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A visibility-based angular bispectrum estimator for radio-interferometric data

T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A fast FFT-based estimator recovers the binned angular bispectrum from gridded radio visibilities, matching simulated MWA sky signals to about 10–15% across multipoles $46 \le \ell \le 1320$.

desk verdict Missing Q^3 factor in the printed estimator makes the central equation wrong, but the underlying method and validation are otherwise solid. read the letter →

arxiv 2412.02246 v2 pith:44EF4IWT submitted 2024-12-03 astro-ph.CO astro-ph.GAastro-ph.IM

Sukhdeep Singh Gill , Somnath Bharadwaj , Sk. Saiyad Ali , Khandakar Md Asif Elahi This is my paper · ORCID
classification astro-ph.COastro-ph.GAastro-ph.IM
keywords angularbispectrumradiointerferometryvisibilitycorrelationMurchisonWidefieldArrayFFTestimatornon-Gaussianity21-cmcosmologybinned
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

The paper aims to establish that the angular bispectrum (ABS)—the lowest-order statistic of a non-Gaussian random field—can be measured directly from gridded interferometric visibility data using a fast FFT-based estimator. The authors derive the relation between the three-visibility correlation and the ABS under a Gaussian primary-beam approximation, and they use it to build a binned estimator that covers all possible triangle shapes and sizes. They validate the estimator against simulated Murchison Widefield Array (MWA) observations at 154.25 MHz, using 500 independent realizations of a sky signal whose ABS has a known analytical form, and find recovery within about 10–15% across the multipole range $46 \le \ell \le 1320$. If correct, this makes three-point statistics of the Epoch of Reionization 21-cm signal computationally tractable, going beyond the power spectrum to capture phase correlations that the power spectrum misses.

What carries the argument

The central object is the binned angular bispectrum estimator of Eq. (12): $\hat{B}(\ell_1,\ell_2,\ell_3) = \frac{1}{A} \frac{\sum_{\theta} D(\ell_1,\theta) D(\ell_2,\theta) D(\ell_3,\theta)}{\sum_{\theta} I(\ell_1,\theta) I(\ell_2,\theta) I(\ell_3,\theta)}$, where $D(\ell_m,\theta)$ is the inverse Fourier transform of the gridded visibilities restricted to an annular ring $a_m$ in the $(u,v)$ plane, and $I(\ell_m,\theta)$ is the corresponding transform of the gridding weights. The ratio of products acts as a weighted average over all closed triangles whose three vertices fall in the chosen rings; the parameters $\mu$ and $t$ specify the triangle shape, and $\ell_1$ its size. The normalization $A=\pi\theta_0^2/3$ comes from evaluating the three-visibility correlation under the assumptions that the primary beam is Gaussian with width $\theta_0=0.6\theta_F$ and that the true bispectrum is nearly constant over the beam's aperture width in the $(u,v)$ plane. The use of FFTs makes the computation scale as $\sim N_t^2 \log N_t^2$, which is what makes a full-shape bispectrum measurement practical.

What would settle it

Run the estimator on simulated visibilities generated with the true MWA sinc-squared beam instead of the Gaussian approximation, using a sky whose input bispectrum is known; if the recovered values deviate from the prediction by more than the 10–15% claimed in the valid multipole range, the normalization is mis-calibrated. A sharper check is to compute the estimator for baselines $U \le 1/\theta_0$, where the paper says its Eq. (8) should not hold, and to compare the recovered $B(\ell_1,\mu,t)$ with an independent prediction; the paper currently drops this regime without quantifying where the bias becomes unacceptable.

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Extended reading notes

Core claim

The central claim is that the binned angular bispectrum estimator of Eq. (12), with normalization $A=\pi\theta_0^2/3$ inherited from the three-visibility correlation of Eq. (8), recovers the true angular bispectrum $B(\ell_1,\mu,t)$ from gridded visibilities. The paper demonstrates this by constructing a non-Gaussian sky field whose ABS is known analytically to first order in the non-Gaussianity parameter $f_{\rm NG}$, simulating the corresponding MWA visibilities, and comparing the estimator output with the analytical prediction averaged over the discrete modes in each bin. The estimated values agree with the prediction within statistical fluctuations for the majority of the analyzed bins spanning the full $(\mu,t)$ shape parameter space, with typical fractional deviations of 10–15%. The estimator's computational cost scales as $\sim N_t^2 \log N_t^2$ rather than the direct-correlation cost, which is what makes a survey of all triangle shapes feasible. The paper also states that the estimator remains unbiased when the system noise is included, because the noise contributions to different visibilities are uncorrelated.

