pith. machine review for the scientific record. sign in

arxiv: 2604.15440 · v1 · submitted 2026-04-16 · 🌌 astro-ph.CO

Recognition: unknown

Probing the large-scale structure with 21cm-galaxy cross-bispectrum: Estimates from simulations and forecasts for upcoming cosmological surveys

Leon Noble , Suman Majumdar , Matteo Viel , Fabio Fontanot , Gabriella De Lucia , Abinash Kumar Shaw , Marta Spinelli , Mohd Kamran
show 2 more authors
Lizhi Xie Michaela Hirschmann
Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:25 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords 21cm signalcross-bispectrumlarge-scale structureSKA-MidEuclidcosmological forecastsnon-Gaussianitypost-reionization
0
0 comments X

The pith

The 21cm-galaxy cross-bispectrum provides enhanced detectability over the auto-bispectrum for probing large-scale structure with future surveys.

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

The paper investigates the use of the cross-bispectrum between the 21cm signal from neutral hydrogen and galaxy distributions to extract non-Gaussian information from the post-reionization universe. It employs theoretical models of galaxy evolution on large volumes to compute this statistic for different triangle configurations and assesses its observability with SKA-Mid and Euclid-like surveys. The analysis shows that cross-correlating these fields improves the signal-to-noise ratio in the presence of instrumental noise compared to using the 21cm signal alone. This approach matters for cosmologists because it offers a way to access higher-order statistics that reveal more about structure formation while reducing the impact of systematics in radio observations.

Core claim

The 21cm-galaxy cross-bispectrum shows enhanced detectability compared to the 21cm auto-bispectrum for all unique triangles in the presence of instrumental noise for observations in interferometric mode. Forecasts indicate a 10σ detection for squeezed-limit triangles and a 100σ detection for all shapes combined on scales 0.2 Mpc^{-1} ≤ k1 ≤ 0.9 Mpc^{-1} with 100 hours of SKA-Mid observations per pointing. Detectability on large scales with single-dish mode is limited by cosmic variance. The work represents an initial step toward an end-to-end analysis pipeline for future observations.

What carries the argument

The 21cm-galaxy cross-bispectrum, which measures three-point correlations between the redshifted 21 cm signal and galaxy positions using predictions from galaxy evolution models.

If this is right

  • The cross-bispectrum allows higher significance detections than the auto-bispectrum when instrumental noise is accounted for in interferometric observations.
  • High detection significances of 10σ for squeezed triangles and 100σ for combined shapes are expected on intermediate scales with SKA-Mid.
  • Single-dish observations are restricted by cosmic variance on very large scales.
  • This sets the foundation for comprehensive analysis methods for cross-bispectrum data from upcoming surveys.

Where Pith is reading between the lines

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

  • Such cross-correlations might help break degeneracies in cosmological parameters by providing complementary information to power spectra.
  • Future work could test these forecasts against mock observations from current telescopes to improve model accuracy.
  • The technique may generalize to cross-bispectra with other tracers like weak lensing for broader applications in cosmology.

Load-bearing premise

The theoretical galaxy evolution models must accurately represent the clustering and bias of the neutral hydrogen and galaxy fields on cosmological scales, otherwise the predicted detection strengths will not apply to real observations.

What would settle it

Observations from SKA-Mid in interferometric mode that yield signal-to-noise ratios much lower than 10σ for squeezed triangles or 100σ overall on scales from 0.2 to 0.9 per Mpc would indicate the forecasts are not accurate.

Figures

Figures reproduced from arXiv: 2604.15440 by Abinash Kumar Shaw, Fabio Fontanot, Gabriella De Lucia, Leon Noble, Lizhi Xie, Marta Spinelli, Matteo Viel, Michaela Hirschmann, Mohd Kamran, Suman Majumdar.

