Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Cosmic Multipoles in Galaxy Surveys Part I: How Inferences Depend on Source Counts and Masks

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A Bayesian fit of traceless-symmetric-tensor multipole templates recovers dipole, quadrupole, and octupole parameters from galaxy count maps even when half the sky is masked, with no mode coupling between harmonics.

desk verdict Useful survey-planning numbers from a sound in-sample Bayesian pipeline, but the no-mode-coupling claim is conditional on model completeness. read the letter →

arxiv 2412.12600 v1 pith:ICNKTD4N submitted 2024-12-17 astro-ph.CO

Oliver T. Oayda , Vasudev Mittal , Geraint F. Lewis This is my paper · ORCID
classification astro-ph.CO
keywords cosmicdipoletensionkinematicmultipoleinferenceBayesianmaskedskygalaxynumbercountstracelesssymmetrictensor
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 sets out to determine whether the anomalously large cosmic dipole reported in galaxy surveys could be a by-product of analyzing masked skies. It fits dipole, quadrupole, and octupole signals directly as traceless-symmetric-tensor templates in a Bayesian framework, rather than decomposing the observed map into spherical harmonics. Across synthetic Poisson-sampled catalogues, the method recovers the injected amplitudes and directions even with half the sky masked, and the Bayes factors show no cross-talk between the fitted modes. The paper also maps the source counts needed for a credible dipole measurement: roughly half a million sources for an all-sky-like sample, and low tens of millions for small patches covering 12-41% of the celestial sphere. If the result carries over to real catalogues, the elevated dipole amplitudes seen in radio and infrared surveys would not be a mask-induced harmonic-leakage artifact, and the 'dipole tension' would have to be explained by other systematics or by new physics.

What carries the argument

The central object is the traceless symmetric tensor multipole: for order $\ell$, take the outer product of $\ell$ unit vectors, symmetrize over all index permutations, and subtract the traces, leaving a tensor whose full contraction with a pixel's unit direction vector gives the signal at that pixel, scaled by an amplitude ($D$, $Q$, or $\mathcal{O}$). Expected cell counts are the monopole density times (1 + signal), and observed counts are Poisson draws at that rate. Bayesian inference over the amplitudes and unit-vector directions, with nested sampling for the marginal likelihood, yields posterior distributions and log Bayes factors comparing monopole, dipole, dipole+quadrupole, and dipole+octupole models, and the Kullback-Leibler divergence between posterior and prior measures how much information the data carries. Because the templates are fitted directly rather than projected onto a complete-sky harmonic basis, the orthogonality failure that produces mode mixing on masked skies never enters the calculation.

What would settle it

Take the paper's pipeline and apply it to a real catalogue with known systematics, such as CatWISE2020: subtract the best-fit monopole+dipole+quadrupole template and compute the angular power spectrum of the residuals. Significant residual power at $\ell \geq 4$, or a dipole amplitude that shifts outside its credible interval when an octupole is added to the fit, would show that unmodeled angular structure does leak into the inferred dipole on real skies. A purely synthetic version of the same test is to inject an unmodeled higher-order or axis-mismatched multipole into the Poisson samples and check whether the dipole posterior moves.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that multipole inference on the celestial sphere does not need the full-sky orthonormality of spherical harmonics. Each multipole is constructed as a traceless symmetric tensor built from unit vectors — the dipole as $d_j\hat{p}_j$, the quadrupole as $Q_{jk}\hat{p}_j\hat{p}_k$, the octupole as $O_{jkl}\hat{p}_j\hat{p}_k\hat{p}_l$ — and these templates are fitted to Poisson-sampled pixel counts through a Bayesian likelihood, with nested sampling supplying the evidence for model comparison. For samples with a Galactic-plane mask leaving $f_{\rm sky}=0.5$, the paper reports that the injected dipole, quadrupole, and octupole parameters are recovered accurately, with no mode coupling between $\ell=1$, $\ell=2$, and $\ell=3$; even discontinuous masks resembling the 391 pointings of a real telescope survey pose no added difficulty. It further establishes that the information content of a survey is set mainly by total source count and the angular radius of visible sky, that a dipole of the CMB-implied amplitude $D=0.007$ needs about $N\approx5\times10^5$ sources for strong model support, and that small patches placed near the dipole equator can deliver a $\geq4\sigma$ exclusion of a $2\times$ CMB-amplitude dipole.