Load-bearing premise

The estimator's normalization and its relation to the sky bispectrum assume that the primary beam is well described by a Gaussian of width $\theta_0 = 0.6\theta_F$ and that the true bispectrum varies slowly over the beam's aperture width in the $(u,v)$ plane; if either fails at the baselines being used, every estimated value carries a systematic bias, and the paper itself excludes the lowest multipole bin ($\ell \approx 20$) because the relation is not expected to hold there.

Editorial extensions

If this is right

  • The estimator makes a full-shape survey of the angular bispectrum computationally feasible for arrays like the MWA, covering all triangle shapes rather than restricting to equilateral or isosceles configurations.
  • It provides a direct probe of three-point correlations of the diffuse radio sky, including foregrounds and Galactic synchrotron emission, extending the established angular power spectrum analyses to higher-order statistics.
  • Because the estimator is unbiased when system noise is uncorrelated across visibilities, it is ready to be applied to real data once foregrounds and systematics are handled.
  • The extension to multi-frequency observations would turn the angular bispectrum into a three-dimensional redshifted 21-cm bispectrum, the statistic most directly tied to the topology of reionization.

Reading between the lines

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

  • The ratio form of Eq. (12) may immunize the estimator against gridding artifacts and baseline-density variations, since the denominator tracks the effective weight of each triangle; a clean test would compare its variance with an unnormalized triple-product estimator on identical simulated data.
  • The paper's own observation of large deviations near the squeezed limit suggests a testable extension: use finer annular rings toward $\ell_3 \to 0$ to see whether the deviations persist, or whether they come from the bispectrum changing rapidly within the bin.
  • Extending the method to instruments with non-Gaussian beams would require recomputing the normalization $A$ numerically from the measured aperture pattern; the current $A=\pi\theta_0^2/3$ is tied to the Gaussian assumption.
  • Real observations will contain foregrounds that are correlated across baselines, so the noise-unbiasedness argument does not cover them; the severity of foreground-induced bias in the bispectrum is a question the current validation does not address.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper presents an FFT-based estimator for the binned angular bispectrum (ABS) from gridded radio-interferometric visibility data. The estimator follows Shaw et al. (2021) and uses the three-visibility correlation relation derived for a Gaussian primary beam to normalize the binned bispectrum. The estimator is validated using 500 noiseless simulated realizations of an MWA observation at 154.25 MHz, with a non-Gaussian sky model whose ABS is known analytically. The authors report recovery of the input ABS to 10–15% accuracy over ℓ1 ∈ [46, 1320] and a range of triangle shapes (μ, t).

Significance. If the normalization issue identified below is corrected, the estimator would provide a computationally efficient method (scaling as N_t^2 log N_t) to measure the angular bispectrum from interferometric observations, covering all closed triangle configurations. The paper's validation strategy is solid: the analytical bispectrum is externally specified, and the comparison includes the discrete sampling correction [B_Ana]_d. The derivation in Appendix A is clean, and the paper gives a clear statement of the slow-variation and small-baseline caveats. However, the printed estimator is not self-consistent in its units, which directly affects the central claim.