Figure 1
Figure 1. Figure 1: The unique shapes of the k−triangles distributed over the k2/k1 − cos θ plane. Unique shapes are confined to the region where k2 k1 cos θ ≥ 0.5, shown in grey. We divided the entire k2/k1 − cos θ plane with grid size ∆ k2/k1 = 0.05 and ∆ cos θ = 0.05, represented by the orange colour grids. Fourier space, we followed the bispectrum parameter￾ization introduced in S. Bharadwaj et al. (2020) and S. Majumdar … view at source ↗
Figure 2
Figure 2. Figure 2: The HI and galaxy auto and cross-bispectrum as a function of k1 at z = 0.99 estimated from GAEA simulations ( see Section 2.1) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of RSD on the HI-galaxy auto and cross-bispectrum for squeezed-limit triangles at z = 0.99. The solid lines show auto and cross-bispectrum in the real space, while the dashed line corresponds to the redshift space bispectrum. The left panel presents the HI and galaxy auto-bispectrum. The middle and right panels show HI-galaxy cross-bispectrum with two HI fields (∆3 HI,HI,Gal, ∆3 HI,Gal,HI and ∆3 Gal… view at source ↗
Figure 4
Figure 4. Figure 4: Joint constraints (1σ and 2σ confidence re￾gions) and marginalized posterior distributions for the lin￾ear (b1,HI and b1,Gal) and quadratic (b2,HI and b2,Gal) bias parameters of the HI and galaxy fields. The results correspond to the various dataset combina￾tions listed in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: ID marginalized constraints on HI linear (b1,HI) and quadratic (b2,HI) bias parameters obtained from different dataset listed in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparing the perturbation theory predictions of the HI-galaxy cross-bispectrum with simulations at z = 0.99. The solid line represents the perturbation theory predictions, while the points correspond to bispectrum estimates from simulations. Left, middle, and right panels present comparisons for squeezed-limit, equilateral, and stretched k − triangles as a function of k1. Different colors correspond to di… view at source ↗
Figure 7
Figure 7. Figure 7: The SNR for detecting 21-cm auto bispectrum and 21-cm-galaxy cross-bispectrum for all unique triangles at z = 0.99 with SKA-Mid (interferometric mode) and Euclid-like galaxy survey. Grey, blue, orange and green colours indicate SNR < 2σ, 2σ ≤ SNR < 5σ, 5 ≤ SNR < 10σ and SNR > 10σ respectively. Figure 7a and 7b correspond to SNR estimates for 100 and 200 hours of SKA-Mid observations per pointing (tp), resp… view at source ↗
Figure 8
Figure 8. Figure 8: The SNR for detecting the 21cm auto-bispectrum (orange) and 21cm-galaxy cross-bispectrum (yellow) as function of k1 for different observation time per pointing (tp). The left panel presents results for squeezed-limit k − triangles and right panel corresponds to all shapes (k2/k1, cos θ) combined. without the inclusion of the telescopic beam (∆3 no beam) and with the telescopic beam (∆3 Gaussian beam) in th… view at source ↗
Figure 9
Figure 9. Figure 9: Left: Impact of instrumental beam on the 21cm-galaxy cross-bispectrum for squeezed-limit k − triangles. The solid lines represent the bispectrum estimates without any telescopic beam, and dashed lines correspond to bispectrum estimates from maps where the impact of a Gaussian beam is included. Right: The SNR for detecting 21-cm auto bispectrum for squeezed-limit k − triangles as function of k1 at z = 0.99 … view at source ↗
read the original abstract

The redshifted 21cm signal from the post-reionization epoch is highly non-Gaussian; thus, higher-order statistics, such as the bispectrum, are required to extract this non-Gaussian information. However, high-signal-to-noise ratio detection of the 21cm auto-bispectrum will be hindered by the presence of residual systematics. Cross-correlating the 21cm signal with galaxies offers a promising path to suppress this uncertainty from residual systematics and potentially increase the signal-to-noise ratio. We present a comprehensive analysis of the HI-galaxy cross-bispectrum using the predictions of theoretical galaxy evolution models defined on large cosmological volumes. Our analysis includes the cross-bispectrum for different triangle sizes and shapes, as well as for different combinations of the HI and galaxy fields. We forecast the detectability of the 21cm-galaxy cross-bispectrum at redshift $z\approx1$ with Euclid-like galaxy survey and SKA-Mid observations in both interferometric and single-dish modes of survey. We find that the 21cm-galaxy cross-bispectrum shows enhanced detectability compared to the 21cm auto-bispectrum for all unique triangles in the presence of instrumental noise for observations in interferometric mode. We forecast a 10$\sigma$ detection of cross-bispectrum for squeezed-limit triangles and a 100$\sigma$ detection for all shapes combined for scales $0.2~\text{Mpc}^{-1}\leq k_1 \leq 0.9~\text{Mpc}^{-1}$ with 100 hours of SKA-Mid observations per pointing. However, the detectability of the cross-bispectrum for large scales ($k_1 < 0.1~\text{Mpc}^{-1}$), which is accessible with the single-dish mode of survey, is limited by cosmic variance. Our analysis presents a first step towards an end-to-end analysis pipeline for the future observations of the 21cm-galaxy cross-bispectrum.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the HI-galaxy cross-bispectrum at z≈1 using theoretical galaxy evolution models on large cosmological volumes. It compares the cross-bispectrum to the 21cm auto-bispectrum for various triangle shapes and scales, and forecasts detectability with Euclid-like galaxy surveys combined with SKA-Mid observations in interferometric and single-dish modes. The central claims are enhanced detectability of the cross-bispectrum in the presence of instrumental noise and specific high signal-to-noise ratios (10σ for squeezed-limit triangles, 100σ for all shapes combined) for 0.2 Mpc^{-1} ≤ k1 ≤ 0.9 Mpc^{-1} with 100 hours per pointing, while noting cosmic-variance limits on larger scales.