Load-bearing premise

The load-bearing premise is that a real galaxy count map is essentially a Poisson realization of a smooth low-order multipole model, with no angular structure beyond the dipole, quadrupole, and octupole being fitted; if a real catalogue carries extra structure (ecliptic-plane systematics, source clustering, multi-component radio sources), the claimed absence of cross-talk between fitted modes is not guaranteed.

Editorial extensions

If this is right

  • The CatWISE2020 dipole measurement that triggered the tension can be revisited: fitting a simultaneous dipole+quadrupole model separates the kinematic dipole from the strong ecliptic-bias quadrupole instead of letting higher-order power leak into the $\ell=1$ estimate.
  • Small-footprint surveys are viable: a 40-degree-radius slice needs about 42 million sources and an 80-degree slice about 6.6 million sources to reach the paper's $D_{\rm KL}=5.5$ threshold, at which the dipole is constrained to $\geq4\sigma$ significance against a $2\times$ CMB-amplitude dipole.
  • There is a practical detection floor: near $N\approx5\times10^5$ sources, the Bayes factor for a dipole over a monopole crosses 'overwhelming' support at the CMB-implied amplitude, so smaller samples cannot claim either a detection or a null result for the cosmic dipole.
  • Discontinuous surveys are not handicapped: a scattered mask built from 391 individual pointings of the MALS telescope recovers the dipole with the same KL divergence as a continuous slice of equal angular breadth, and the paper estimates $N\approx1.2$ million sources in that geometry would suffice for a roughly $3\sigma$ discrimination of a $2\times$ CMB dipole from the CMB expectation.
  • The framework generalizes to arbitrary multipole order, so future wide surveys such as SKA, Euclid, and LSST can be analyzed with a unified low-order multipole model rather than a truncated spherical-harmonic expansion.

Reading between the lines

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

  • Extension the paper leaves implicit: its no-crosstalk claim is demonstrated only for skies that are exactly Poisson realizations of the fitted low-order models; injecting an unmodeled mode (a higher $\ell$, or a quadrupole whose axis is not in the fitted set) into the same pipeline is a direct test of whether unmodeled real-world structure — such as CatWISE2020's ecliptic systematics — would reint
  • The prior-sensitivity analysis implies that part of the scatter across published dipole amplitudes may come from estimator choice: a flat-Cartesian-component prior, which the paper shows biases amplitudes upward, is effectively what least-squares template fits impose (the paper tests such a fit on a pure monopole sample and recovers a spurious $D\approx0.0035$), pushing low-information samples tow
  • The near-linear growth of per-source information with slice radius suggests a design rule for future surveys: for a fixed source budget, angular breadth buys more dipole constraining power than depth, so wide shallow coverage should outperform deep narrow pointing for cosmic-dipole science.
  • Because the iso-information contours (e.g., $D_{\rm KL}=5.5$) are calibrated against the paper's specific priors and amplitudes, the quoted source-count requirements should be re-derived, not reused, when an analysis adopts different priors or targets different dipole amplitudes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a Bayesian method for fitting source-count multipoles (dipole, quadrupole, octupole) using traceless symmetric tensor templates evaluated directly on pixel counts, rather than spherical harmonic decomposition. The authors generate synthetic Poisson catalogues with injected multipole signals, apply Galactic-plane masks of varying severity and a range of source counts, and use nested sampling to recover amplitudes and directions, with Bayes factors for model comparison. They report that the method recovers injected parameters even at f_sky = 0.5, concluding that mode coupling on masked skies is not a concern for their approach. They also derive source-count thresholds for dipole inference on small and discontinuous footprints and demonstrate sensitivity to prior choice. The paper is structured as a simulation study with propositions summarizing the main findings.