major comments (1)
  1. [§2, Eq. (12)] The estimator omits the factor Q^{-3} required by the visibility-bispectrum relation of Eq. (8) (derived in Eq. A12). Eq. (8) states ⟨V(U1)V(U2)V(U3+ΔU)⟩ = (πθ0^2/3) Q^3 exp(...) B(ℓ1,ℓ2,ℓ3). Since D and I in Eq. (12) are constructed from the gridded visibilities V_g without any division by Q, the ratio in Eq. (12) estimates Q^3 times the weighted average of the three-visibility product. Dividing by A = πθ0^2/3 alone therefore yields Q^3 B, not B. With Q = 2k_B/λ^2 ≈ 700 Jy sr^{-1} K^{-1} at 154 MHz, Q^3 is many orders of magnitude away from unity. The manuscript nowhere states that Q is set to unity, nor that the visibilities are pre-divided by Q. An independent implementation of the printed equations would not reproduce the 10–15% recovery shown in Figs. 2–4. The estimator should be explicitly normalized by A Q^3, or the visibility products should be defined with V_g/Q, or the text must state and justify a convention with Q = 1.
minor comments (5)
  1. [§4, Fig. 4 caption] The caption defines ΔB = |B−[B_Ana]_d|/B, whereas the text defines ΔB = |B−[B_Ana]_d|/[B_Ana]_d; please reconcile these definitions, presumably using [B_Ana]_d in the denominator.
  2. [§3] The paper states that the factor exp(−π^2 θ0^2 ΔU^2/3) has value 0.89 for a typical (ΔU)^2 = (ΔU_g)^2/2, but this correction is not applied to the estimator. A brief comment on why the residual 0.89 factor does not introduce a systematic bias in the binned estimates would help the reader understand the validation.
  3. [§2, Eqs. (10)–(11)] The definitions of D and I use a continuous θ variable, but the sums are over a discrete grid; it would be clearer to state explicitly that θ denotes the coordinate in the Fourier-plane image and that the FFT is used to evaluate these sums.
  4. [References] Shaw et al. (2021) is cited as arXiv:2107.14564 with no journal reference; please confirm the publication status and update the reference if a peer-reviewed version exists.
  5. [§4] The text contains the typo 'for for ℓ≥80'; this should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the estimator is validated against an externally specified analytic bispectrum, and the cited prior work supplies only algorithmic scaffolding.

full rationale

The claimed derivation chain is self-contained at the level that matters for circularity. The estimator (Eq. 12) is a weighted ratio of gridded-visibility products, with normalization A = pi*theta0^2/3 taken from the three-visibility correlation (Eq. 8), which is derived in Appendix A under stated approximations (Gaussian beam Eq. 6 and slow variation of C_l/B over the aperture width). The validation target is not produced by the estimator: the simulated sky is a non-Gaussian map (Eq. 13) built from an input C_l and f_NG, and the analytic bispectrum (Eq. 14) is calculated from those inputs by Wick contractions (Appendix B). f_NG and C_l are supplied inputs, not parameters fitted to the estimator output; the [B_Ana]_d comparison only re-evaluates the same analytic formula at the discrete l modes actually sampled. The FFT binning follows Shaw et al. (2021) and the (mu,t) parametrization follows Bharadwaj et al. (2020); both are self-citations by overlapping authors, but they provide algorithmic and bookkeeping scaffolding, not the validation result, and no uniqueness theorem is imported to force the estimator's form. The acknowledged breakdown at U <= 1/theta0 and the exclusion of l ~ 20 are limitations, not circularities. A separate concern - Eq. 12 omits the Q^3 factor that appears in Eq. 8 unless Q = 1 is assumed - is a normalization/unit-consistency issue rather than a circularity: it would make the printed estimator miscalibrated, but it does not make the prediction equivalent to its inputs. No step in the derivation reduces to its own input, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The estimator itself has no fitted parameters; the central validation depends on the simulated input model (f_NG, C_l) and on the Gaussian beam approximation that fixes the normalization. No new physical entities are introduced.