Significance. If the underlying galaxy evolution models accurately capture the HI and galaxy clustering and higher-order correlations, the forecasts would usefully demonstrate the advantages of cross-bispectrum measurements for mitigating systematics and extracting non-Gaussian information from upcoming surveys. The use of large-volume simulations to include cosmic variance is a methodological strength that supports the scale-dependent claims.

major comments (2)
  1. [abstract and results] The quantitative forecasts for 10σ and 100σ detections (abstract and results section) are derived entirely from cross-bispectrum amplitudes generated by the external theoretical galaxy evolution models. No calibration against observational HI or galaxy clustering data at z≈1, nor comparison to independent simulations with known bias properties, is presented. Because the S/N ratios scale directly with these amplitudes, any systematic offset in modeled bias or three-point correlations would rescale the claimed significances; this is load-bearing for the central detectability claims.
  2. [methods] No sensitivity analysis or error budget is provided for variations in the galaxy evolution model parameters or assumptions (e.g., astrophysical prescriptions for HI content). Given that the paper's primary output consists of specific numerical forecasts rather than qualitative trends, the absence of such tests leaves the robustness of the 10σ/100σ numbers unassessed.
minor comments (2)
  1. [methods] The definition of 'unique triangles' and the precise binning in k-space for the combined-shape forecast could be clarified with an explicit equation or diagram in the methods section.
  2. [figures] Figure captions for the S/N plots should include the exact noise model parameters, survey area, and integration time assumptions used to generate the quoted values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which help clarify the robustness of our forecasts. We respond to each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [abstract and results] The quantitative forecasts for 10σ and 100σ detections (abstract and results section) are derived entirely from cross-bispectrum amplitudes generated by the external theoretical galaxy evolution models. No calibration against observational HI or galaxy clustering data at z≈1, nor comparison to independent simulations with known bias properties, is presented. Because the S/N ratios scale directly with these amplitudes, any systematic offset in modeled bias or three-point correlations would rescale the claimed significances; this is load-bearing for the central detectability claims.

    Authors: We agree that the absolute S/N values depend on the amplitudes predicted by the galaxy evolution models. These models were previously calibrated to match HI and galaxy clustering observations at z≈1 (as referenced in the Methods section of our manuscript), but we acknowledge that an explicit summary of this calibration and comparisons to independent simulations were not included here. In the revised manuscript we will add a dedicated paragraph in the Methods section that summarizes the model calibration procedures from the original publications, cites relevant validation against observational data and other simulations, and discusses potential systematic uncertainties in the three-point correlations. We emphasize that the key comparative result—the enhanced detectability of the cross-bispectrum relative to the auto-bispectrum in the presence of noise—remains robust under overall amplitude rescalings, since both statistics are affected similarly by bias offsets. revision: yes

  2. Referee: [methods] No sensitivity analysis or error budget is provided for variations in the galaxy evolution model parameters or assumptions (e.g., astrophysical prescriptions for HI content). Given that the paper's primary output consists of specific numerical forecasts rather than qualitative trends, the absence of such tests leaves the robustness of the 10σ/100σ numbers unassessed.

    Authors: We concur that a sensitivity analysis is important for assessing the robustness of the specific numerical forecasts. In the revised manuscript we will add a new subsection (or appendix) that performs a limited sensitivity analysis. We will vary key model parameters, such as the HI-halo mass relation and star-formation efficiency, within their observational uncertainties, recompute the cross-bispectrum amplitudes and S/N ratios for representative triangle configurations, and present the resulting range in the quoted detection significances. This will provide a basic error budget and quantify how the 10σ and 100σ claims respond to plausible variations in the astrophysical prescriptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; forecasts use independent forward modeling from external galaxy evolution simulations.

full rationale

The paper computes cross-bispectrum amplitudes directly from the outputs of theoretical galaxy evolution models run on large cosmological volumes, then derives S/N forecasts by comparing those amplitudes to instrumental noise and cosmic variance for specified k-ranges and triangle shapes. This is standard forward modeling with no parameter fitting to the target observables, no self-definitional reduction of predictions to inputs, and no load-bearing self-citations or uniqueness theorems invoked to justify the central claims. The derivation chain remains independent of the forecasted detectability numbers themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the analysis rests on existing galaxy evolution models whose internal assumptions are not detailed here.

pith-pipeline@v0.9.0 · 5705 in / 1172 out tokens · 38723 ms · 2026-05-10T09:25:40.998394+00:00 · methodology

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

104 extracted references · 103 canonical work pages · 1 internal anchor

  1. [1]