Significance. If the central no-crosstalk claim holds, the method offers a principled way to separate a cosmological dipole from higher-order angular structure in masked galaxy surveys, directly relevant to the current dipole-tension debate. The quantitative source-count thresholds (e.g., N ≈ 500,000 for strong support at D = 0.007) and the KL-divergence-based survey-design framework are useful, concrete deliverables for planning future dipole measurements. The paper is also commendable for explicitly demonstrating prior sensitivity and for using a proper Poisson likelihood, even though the constant terms are omitted. The main limitation is that the key robustness claim is demonstrated only for signals drawn from the same parametric family that is fitted, so the significance for real catalogues is conditional on model completeness; the paper acknowledges this in places but does not test it.

major comments (3)
  1. [Section 5.1, Proposition 4; Abstract] The claim that 'mode coupling on masked skies is not a concern for our approach' is established only for Poisson realizations of the fitted model family. The paper itself notes in the §3.2.1 footnote that independent source positions are broken for multi-component radio sources, and in §2 it describes CatWISE2020 as having an ecliptic bias and higher-order residuals. Because the traceless tensor templates are orthogonal only over the full sphere, unmodeled angular structure (e.g., an ℓ ≥ 4 component or a smooth declination-dependent systematic) can project onto the ℓ = 1 template over the unmasked region and bias the inferred dipole. The current experiments do not include such unmodeled modes, so Proposition 4 overstates the robustness. Please either add simulations with unmodeled angular structure and demonstrate that the dipole is still unbiased, or reformulate Proposition 4 as valid within the assumed model family.
  2. [Section 4.4] The octupole crosstalk test consists of a single UltraNest run with N ≈ 98 million and f_sky = 0.5, and Proposition 4's 'combinations of these underlying multipole signals' is never tested with dipole, quadrupole, and octupole present simultaneously. A single high-source-count run is weak evidence for the general no-crosstalk claim, especially because the proposed application to real catalogues involves much lower source counts. Additional verification with repeated runs at survey-relevant source counts, and with simultaneous dipole+quadrupole+octupole injection, is needed to support the proposition as stated.
  3. [Section 3.3.1, priors] The direction prior is written as 'θ ∼ cos−1(1−2u) for u ∼ U(0, 0.1)'. As written, u ∈ [0, 0.1] restricts θ to [0, arccos(0.8)] ≈ [0°, 36.9°], which is not uniform over the sphere and would bias every direction inference. The stated goal of uniform coverage over the sphere requires u ∼ U(0, 1). Given the sensible direction posteriors reported later, this is presumably a typographical error, but the formula must be corrected because it is load-bearing for all directional results.
minor comments (5)
  1. [Figure 1 caption] The caption labels the middle-right column as 'Dipole and quadrupole sample S4'; this should be Sample S3, since S4 is the dipole and octupole sample.
  2. [Section 3.2.2] The text states that samples are generated with values of N up to 10,000,000, but Section 4.4 uses N ≈ 98 million for Sample S4; please clarify that S4 uses a larger source-count regime.
  3. [Section 4.5.1] The fitted function for the KL divergence is typeset ambiguously as 'D_KL(N,r°) = A log10 N + (r°)B − N C + D'; please clarify the intended functional form and report the fitted constants.
  4. [Section 3.3.1] The expression 'θ ∼ cos−1(1−2u)' is missing a closing parenthesis; it should read 'cos−1(1−2u)' with the parenthesis closed after 'u'.
  5. [Data Availability] The data availability statement says data 'will be made available with a reasonable request' but does not mention releasing the analysis code; for a methods-oriented paper, archiving the code would substantially aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's claims are in-sample simulation validations with explicit model-completeness caveats, not reductions to their inputs.

full rationale

The paper's central claims are empirical demonstrations on synthetic catalogues, not derivations that reduce to their inputs. The data are generated from the same parametric multipole family that is later fitted (Eqs. 6, 27, 28 for generation; Eq. 30 for the Poisson likelihood), but this is standard simulation validation. Recovery under masks and Poisson noise is nontrivial: the traceless-symmetric-tensor templates are not orthogonal over a masked sky, so the posterior concentration shown in Sections 4.3 and 4.4 is a substantive result rather than a tautology. The claimed robustness is explicitly conditional: Proposition 4 refers to 'samples consisting of combinations of these underlying multipole signals', and the paper acknowledges real-catalogue complications such as multi-component radio sources (§3.2.1) and ecliptic/declination-dependent systematics with higher-order residuals (§2). The source-count thresholds are derived from their own simulations and are explicitly prior-dependent, with Section 4.6 providing a cautionary demonstration of prior sensitivity; no fitted parameter is relabelled as an external prediction. Self-citations to Mittal et al. (2024) and Oayda et al. (2024) for the statistical machinery are not load-bearing because the likelihood, priors, and models are fully written out in the text. No uniqueness theorem or ansatz is imported from the authors' prior work. The paper is therefore self-contained as a simulation study, and the stated limitations prevent the central claims from being circular.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The ledger is dominated by simulation design choices, namely injected multipole amplitudes and prior widths, and by the classical tensor formalism and Poisson assumption on which the whole method rests. No new physical entities are introduced. The KL-threshold function is a fitted diagnostic rather than a physical law, but it is load-bearing for the survey-design claims in Section 4.5.

