

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

    work page 2011

  16. [16]

    S., & White , S

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

    work page 1985

  17. [17]

    S., & Novaes , C

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

    work page 2020

  18. [18]

    S., Agertz , O., Berbel , A

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

    work page 2019

  19. [19]

    The DESI Experiment Part I: Science,Targeting, and Survey Design

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

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  20. [20]

    2018, , 733, 1

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

    work page 2018

  21. [21]

    2019, , 51, 341

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

    work page 2019

  22. [22]

    E., et al

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

    work page 2025

  23. [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

    work page arXiv 2025

  24. [24]

    A., et al

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

    work page 2025

  25. [25]

    2022, , 662, A112

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

    work page 2022

  26. [26]

    2020, , 497, 2699

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

    work page 2020

  27. [27]

    2024, arXiv e-prints, arXiv:2408.03036

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

    work page arXiv 2024

  28. [28]

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

    work page 1995

  29. [29]

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

    work page 1994

  30. [30]

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

    work page 1984

  31. [31]

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

    work page 1994

  32. [32]

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

    work page 1993

  33. [33]

    1998, , 81, 1562

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

    work page 1998

  34. [34]

    2009, , 399, 801

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

    work page 2009

  35. [35]

    M., & Croton , D

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

    work page 2005

  36. [36]

    N., S \'a nchez , A

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

    work page 2017

  37. [37]

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

    work page 1977

  38. [38]

    2023, , 2023, 066

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

    work page 2023

  39. [39]

    & Villaescusa-Navarro , F

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

    work page 2021

  40. [40]

    2020, , 2020, 040

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

    work page 2020

  41. [41]

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

    work page 1992

  42. [42]

    2007, , 464, 399

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

    work page 2007

  43. [43]

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

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

    work page 1995

  44. [44]

    J., & Tegmark , M

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

    work page 1998

  45. [45]

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

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

    work page 2020

  46. [46]

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

    work page 2004

  47. [47]

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

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

    work page 1995

  48. [48]

    1984, , 284, L9

    Kaiser , N. 1984, , 284, L9

    work page 1984

  49. [49]

    1987, , 227, 1

    Kaiser , N. 1987, , 227, 1

    work page 1987

  50. [50]

    & Slepian , Z

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

    work page arXiv 2025

  51. [51]

    & Slepian , Z

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

    work page 2025

  52. [52]

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

    work page 1993

  53. [53]

    Euclid Definition Study Report

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

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  54. [54]

    & Pastor , S

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

    work page 2006

  55. [55]

    Mainieri, R

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

    work page arXiv 2024

  56. [56]

    A., Blake , C., Poole , G

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

    work page 2013

  57. [57]

    2011, , 418, 346

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

    work page 2011

  58. [58]

    K., Connolly , A

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

    work page 2011

  59. [59]

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

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

    work page 2019

  60. [60]

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

    work page 1996

  61. [61]

    2014, , 443, 2874

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

    work page 2014

  62. [62]

    2021, , 919, 144

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

    work page 2021

  63. [63]

    2023, , 2023, 025

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

    work page 2023

  64. [64]

    2025, , 2025, 045

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

    work page 2025

  65. [65]

    2024, , 110, 030001

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

    work page 2024

  66. [66]

    2021, , 2021, 038

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

    work page 2021

  67. [67]

    2022, , 2022, 066

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

    work page 2022

  68. [68]

    2021, , 2021, 009

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

    work page 2021

  69. [69]

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

    work page 1973

  70. [70]

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

    work page 1980

  71. [71]

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

    work page 1975

  72. [72]

    2015, , 2015, 001

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

    work page 2015

  73. [73]

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

    work page 2021

  74. [74]

    2020, , 641, A6

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

    work page 2020

  75. [75]

    2025, , 2025, 075

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

    work page 2025

  76. [76]

    2018, , 2018, 003

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

    work page 2018

  77. [77]

    G., Montesano , F., Kazin , E

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

    work page 2014

  78. [78]

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

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

    work page 2017

  79. [79]

    2004, , 70, 083007

    Scoccimarro , R. 2004, , 70, 083007

    work page 2004

  80. [80]

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

    work page 1999

Showing first 80 references.