    Abdalla, E., Ferreira, E. G. M., Landim, R. G., et al. 2022, A&A, 664, A14, doi: 10.1051/0004-6361/202140883

    work page doi:10.1051/0004-6361/202140883 2022

  2. [2]

    G., & Santos, M

    Alonso, D., Bull, P., Ferreira, P. G., & Santos, M. G. 2015, MNRAS, 447, 400, doi: 10.1093/mnras/stu2474

    work page doi:10.1093/mnras/stu2474 2015

  3. [3]

    2023, ApJ, 947, 16, doi: 10.3847/1538-4357/acb13f

    Amiri, M., Bandura, K., Chen, T., et al. 2023, ApJ, 947, 16, doi: 10.3847/1538-4357/acb13f

    work page doi:10.3847/1538-4357/acb13f 2023

  4. [4]

    2024, ApJ, 963, 23, doi: 10.3847/1538-4357/ad0f1d

    Amiri, M., Bandura, K., Chakraborty, A., et al. 2024, ApJ, 963, 23, doi: 10.3847/1538-4357/ad0f1d

    work page doi:10.3847/1538-4357/ad0f1d 2024

  5. [5]

    J., Luciw, N

    Anderson, C. J., Luciw, N. J., Li, Y. C., et al. 2018, MNRAS, 476, 3382, doi: 10.1093/mnras/sty346

    work page doi:10.1093/mnras/sty346 2018

  6. [6]

    S., & Viel, M

    Autieri, G., Berti, M., Spinelli, M., Haridasu, B. S., & Viel, M. 2026, JCAP, 2026, 050, doi: 10.1088/1475-7516/2026/01/050

    work page doi:10.1088/1475-7516/2026/01/050 2026

  7. [7]

    2018, Monthly Notices of the Royal Astronomical Society, 477, 1984, doi: 10.1093/mnras/sty714

    Bag, S., Mondal, R., Sarkar, P., Bharadwaj, S., & Sahni, V. 2018, Monthly Notices of the Royal Astronomical Society, 477, 1984, doi: 10.1093/mnras/sty714

    work page doi:10.1093/mnras/sty714 2018

  8. [8]

    , keywords =

    Barberi-Squarotti, M., et al. 2025, Mon. Not. Roy. Astron. Soc., 537, 3632, doi: 10.1093/mnras/staf195

    work page doi:10.1093/mnras/staf195 2025

  9. [9]

    and Komatsu, E

    Bartolo, N., Komatsu, E., Matarrese, S., & Riotto, A. 2004, PhR, 402, 103, doi: 10.1016/j.physrep.2004.08.022

    work page doi:10.1016/j.physrep.2004.08.022 2004

  10. [10]

    2004, MNRAS, 351, 1379, doi: 10.1111/j.1365-2966.2004.07876.x

    Battye, R. A., Davies, R. D., & Weller, J. 2004, MNRAS, 355, 1339, doi: 10.1111/j.1365-2966.2004.08416.x

    work page doi:10.1111/j.1365-2966.2004.08416.x 2004

  11. [11]

    , keywords =

    Baugh, C. M., Gonzalez-Perez, V., Lagos, C. D. P., et al. 2019, MNRAS, 483, 4922, doi: 10.1093/mnras/sty3427

    work page doi:10.1093/mnras/sty3427 2019

  12. [12]

    Bernal, J. L. 2024, Phys. Rev. D, 109, 043517, doi: 10.1103/PhysRevD.109.043517

    work page doi:10.1103/physrevd.109.043517 2024

  13. [13]

    and Colombi, S

    Scoccimarro, R. 2002, PhR, 367, 1, doi: 10.1016/S0370-1573(02)00135-7

    work page doi:10.1016/s0370-1573(02)00135-7 2002

  14. [14]

    2024, MNRAS, 529, 4803, doi: 10.1093/mnras/stae755

    Berti, M., Spinelli, M., & Viel, M. 2024, MNRAS, 529, 4803, doi: 10.1093/mnras/stae755

    work page doi:10.1093/mnras/stae755 2024

  15. [15]

    2020, MNRAS, 493, 594, doi: 10.1093/mnras/staa279

    Bharadwaj, S., Mazumdar, A., & Sarkar, D. 2020, MNRAS, 493, 594, doi: 10.1093/mnras/staa279

    work page doi:10.1093/mnras/staa279 2020

  16. [16]

    B., & Sethi, S

    Bharadwaj, S., Nath, B. B., & Sethi, S. K. 2001, Journal of Astrophysics and Astronomy, 22, 21, doi: 10.1007/BF02933588

    work page doi:10.1007/bf02933588 2001

  17. [17]

    Bharadwaj, S., & Sethi, S. K. 2001, Journal of Astrophysics and Astronomy, 22, 293, doi: 10.1007/BF02702273

    work page doi:10.1007/bf02702273 2001

  18. [18]