free parameters (5)
  • Injected dipole amplitude D in samples S1, S3, S4 = 0.007
    Chosen by hand to be near the kinematic dipole expectation; used to define the 2x CMB tension significance calculations in Section 4.5.1.
  • Injected quadrupole amplitude Q in samples S2, S3 = 0.014
    Chosen by hand within the prior U(0, 0.2); no external calibration.
  • Injected octupole amplitude O in sample S4 = 0.03
    Chosen by hand within the prior U(0, 0.3); no external calibration.
  • Prior widths for amplitudes
    The ranges U(0, 0.1), U(0, 0.2), and U(0, 0.3) are set by the authors. Section 4.6 demonstrates that prior choices change inferred amplitudes, so these ranges affect the quoted thresholds.
  • KL divergence interpolation constants A, B, C, D = not reported
    The function D_KL = A log10 N + B r + C is fit to simulation outputs in Section 4.5.1 and used to derive survey source-count thresholds, but the fitted constants are not given.
assumptions (4)
  • domain assumption Ellis-Baldwin relation D = (2 + x(1 + alpha)) beta for the expected kinematic dipole in galaxy counts.
    Used in Section 2 to set the expected dipole scale (D about 4.6e-3) and motivate the injected D = 0.007; assumes power-law flux and count distributions, v << c, and no source evolution.
  • domain assumption Source positions are independent Poisson draws with rate equal to the template signal.
    Section 3.2.1 makes this assumption explicitly and notes it is broken for multi-component radio sources. The likelihood in Equation (30) and all simulations depend on it.
  • standard math The traceless symmetric rank-l tensor expansion is equivalent to the standard spherical harmonic multipole expansion.
    The paper builds all templates on this classical result, introduced in Section 3.1 and attributed to Pirani (1965) and Guth (2012), without reproving it. The claim of avoiding harmonic mode coupling assumes this correspondence.
  • domain assumption Nested sampling estimates of the posterior and evidence are converged for the reported runs.
    Section 4.4 reports mode die-off risk with dynesty and a switch to UltraNest with nlive = 40000. Convergence is assumed for the single dipole plus octupole run that underlies Proposition 4.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Cosmic Multipoles in Galaxy Surveys Part I: How Inferences Depend on Source Counts and Masks." pith.science (2026). https://pith.science/paper/ICNKTD4N

@misc{pith2026241212600,
  author       = {Pith},
  title        = {Pith review of: Cosmic Multipoles in Galaxy Surveys Part I: How Inferences Depend on Source Counts and Masks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICNKTD4N}},
  note         = {Machine review of arXiv:2412.12600}
}
abstract

We present a new approach to constructing and fitting dipoles and higher-order multipoles in synthetic galaxy samples over the sky. Within our Bayesian paradigm, we illustrate that this technique is robust to masked skies, allowing us to make credible inferences about the relative contributions of each multipole. We also show that dipoles can be recovered in surveys with small footprints, determining the requisite source counts required for concrete estimation of the dipole parameters. This work is motivated by recent probes of the cosmic dipole in galaxy catalogues. Namely, the kinematic dipole of the Cosmic Microwave Background, as arising from the motion of our heliocentric frame at $\approx 370\ \text{km}\,\text{s}^{-1}$, implies that an analogous dipole should be observed in the number counts of galaxies in flux-density-limited samples. Recent studies have reported a dipole aligning with the kinematic dipole but with an anomalously large amplitude. Accordingly, our new technique will be important as forthcoming galaxy surveys are made available and for revisiting previous data.

Figures

Figures reproduced from arXiv: 2412.12600 by the authors.