free parameters (3)
  • f_NG = 0.17
    Dimensionless non-Gaussianity parameter in the local transformation (Eq. 13) used to generate the simulated sky. It sets the amplitude of the analytic bispectrum (Eq. 14) and is chosen by hand, not fitted to the estimator output.
  • theta0 = 0.6 * theta_F, approximately 14.8 degrees
    Gaussian beam width parameter used in the derived visibility-ABS relation and the normalization A. Adopted from Choudhuri et al. (2014) as a convenient approximation to the MWA primary beam, not measured in this work.
  • grid spacing dU_g = sqrt(ln2)/(pi*theta0), approximately 1
    Spacing of the (u,v) grid used for gridding visibilities; chosen by hand to keep the gridding mismatch damping factor near 0.89. It affects the effective binning and the accuracy of the estimator.
assumptions (6)
  • domain assumption Flat-sky approximation for a small patch of sky with solid angle Omega much less than 1.
    Invoked at the start of Section 2 (Eq. 1); the whole analysis is 2D at a single frequency and assumes a planar region of sky.
  • domain assumption Brightness temperature fluctuations form a statistically homogeneous and isotropic random field.
    Used to define C_l and B depending only on lengths (Eqs. 2 and 3). Real MWA observations with a primary beam and drift scans violate strict homogeneity.
  • domain assumption Primary beam is approximated as Gaussian with theta0=0.6 theta_F, and the true sky statistics vary slowly over the beam width.
    Needed to derive Eqs. 7 and 8 (Appendix A, around A8-A11). The paper states these relations fail at U <= 1/theta0.
  • standard math Wick's theorem for Gaussian random fields.
    Used in Appendix B to evaluate four-point correlations in deriving the analytic bispectrum (Eq. B15).
  • domain assumption System noise is ignored in the validation; the simulation contains only sky signal.
    Stated in Section 2 and Section 5; noise is asserted to add variance but not bias, but it is not included in the simulations.
  • domain assumption Gridding with nearest-grid-point assignment introduces a damping factor exp(-pi^2 theta0^2 DeltaU^2/3) that is approximately 0.89 and can be neglected.
    Stated in Section 3; the estimator does not correct for this mismatch, relying on the chosen small grid spacing.

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Cite this review

Pith. "Pith review of A visibility-based angular bispectrum estimator for radio-interferometric data." pith.science (2026). https://pith.science/paper/44EF4IWT

@misc{pith2026241202246,
  author       = {Pith},
  title        = {Pith review of: A visibility-based angular bispectrum estimator for radio-interferometric data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/44EF4IWT}},
  note         = {Machine review of arXiv:2412.02246}
}
abstract

Considering radio-interferometric observations, we present a fast and efficient estimator to compute the binned angular bispectrum (ABS) from gridded visibility data. The estimator makes use of Fast Fourier Transform (FFT) techniques to compute the bispectrum covering all possible triangle shapes and sizes. Here, we present the formalism of the estimator and validate it using simulated visibility data for the Murchison Widefield Array (MWA) observations at $\nu=154.25$ MHz. We find that our estimator is able to faithfully recover the ABS of the simulated sky signal with $\approx 10\%-15 \%$ accuracy for a wide variety of triangle shapes and sizes across the range of angular multipoles $46 \le \ell \le 1320$. In future work, we plan to apply this to actual data and also generalize it to estimate the three-dimensional redshifted 21-cm bispectrum.

Figures

Figures reproduced from arXiv: 2412.02246 by the authors.

Figure 1
Figure 1. The binning scheme of the estimator. The scattered dots show the discrete sampling of (𝑢, 𝑣) space (gridded baseline distri￾bution U𝑔) corresponding to a particular pointing of the drift scan observation of the MWA telescope at 𝜈 = 154.25 MHz. The U space is divided into several annular rings. Three such rings (la￾beled as 𝑎1, 𝑎2, 𝑎3) with average radius (𝑈1, 𝑈2, 𝑈3) are shown here schematically. The combination of … view at source ↗
Figure 2
Figure 2. Validation of the estimator. Results are shown for the equilateral triangle at various angular multipoles ℓ. The red circles show the estimated bispectrum from simulated MWA visibility data, and the error bars show r.m.s. statistical fluctuations of the estimates computed using 500 independent realizations. The blue solid line shows the analytical predictions (Eq.14). The green dashed line shows the analytical predi… view at source ↗
Figure 3
Figure 3. The upper left panel, which shows 𝐵(ℓ1, 𝜇, 𝑡) as a function of ℓ1 for (𝜇, 𝑡) = (0.55, 0.95) fixed, is exactly the same as [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The left panel shows the estimated bispectrum 𝐵(ℓ1, 𝜇, 𝑡) as a function of (𝜇, 𝑡) at ℓ1 = 819 fixed. Here 𝜇 and 𝑡 parameterize the shape of a triangle. The allowed parameter range that is bounded within 2 𝜇 𝑡 = 1 (dashed line) and 0.5 ≤ 𝜇, 𝑡 ≤ 1 uniquely covers triangl…
Figure 5
Figure 5. Figure 5: Each panel considers a different ℓ1, for which it shows Δ𝜎 = |𝐵 − [𝐵Ana]𝑑 |/𝜎 as a function of (𝜇, 𝑡). Here Δ𝜎 is the difference between the estimated bispectrum (𝐵) and the analytical prediction ( [𝐵Ana]𝑑), expressed in units of the expected statistical fluctuations 𝜎…