    Bharadwaj, S., & Srikant, P. S. 2004, Journal of Astrophysics and Astronomy, 25, 67, doi: 10.1007/BF02702289

    work page doi:10.1007/bf02702289 2004

  19. [19]

    2020, Astron

    Blanchard, A., et al. 2020, Astron. Astrophys., 642, A191, doi: 10.1051/0004-6361/202038071

    work page doi:10.1051/0004-6361/202038071 2020

  20. [20]

    and Bonaldi, A

    Braun, R., Bonaldi, A., Bourke, T., Keane, E., & Wagg, J. 2019, arXiv e-prints, arXiv:1912.12699, doi: 10.48550/arXiv.1912.12699 19

    work page doi:10.48550/arxiv.1912.12699 2019

  21. [21]

    2017, Mon

    Kamionkowski, M. 2017, Mon. Not. Roy. Astron. Soc., 467, 2996, doi: 10.1093/mnras/stx203

    work page doi:10.1093/mnras/stx203 2017

  22. [22]

    2015 , pages =

    Bull, P., Ferreira, P. G., Patel, P., & Santos, M. G. 2015, ApJ, 803, 21, doi: 10.1088/0004-637X/803/1/21

    work page doi:10.1088/0004-637x/803/1/21 2015

  23. [23]

    P., Irfan, M

    Carucci, I. P., Irfan, M. O., & Bobin, J. 2020, MNRAS, 499, 304, doi: 10.1093/mnras/staa2854

    work page doi:10.1093/mnras/staa2854 2020

  24. [24]

    P., Bernal, J

    Carucci, I. P., Bernal, J. L., Cunnington, S., et al. 2025, A&A, 703, A222, doi: 10.1051/0004-6361/202453461

    work page doi:10.1051/0004-6361/202453461 2025

  25. [25]

    2025, MNRAS, 538, 2204, doi: 10.1093/mnras/staf433

    Villaescusa-Navarro, F. 2025, MNRAS, 538, 2204, doi: 10.1093/mnras/staf433

    work page doi:10.1093/mnras/staf433 2025

  26. [26]

    Chang, T.-C., Pen, U.-L., Bandura, K., & Peterson, J. B. 2010, arXiv e-prints, arXiv:1007.3709, doi: 10.48550/arXiv.1007.3709

    work page Pith review doi:10.48550/arxiv.1007.3709 2010

  27. [27]

    Physical Review Letters , author =

    Chang, T.-C., Pen, U.-L., Peterson, J. B., & McDonald, P. 2008, PhRvL, 100, 091303, doi: 10.1103/PhysRevLett.100.091303

    work page doi:10.1103/physrevlett.100.091303 2008

  28. [28]

    Pontzen and F

    Chapman, E., Abdalla, F. B., Harker, G., et al. 2012, MNRAS, 423, 2518, doi: 10.1111/j.1365-2966.2012.21065.x

    work page doi:10.1111/j.1365-2966.2012.21065.x 2012

  29. [29]

    B., Bobin, J., et al

    Chapman, E., Abdalla, F. B., Bobin, J., et al. 2013, MNRAS, 429, 165, doi: 10.1093/mnras/sts333

    work page doi:10.1093/mnras/sts333 2013

  30. [30]

    2025, arXiv e-prints, arXiv:2508.19126, doi: 10.48550/arXiv.2508.19126 CHIME Collaboration, Amiri, M., Bandura, K., et al

    Chhabra, M., & Bharadwaj, S. 2025, arXiv e-prints, arXiv:2508.19126, doi: 10.48550/arXiv.2508.19126 CHIME Collaboration, Amiri, M., Bandura, K., et al. 2025, arXiv e-prints, arXiv:2511.19620, doi: 10.48550/arXiv.2511.19620 CHIME Collaboration, Chakraborty, A., Dobbs, M., et al. 2026, arXiv e-prints, arXiv:2601.03240, doi: 10.48550/arXiv.2601.03240

    work page doi:10.48550/arxiv.2508.19126 2025

  31. [31]

    2022, Mon

    Cunnington, S. 2022, Mon. Not. Roy. Astron. Soc., 512, 2408, doi: 10.1093/mnras/stac576

    work page doi:10.1093/mnras/stac576 2022

  32. [32]

    2021, MNRAS, 507, 1623, doi: 10.1093/mnras/stab2200

    Cunnington, S., Watkinson, C., & Pourtsidou, A. 2021, MNRAS, 507, 1623, doi: 10.1093/mnras/stab2200

    work page doi:10.1093/mnras/stab2200 2021

  33. [33]

    2019, Mon

    Cunnington, S., Wolz, L., Pourtsidou, A., & Bacon, D. 2019, Mon. Not. Roy. Astron. Soc., 488, 5452, doi: 10.1093/mnras/stz1916

    work page doi:10.1093/mnras/stz1916 2019

  34. [34]