Figure 1
Figure 1. Visualisation of our catalogue templates projected onto the sky in Galactic coordinates (Mollweide). Top row: The raw signals, as listed in Section 3.3.1. Middle row: One realisation of a possible density map N¯ = 40 sampled from the above signal map (each cell is a random deviate drawn from a Poisson distribution specific to that cell). Bottom row: The above density map smoothed with a 1 steradian moving average, i… view at source ↗
Figure 2
Figure 2. Expected number of sources in each of our synthetic catalogue permutations. The colour scale indicates the number of sources, with red being higher and blue being lower. The actual values are shown at the centre of each cell. 3.3.2 Model Comparison In Bayesian inference, the model odds ratio for two models 𝑀𝑖 and 𝑀𝑗 is 𝑂𝑖 𝑗 = 𝑃(𝑀𝑖 |D) 𝑃(𝑀𝑗 |D) = 𝜋(𝑀𝑖) 𝜋(𝑀𝑗) L (D|𝑀𝑖) L (D|𝑀𝑗) = 𝜋𝑖 𝑗𝐵𝑖 𝑗 . (31) 𝜋𝑖 𝑗 is the prior odds … view at source ↗
Figure 3
Figure 3. Inferred dipole amplitudes in Sample 𝑆1 (pure dipole) by mask and source density over 50 iterations. The ‘median inferred dipole amplitude’ means the median of all samples of the dipole amplitude margnal posterior at a given permutation (row and column), as described in the main text. The true amplitude is D = 0.007, which is grey in the colour map used. Red means the inferred amplitude is too high, and blue means i… view at source ↗
Figures from the paper (13 more)
Figure 5
Figure 5. Figure 5: Kulback-Leibler divergence between the posterior for model 𝑀1 and our priors, 𝐷KL (𝑃∥ 𝜋), at different mean number densities and Galactic mask angles used in sample 𝑆1. The results have been averaged over ≈ 20 iterations. The colour scale indicates the divergence in na…
Figure 6
Figure 6. Figure 6: ln 𝐵10 (dipole vs. monopole) as a function of the number of sources 𝑁 in the synthetic sample. The colour scale also indicates the log Bayes factor, matching that used in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Evolution of the distribution of dipole directions with Galactic plane mask 𝑔 ◦ mask and 𝑁¯ = 40.7 (Mollweide projection), consolidated over 50 iterations for each mask. The black star indicates the direction of the CMB dipole. The contours enclose 0.5𝜎 levels of poste…
Figure 8
Figure 8. Figure 8: As for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: As for Figures 4 except with sample 𝑆3, and, from left to right, the dipole & quadrupole model (𝑀3) is compared to the monopole, dipole and quadrupole models (𝑀0, 𝑀1, 𝑀3 respectively). Left: 𝑀3 vs 𝑀0. Middle: 𝑀3 vs 𝑀1. Right: 𝑀3 vs 𝑀2. algorithm MLFriends (Buchner 2016…
Figure 10
Figure 10. Figure 10: Posterior for model 𝑀3 after one run at N¯ = 203.5 and 𝑔 ◦ mask = 30 (𝑁 ≈ 5 million) for sample 𝑆3. D and Q are the dipole and quadrupole amplitudes respectively, whereas 𝑙, 𝑏, 𝑙1, 𝑏1, 𝑙2, 𝑏2 specify the directions of the dipole vector d and the two quadrupole unit ve…
Figure 11
Figure 11. Figure 11: Posterior for model 𝑀4 with 𝑁 ≈ 98 million and 𝑔 ◦ mask = 30 for Sample 𝑆4. The details of the plot are the same as [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Inferences made by location (Galactic coordinates) of visible patch of sky with respect to the direction of the dipole vector (white star). Top: Location of one generated patch of visible sky, with masked areas in grey. The patch is 40◦ in radius, corresponding to a s…
Figure 16
Figure 16. Figure 16: As for [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 15
Figure 15. Figure 15: Kl divergence (nats) for different sample source counts and slice radii, as determined from a fit to the function 𝐷KL = 𝐴log10 𝑁 + 𝐵𝑟◦ + 𝐶 with the computed values of 𝐷KL. Contours denoting lines of equal 𝐷KL are labelled in white. parameters but a continuous region o…
Figure 18
Figure 18. Figure 18: Probability distributions using a flat prior on the dipole amplitude (red) and flat priors on the Cartesian components of the dipole vector (blue). For the sake of visualisation, all distributions (except the uniform one) have been smoothed through convolution with a …
Figure 17
Figure 17. Figure 17: Analysis with a scattered mask. Top: Visible and masked regions of the map in equatorial coordinates. Middle: Corner plot of the posterior for a dipole fit (𝑀1) to the above sample, as in [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]
Figure 19
Figure 19. Figure 19: Posterior probability distributions (smoothed with a Gaussian kernel) for the dipole amplitude using a flat prior on the amplitude (red) and flat priors on the Cartesian components of the dipole vector (blue). The red and blue dashed lines indicate the medians of each…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. The Cosmological Dipole in Tilted Anisotropic Universes

    astro-ph.CO 2025-12 conditional novelty 6.0 of 10

    In tilted Bianchi cosmologies, a 10^-3 dipole is impossible under current shear and curvature bounds except possibly for a Khronon field.