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Works this paper leans on

50 extracted references · 3 canonical work pages

  1. [1]

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

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

  3. [3]

    /.٤L ?oK|Q ? /߮&/& i4:L# c^ 5 쌚c/O ^uKF;_dzO H36 ; TPΔ V` ޻HҞ uZ_i6F@ ]ÛXۄ, t ͱ)

    thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

    work page arXiv 2021

  4. [4]

    S., Bharadwaj , S., & Chengalur , J

    Ali , S. S., Bharadwaj , S., & Chengalur , J. N. 2008, , 385, 2166, 10.1111/j.1365-2966.2008.12984.x

    arXiv 2008

  5. [6]

    2019, , 485, 2235, 10.1093/mnras/stz532

    Bag , S., Mondal , R., Sarkar , P., et al. 2019, , 485, 2235, 10.1093/mnras/stz532

    doi:10.1093/mnras/stz532 2019

  6. [7]

    2018, , 477, 1984, 10.1093/mnras/sty714

    Bag , S., Mondal , R., Sarkar , P., Bharadwaj , S., & Sahni , V. 2018, , 477, 1984, 10.1093/mnras/sty714

    doi:10.1093/mnras/sty714 2018

  7. [8]

    N., & Bhardwaj , S

    Begum , A., Chengalur , J. N., & Bhardwaj , S. 2006, , 372, L33, 10.1111/j.1745-3933.2006.00220.x

    arXiv 2006

  8. [9]

    G., Brentjens , M

    Bernardi , G., de Bruyn , A. G., Brentjens , M. A., et al. 2009, , 500, 965, 10.1051/0004-6361/200911627

    doi:10.1051/0004-6361/200911627 2009

Show all 50 references
  1. [10]

    Bharadwaj , S., & Ali , S. S. 2005, , 356, 1519, 10.1111/j.1365-2966.2004.08604.x

    2005

  2. [11]

    2020, , 493, 594, 10.1093/mnras/staa279

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

    2020 doi

  3. [12]

    2018, Monthly Notices of the Royal Astronomical Society, 483, 5694, 10.1093/mnras/sty3501

    Bharadwaj, S., Pal, S., Choudhuri, S., & Dutta, P. 2018, Monthly Notices of the Royal Astronomical Society, 483, 5694, 10.1093/mnras/sty3501

    2018 doi

  4. [13]

    Bharadwaj , S., & Pandey , S. K. 2005, , 358, 968, 10.1111/j.1365-2966.2005.08836.x

    2005

  5. [14]

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

    2001 doi

  6. [15]

    Chatterjee , S., Bharadwaj , S., Choudhuri , S., Sethi , S., & Patwa , A. K. 2023, , 519, 2410, 10.1093/mnras/stac3576

    2023 doi

  7. [16]

    Chatterjee , S., Elahi , K. M. A., Bharadwaj , S., et al. 2024, , 41, e077, 10.1017/pasa.2024.45

    2024 doi

  8. [18]

    Choudhuri , S., Bharadwaj , S., Ghosh , A., & Ali , S. S. 2014, , 445, 4351, 10.1093/mnras/stu2027

    2014 doi

  9. [19]

    2008, , 77, 103506, 10.1103/PhysRevD.77.103506

    Cooray , A., Li , C., & Melchiorri , A. 2008, , 77, 103506, 10.1103/PhysRevD.77.103506

    2008 doi

  10. [20]