    , keywords =

    Cunnington, S., Li, Y., Santos, M. G., et al. 2023, MNRAS, 518, 6262, doi: 10.1093/mnras/stac3060 de Jong, R. S., Agertz, O., Berbel, A. A., et al. 2019, The Messenger, 175, 3, doi: 10.18727/0722-6691/5117 De Lucia, G., Fontanot, F., Xie, L., & Hirschmann, M. 2024, A&A, 687, A68, doi: 10.1051/0004-6361/202349045 De Lucia, G., Tornatore, L., Frenk, C. S., ...

    work page doi:10.1093/mnras/stac3060 2023

  35. [35]

    Physics Reports , author =

    Desjacques, V., Jeong, D., & Schmidt, F. 2018, PhR, 733, 1, doi: 10.1016/j.physrep.2017.12.002

    work page doi:10.1016/j.physrep.2017.12.002 2018

  36. [36]

    M., Majumdar, S., Shekhar Murmu, C., et al

    Dosibhatla, M. M., Majumdar, S., Shekhar Murmu, C., et al. 2025, arXiv e-prints, arXiv:2508.09112, doi: 10.48550/arXiv.2508.09112

    work page doi:10.48550/arxiv.2508.09112 2025

  37. [37]

    2020, JCAP, 2020, 003, doi: 10.1088/1475-7516/2020/12/003 Euclid Collaboration, Pezzotta, A., Moretti, C., et al

    Montanari, F. 2020, JCAP, 2020, 003, doi: 10.1088/1475-7516/2020/12/003 Euclid Collaboration, Pezzotta, A., Moretti, C., et al. 2024, A&A, 687, A216, doi: 10.1051/0004-6361/202348939 Euclid Collaboration, Castander, F. J., Fosalba, P., et al. 2025, A&A, 697, A5, doi: 10.1051/0004-6361/202450853

    work page doi:10.1088/1475-7516/2020/12/003 2020

  38. [38]

    2025, Astron

    Fontanot, F., De Lucia, G., Xie, L., et al. 2025, Astron. Astrophys., 699, A108, doi: 10.1051/0004-6361/202452029

    work page doi:10.1051/0004-6361/202452029 2025

  39. [39]

    2020, MNRAS, 496, 3943, doi: 10.1093/mnras/staa1716

    Fontanot, F., De Lucia, G., Hirschmann, M., et al. 2020, MNRAS, 496, 3943, doi: 10.1093/mnras/staa1716

    work page doi:10.1093/mnras/staa1716 2020

  40. [40]

    and Lang, Dustin and Goodman, Jonathan , title =

    Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067

    work page doi:10.1086/670067 2013

  41. [41]

    Fry, J. N. 1984, ApJ, 279, 499, doi: 10.1086/161913

    work page doi:10.1086/161913 1984

  42. [42]

    R., Oh, S

    Furlanetto, S., Oh, S. P., & Briggs, F. 2006, Phys. Rept., 433, 181, doi: 10.1016/j.physrep.2006.08.002 Gil-Mar´ ın, H., Wagner, C., Fragkoudi, F., Jimenez, R., &

    work page doi:10.1016/j.physrep.2006.08.002 2006

  43. [43]

    2012, JCAP, 2012, 047, doi: 10.1088/1475-7516/2012/02/047

    Verde, L. 2012, JCAP, 2012, 047, doi: 10.1088/1475-7516/2012/02/047

    work page doi:10.1088/1475-7516/2012/02/047 2012

  44. [44]

    P., Alonso, D., & Moodley, K

    Guandalin, C., Carucci, I. P., Alonso, D., & Moodley, K. 2022, MNRAS, 516, 3029, doi: 10.1093/mnras/stac2343 Guha Sarkar, T., & Hazra, D. K. 2013, JCAP, 2013, 002, doi: 10.1088/1475-7516/2013/04/002

    work page doi:10.1093/mnras/stac2343 2022

  45. [45]

    S., et al

    Gupta, Y., Ajithkumar, B., Kale, H. S., et al. 2017, Current Science, 113, 707, doi: 10.18520/cs/v113/i04/707-714

    work page doi:10.18520/cs/v113/i04/707-714 2017

  46. [46]

    2016 , journal =

    Hirschmann, M., De Lucia, G., & Fontanot, F. 2016, MNRAS, 461, 1760, doi: 10.1093/mnras/stw1318

    work page doi:10.1093/mnras/stw1318 2016

  47. [47]

    M., et al

    Jolicoeur, S., Maartens, R., De Weerd, E. M., et al. 2021, JCAP, 2021, 039, doi: 10.1088/1475-7516/2021/06/039

    work page doi:10.1088/1475-7516/2021/06/039 2021

  48. [48]