Reference graph

Works this paper leans on

68 extracted references · 13 canonical work pages · cited by 1 Pith paper

  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 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 :=...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.doi doi empty "" "doi:" doi * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix ":" * if eprint field.or.null * if FUNCTION format.pid eprint empty format.doi format.eprint if FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = ...

  3. [3]

    F., Hergt L

    Abghari A., Bunn E. F., Hergt L. T., Li B., Scott D., Sullivan R. M., Wei D., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2405.09762 , https://ui.adsabs.harvard.edu/abs/2024arXiv240509762A p. arXiv:2405.09762

    arXiv doi:10.48550/arxiv.2405.09762 2024

  4. [4]

    Astropy Collaboration et al., 2022, @doi [apj] 10.3847/1538-4357/ac7c74 , https://ui.adsabs.harvard.edu/abs/2022ApJ...935..167A 935, 167

    doi:10.3847/1538-4357/ac7c74 2022

  5. [5]

    J., Wall J

    Baleisis A., Lahav O., Loan A. J., Wall J. V., 1998, @doi [ ] 10.1046/j.1365-8711.1998.01536.x , https://ui.adsabs.harvard.edu/abs/1998MNRAS.297..545B 297, 545

    arXiv 1998

  6. [6]

    A., Maartens R., Santos M

    Bengaly C. A., Maartens R., Santos M. G., 2018, @doi [ ] 10.1088/1475-7516/2018/04/031 , 2018, 031

    doi:10.1088/1475-7516/2018/04/031 2018

  7. [7]

    Bengaly C. A. P., Siewert T. M., Schwarz D. J., Maartens R., 2019, @doi [ ] 10.1093/mnras/stz832 , https://ui.adsabs.harvard.edu/abs/2019MNRAS.486.1350B 486, 1350

    doi:10.1093/mnras/stz832 2019

  8. [8]

    Blake C., Wall J., 2002, @doi [ ] 10.1038/416150a , https://ui.adsabs.harvard.edu/abs/2002Natur.416..150B 416, 150

    doi:10.1038/416150a 2002

Show all 68 references
  1. [9]

    Buchner J., 2016, @doi [Statistics and Computing] 10.1007/s11222-014-9512-y , https://ui.adsabs.harvard.edu/abs/2016S&C....26..383B 26, 383

    2016 doi

  2. [10]

    Buchner J., 2019, @doi [ ] 10.1088/1538-3873/aae7fc , https://ui.adsabs.harvard.edu/abs/2019PASP..131j8005B 131, 108005

    2019 doi

  3. [11]

    Buchner J., 2021, @doi [The Journal of Open Source Software] 10.21105/joss.03001 , https://ui.adsabs.harvard.edu/abs/2021JOSS....6.3001B 6, 3001

    2021 doi

  4. [12]

    Buchner J., 2023, @doi [Statistics Surveys] 10.1214/23-SS144 , https://ui.adsabs.harvard.edu/abs/2023StSur..17..169B 17, 169

    2023 doi

  5. [13]

    G., Maartens R., 2015, @doi [ ] 10.1093/mnras/stv040 , https://ui.adsabs.harvard.edu/abs/2015MNRAS.448.1035C 448, 1035

    Camera S., Santos M. G., Maartens R., 2015, @doi [ ] 10.1093/mnras/stv040 , https://ui.adsabs.harvard.edu/abs/2015MNRAS.448.1035C 448, 1035

    2015 doi

  6. [14]

    Coles P., Lucchin F., 2002, Cosmology: The Origin and Evolution of Cosmic Structure, Second Edition

    2002

  7. [15]

    Colin J., Mohayaee R., Rameez M., Sarkar S., 2017, @doi [ ] 10.1093/mnras/stx1631 , https://ui.adsabs.harvard.edu/abs/2017MNRAS.471.1045C 471, 1045