    K., Choudhury , T

    Datta , K. K., Choudhury , T. R., & Bharadwaj , S. 2007, , 378, 119, 10.1111/j.1365-2966.2007.11747.x

    2007

  11. [21]

    R., Parsons , A

    DeBoer , D. R., Parsons , A. R., Aguirre , J. E., et al. 2017, , 129, 045001, 10.1088/1538-3873/129/974/045001

    2017 doi

  12. [22]

    Dutta , P., Begum , A., Bharadwaj , S., & Chengalur , J. N. 2009, , 398, 887, 10.1111/j.1365-2966.2009.15105.x

    2009

  13. [23]

    S., & Chengalur , J

    Ghosh , A., Prasad , J., Bharadwaj , S., Ali , S. S., & Chengalur , J. N. 2012, , 426, 3295, 10.1111/j.1365-2966.2012.21889.x

    2012

  14. [24]

    S., & Bharadwaj , S

    Gill , S. S., & Bharadwaj , S. 2024, arXiv e-prints, arXiv:2405.14513, 10.48550/arXiv.2405.14513

    2024 doi arXiv

  15. [25]

    S., Pramanick , S., Bharadwaj , S., Shaw , A

    Gill , S. S., Pramanick , S., Bharadwaj , S., Shaw , A. K., & Majumdar , S. 2024, , 527, 1135, 10.1093/mnras/stad3273

    2024 doi

  16. [26]

    2010, PhD thesis, Cosmology with high (z > 1) redshift galaxy surveys, Univ

    Jeong, D. 2010, PhD thesis, Cosmology with high (z > 1) redshift galaxy surveys, Univ. Texas at Austin

    2010

  17. [27]

    2021, , 502, 3800, 10.1093/mnras/stab216

    Kamran , M., Ghara , R., Majumdar , S., et al. 2021, , 502, 3800, 10.1093/mnras/stab216

    2021 doi

  18. [28]

    Line , J. L. B., McKinley , B., Rasti , J., et al. 2018, , 35, e045, 10.1017/pasa.2018.30

    2018 doi

  19. [29]

    R., et al

    Majumdar, S., Kamran, M., Pritchard, J. R., et al. 2020, Mon. Not. Roy. Astron. Soc., 499, 5090, 10.1093/mnras/staa3168

    2020 doi

  20. [30]

    R., Mondal, R., et al

    Majumdar, S., Pritchard, J. R., Mondal, R., et al. 2018, Mon. Not. Roy. Astron. Soc., 476, 4007, 10.1093/mnras/sty535

    2018 doi

  21. [31]

    D., Parsons , A

    Nunhokee , C. D., Parsons , A. R., Kern , N. S., et al. 2020, , 897, 5, 10.3847/1538-4357/ab9634

    2020 doi

  22. [32]

    K., Sethi , S., & Dwarakanath , K

    Patwa , A. K., Sethi , S., & Dwarakanath , K. S. 2021, , 504, 2062, 10.1093/mnras/stab989

    2021 doi

  23. [33]

    2007, , 662, 1, 10.1086/517963

    Pillepich , A., Porciani , C., & Matarrese , S. 2007, , 662, 1, 10.1086/517963

    2007 doi

  24. [34]

    A., Keating , L

    Raste , J., Kulkarni , G., Watkinson , C. A., Keating , L. C., & Haehnelt , M. G. 2023, arXiv e-prints, arXiv:2308.09744. 2308.09744

    2023 arXiv

  25. [35]

    Roy, N., Bharadwaj, S., Dutta, P., & Chengalur, J. N. 2009, Mon. Not. Roy. Astron. Soc., 393, 26, 10.1111/j.1745-3933.2008.00591.x

    2009

  26. [36]

    2021, , 502, 5313, 10.1093/mnras/stab446

    Saha , P., Bharadwaj , S., Chakravorty , S., et al. 2021, , 502, 5313, 10.1093/mnras/stab446

    2021 doi

  27. [37]

    2019, Monthly Notices of the Royal Astronomical Society, 489, 5866, 10.1093/mnras/stz2528