    2025, JCAP, 08, 047, doi: 10.1088/1475-7516/2025/08/047

    Joshi, B., & Kothari, R. 2025, JCAP, 08, 047, doi: 10.1088/1475-7516/2025/08/047

    work page doi:10.1088/1475-7516/2025/08/047 2025

  49. [49]

    , keywords =

    Kaiser, N. 1987, MNRAS, 227, 1, doi: 10.1093/mnras/227.1.1

    work page doi:10.1093/mnras/227.1.1 1987

  50. [50]

    2025, JCAP, 2025, 054, doi: 10.1088/1475-7516/2025/07/054

    Kamran, M., Sahl´ en, M., Sarkar, D., & Majumdar, S. 2025, JCAP, 2025, 054, doi: 10.1088/1475-7516/2025/07/054

    work page doi:10.1088/1475-7516/2025/07/054 2025

  51. [51]

    2021, Phys

    Karagiannis, D., Fonseca, J., Maartens, R., & Camera, S. 2021, Phys. Dark Univ., 32, 100821, doi: 10.1016/j.dark.2021.100821

    work page doi:10.1016/j.dark.2021.100821 2021

  52. [52]

    2024, JCAP, 03, 034, doi: 10.1088/1475-7516/2024/03/034

    Clarkson, C. 2024, JCAP, 03, 034, doi: 10.1088/1475-7516/2024/03/034

    work page doi:10.1088/1475-7516/2024/03/034 2024

  53. [53]

    Karagiannis, D., Maartens, R., & Randrianjanahary, L. F. 2022, JCAP, 11, 003, doi: 10.1088/1475-7516/2022/11/003

    work page doi:10.1088/1475-7516/2022/11/003 2022

  54. [54]

    2020, JCAP, 11, 052, doi: 10.1088/1475-7516/2020/11/052 20

    Karagiannis, D., Slosar, A., & Liguori, M. 2020, JCAP, 11, 052, doi: 10.1088/1475-7516/2020/11/052 20

    work page doi:10.1088/1475-7516/2020/11/052 2020

  55. [55]

    2025, Eur

    Kopana, M., Jolicoeur, S., & Maartens, R. 2025, Eur. Phys. J. C, 85, 538, doi: 10.1140/epjc/s10052-025-14241-7

    work page doi:10.1140/epjc/s10052-025-14241-7 2025

  56. [56]

    E., et al

    Levi, M., Allen, L. E., Raichoor, A., et al. 2019, in Bulletin of the American Astronomical Society, Vol. 51, 57, doi: 10.48550/arXiv.1907.10688

    work page doi:10.48550/arxiv.1907.10688 2019

  57. [57]

    2021, Research in Astronomy and Astrophysics, 21, 030, doi: 10.1088/1674-4527/21/2/30

    Li, L.-C., Staveley-Smith, L., & Rhee, J. 2021, Research in Astronomy and Astrophysics, 21, 030, doi: 10.1088/1674-4527/21/2/30

    work page doi:10.1088/1674-4527/21/2/30 2021

  58. [58]

    R., et al

    Majumdar, S., Kamran, M., Pritchard, J. R., et al. 2020, MNRAS, 499, 5090, doi: 10.1093/mnras/staa3168

    work page doi:10.1093/mnras/staa3168 2020

  59. [59]

    2018, doi: 10.2172/1471560

    Mandelbaum, R., et al. 2018, doi: 10.2172/1471560

    work page doi:10.2172/1471560 2018

  60. [60]

    2023, The Astrophysical Journal, 951, 70, doi: 10.3847/1538-4357/acd44d

    Massara, E., Villaescusa-Navarro, F., Hahn, C., et al. 2023, The Astrophysical Journal, 951, 70, doi: 10.3847/1538-4357/acd44d

    work page doi:10.3847/1538-4357/acd44d 2023

  61. [61]

    W., Switzer, E

    Masui, K. W., Switzer, E. R., Banavar, N., et al. 2013, ApJL, 763, L20, doi: 10.1088/2041-8205/763/1/L20

    work page doi:10.1088/2041-8205/763/1/l20 2013

  62. [62]

    Matarrese, S., Verde, L., & Heavens, A. F. 1997, MNRAS, 290, 651, doi: 10.1093/mnras/290.4.651

    work page doi:10.1093/mnras/290.4.651 1997

  63. [63]

    D., Spinelli, M., Santos, M

    Matshawule, S. D., Spinelli, M., Santos, M. G., & Ngobese, S. 2021, Mon. Not. Roy. Astron. Soc., 506, 5075, doi: 10.1093/mnras/stab1688 MeerKLASS Collaboration, et al. 2025, Mon. Not. Roy. Astron. Soc., 537, 3632, doi: 10.1093/mnras/staf195

    work page doi:10.1093/mnras/stab1688 2021

  64. [64]

    2026, MNRAS, 545, staf2071, doi: 10.1093/mnras/staf2071

    Mishra, S., Trotta, R., & Viel, M. 2026, MNRAS, 545, staf2071, doi: 10.1093/mnras/staf2071

    work page doi:10.1093/mnras/staf2071 2026

  65. [65]

    2015, MNRAS, 449, L41, doi: 10.1093/mnrasl/slv015

    Acharyya, A. 2015, MNRAS, 449, L41, doi: 10.1093/mnrasl/slv015

    work page doi:10.1093/mnrasl/slv015 2015

  66. [66]

    2023, arXiv e-prints, arXiv:2311.05904, doi: 10.48550/arXiv.2311.05904

    Moodley, K., Naidoo, W., Prince, H., & Penin, A. 2023, arXiv e-prints, arXiv:2311.05904, doi: 10.48550/arXiv.2311.05904

    work page doi:10.48550/arxiv.2311.05904 2023

  67. [67]

    2011, International Journal of Modern Physics D, 20, 989, doi: 10.1142/S0218271811019335

    Nan, R., Li, D., Jin, C., et al. 2011, International Journal of Modern Physics D, 20, 989, doi: 10.1142/S0218271811019335

    work page doi:10.1142/s0218271811019335 2011

  68. [68]

    F., Frenk, C

    Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563, doi: 10.1086/177173

    work page doi:10.1086/177173 1996

  69. [69]

    B., Bandura K., Bucher M

    Newburgh, L. B., Bandura, K., Bucher, M. A., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9906, Ground-based and Airborne Telescopes VI, ed. H. J. Hall, R. Gilmozzi, & H. K. Marshall, 99065X, doi: 10.1117/12.2234286

    work page doi:10.1117/12.2234286 2016

  70. [70]

    C., Remazeilles, M., & Dickinson, C

    Olivari, L. C., Remazeilles, M., & Dickinson, C. 2016, MNRAS, 456, 2749, doi: 10.1093/mnras/stv2884

    work page doi:10.1093/mnras/stv2884 2016

  71. [71]

    Pal and D

    Pal, S., & Sarkar, D. 2026, https://arxiv.org/abs/2601.18862

    work page arXiv 2026

  72. [72]

    2022, Journal of Cosmology and Astroparticle Physics, 2022, 027, doi: 10.1088/1475-7516/2022/11/027

    Pathak, A., Bag, S., Dasgupta, S., et al. 2022, Journal of Cosmology and Astroparticle Physics, 2022, 027, doi: 10.1088/1475-7516/2022/11/027

    work page doi:10.1088/1475-7516/2022/11/027 2022

  73. [73]

    G., Chen, Z., et al

    Paul, S., Santos, M. G., Chen, Z., & Wolz, L. 2023, arXiv e-prints, arXiv:2301.11943, doi: 10.48550/arXiv.2301.11943

    work page doi:10.48550/arxiv.2301.11943 2023

  74. [74]

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

    1980

  75. [75]

    F., Costa, A

    Pinheiro, R. F., Costa, A. A., & Sang, Y. 2026, https://arxiv.org/abs/2603.20160

    work page arXiv 2026

  76. [76]

    M., Geach, J

    Pozzetti, L., Hirata, C. M., Geach, J. E., et al. 2016, A&A, 590, A3, doi: 10.1051/0004-6361/201527081

    work page doi:10.1051/0004-6361/201527081 2016

  77. [77]

    F., Karagiannis, D., & Maartens, R

    Randrianjanahary, L. F., Karagiannis, D., & Maartens, R. 2024, Physics of the Dark Universe, 45, 101530, doi: 10.1016/j.dark.2024.101530 Saiyad Ali, S., Bharadwaj, S., & Pandey, S. K. 2006, MNRAS, 366, 213, doi: 10.1111/j.1365-2966.2005.09847.x

    work page doi:10.1016/j.dark.2024.101530 2024

  78. [78]

    2016, in MeerKAT Science: On the Pathway to the SKA, 32, doi: 10.22323/1.277.0032

    Santos, M., Bull, P., Camera, S., et al. 2016, in MeerKAT Science: On the Pathway to the SKA, 32, doi: 10.22323/1.277.0032

    work page doi:10.22323/1.277.0032 2016

  79. [79]

    G., et al

    Santos, M. G., et al. 2015, PoS, AASKA14, 019, doi: 10.22323/1.215.0019

    work page doi:10.22323/1.215.0019 2015

  80. [80]

    2019, MNRAS, 490, 2880, doi: 10.1093/mnras/stz2799

    Sarkar, D., Majumdar, S., & Bharadwaj, S. 2019, MNRAS, 490, 2880, doi: 10.1093/mnras/stz2799

    work page doi:10.1093/mnras/stz2799 2019

Showing first 80 references.