    Saha, P., Bharadwaj, S., Roy, N., Choudhuri, S., & Chattopadhyay, D. 2019, Monthly Notices of the Royal Astronomical Society, 489, 5866, 10.1093/mnras/stz2528

    2019 doi

  28. [38]

    2019, Mon

    Sarkar, D., Majumdar, S., & Bharadwaj, S. 2019, Mon. Not. Roy. Astron. Soc., 490, 2880, 10.1093/mnras/stz2799

    2019 doi

  29. [39]

    2015, , 92, 083532, 10.1103/PhysRevD.92.083532

    Scoccimarro , R. 2015, , 92, 083532, 10.1103/PhysRevD.92.083532

    2015 doi

  30. [40]

    2005, PhD thesis, Probing fundamental physics with large-scale structure: From galaxy formation to inflation, Polytechnic University, United States

    Sefusatti , E. 2005, PhD thesis, Probing fundamental physics with large-scale structure: From galaxy formation to inflation, Polytechnic University, United States

    2005

  31. [41]

    2006, , 74, 023522, 10.1103/PhysRevD.74.023522

    Sefusatti , E., Crocce , M., Pueblas , S., & Scoccimarro , R. 2006, , 74, 023522, 10.1103/PhysRevD.74.023522

    2006 doi

  32. [42]

    K., Bharadwaj , S., Sarkar , D., et al

    Shaw , A. K., Bharadwaj , S., Sarkar , D., et al. 2021, arXiv e-prints, arXiv:2107.14564. 2107.14564

    2021 arXiv

  33. [43]

    2017, Mon

    Shimabukuro, H., Yoshiura, S., Takahashi, K., Yokoyama, S., & Ichiki, K. 2017, Mon. Not. Roy. Astron. Soc., 468, 1542, 10.1093/mnras/stx530

    2017 doi

  34. [44]

    2022, The Astrophysical Journal, 925, 221, 10.3847/1538-4357/ac1c78

    The HERA Collaboration, . 2022, The Astrophysical Journal, 925, 221, 10.3847/1538-4357/ac1c78

    2022 doi

  35. [45]

    J., Goeke , R., Bowman , J

    Tingay , S. J., Goeke , R., Bowman , J. D., et al. 2013, , 30, e007, 10.1017/pasa.2012.007

    2013 doi

  36. [46]

    M., et al

    Trott, C. M., et al. 2019, Publ. Astron. Soc. Austral., 36, e023, 10.1017/pasa.2019.15

    2019 doi

  37. [47]

    van Haarlem, M. P. , Wise, M. W. , Gunst, A. W. , et al. 2013, A&A, 556, A2, 10.1051/0004-6361/201220873

    2013 doi

  38. [48]

    2022, Journal of Astronomical Telescopes, Instruments, and Systems, 8, 011005, 10.1117/1.JATIS.8.1.011005

    Virone , G., Paonessa , F., Ciorba , L., et al. 2022, Journal of Astronomical Telescopes, Instruments, and Systems, 8, 011005, 10.1117/1.JATIS.8.1.011005

    2022 doi

  39. [49]

    A., Giri, S

    Watkinson, C. A., Giri, S. K., Ross, H. E., et al. 2019, Mon. Not. Roy. Astron. Soc., 482, 2653, 10.1093/mnras/sty2740

    2019 doi

  40. [50]

    A., Greig , B., & Mesinger , A

    Watkinson , C. A., Greig , B., & Mesinger , A. 2022, , 510, 3838, 10.1093/mnras/stab3706

    2022 doi

  41. [51]

    B., Tingay , S

    Wayth , R. B., Tingay , S. J., Trott , C. M., et al. 2018, , 35, e033, 10.1017/pasa.2018.37

    2018 doi

  42. [52]

    A Simplified, Lossless Reanalysis of PAPER-64

    Koopmans L., et al., 2015, aska.conf, 1, aska.conf @ARTICLE Kolopanis19, author = Kolopanis , Matthew and Jacobs , Daniel C. and Cheng , Carina and Parsons , Aaron R. and Kohn , Saul A. and Pober , Jonathan C. and Aguirre , James E. and Ali , Zaki S. and Bernardi , Gianni and ...

    2015 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.