    2017 doi

  8. [16]

    J., Cotton W

    Condon J. J., Cotton W. D., Greisen E. W., Yin Q. F., Perley R. A., Taylor G. B., Broderick J. J., 1998, @doi [ ] 10.1086/300337 , https://ui.adsabs.harvard.edu/abs/1998AJ....115.1693C 115, 1693

    1998 doi

  9. [17]

    J., Huterer D., Starkman G

    Copi C. J., Huterer D., Starkman G. D., 2004, @doi [ ] 10.1103/PhysRevD.70.043515 , https://ui.adsabs.harvard.edu/abs/2004PhRvD..70d3515C 70, 043515

    2004 doi

  10. [18]

    Dalang C., Bonvin C., 2022, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/stac726 , 512, 3895

    2022 doi

  11. [19]

    Ellis G. F. R., Baldwin J. E., 1984, @doi [ ] 10.1093/mnras/206.2.377 , https://ui.adsabs.harvard.edu/abs/1984MNRAS.206..377E 206, 377

    1984 doi

  12. [20]

    arXiv:2405.18126

    Euclid Collaboration et al., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2405.18126 , https://ui.adsabs.harvard.edu/abs/2024arXiv240518126E p. arXiv:2405.18126

    2024 doi

  13. [21]

    Gibelyou C., Huterer D., 2012, @doi [ ] 10.1111/j.1365-2966.2012.22032.x , https://ui.adsabs.harvard.edu/abs/2012MNRAS.427.1994G 427, 1994

    2012

  14. [22]

    M., Hivon E., Banday A

    G \'o rski K. M., Hivon E., Banday A. J., Wandelt B. D., Hansen F. K., Reinecke M., Bartelmann M., 2005, @doi [ ] 10.1086/427976 , http://adsabs.harvard.edu/abs/2005ApJ...622..759G 622, 759

    2005 doi

  15. [23]

    Guandalin C., Piat J., Clarkson C., Maartens R., 2023, @doi [ ] 10.3847/1538-4357/acdf46 , https://ui.adsabs.harvard.edu/abs/2023ApJ...953..144G 953, 144

    2023 doi

  16. [24]

    Gupta N., et al., 2016, in MeerKAT Science: On the Pathway to the SKA. p. 14 ( @eprint arXiv 1708.07371 ), @doi 10.22323/1.277.0014

    2016 arXiv

  17. [25]

    Guth A., 2012a, LECTURE NOTES 8: THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS, https://ocw.mit.edu/courses/8-07-electromagnetism-ii-fall-2012/6a67040c6a90f70c8164cdea63473a30_MIT8_07F12_ln8.pdf

    2012

  18. [26]

    Guth A., 2012b, LECTURE NOTES 9: TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS, https://ocw.mit.edu/courses/8-07-electromagnetism-ii-fall-2012/0ccc978da12536a8d95333ec7726c555_MIT8_07F12_ln9.pdf

    2012

  19. [27]

    Handley W., Lemos P., 2019a, @doi [ ] 10.1103/PhysRevD.100.023512 , https://ui.adsabs.harvard.edu/abs/2019PhRvD.100b3512H 100, 023512

    doi

  20. [28]

    Handley W., Lemos P., 2019b, @doi [ ] 10.1103/PhysRevD.100.043504 , https://ui.adsabs.harvard.edu/abs/2019PhRvD.100d3504H 100, 043504

    doi

  21. [29]

    R., et al., 2020, @doi [Nature] 10.1038/s41586-020-2649-2 , 585, 357

    Harris C. R., et al., 2020, @doi [Nature] 10.1038/s41586-020-2649-2 , 585, 357

    2020 doi

  22. [30]

    R., 2000, Cosmology

    Harrison E. R., 2000, Cosmology. The science of the universe. . Cambridge University Press

    2000

  23. [31]

    v1., ttp://hdl.handle.net/102.100.100/479788?index=1

    Hopkins A., Norris R., Vernstrom T., Kapinska A., Marvil J., 2022, ASKAP Data Products for Project AS201 (EMU): images and visibilities. v1., ttp://hdl.handle.net/102.100.100/479788?index=1

    2022

  24. [32]

    P., Jia X

    Hu J. P., Jia X. D., Hu J., Wang F. Y., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2410.06450 , https://ui.adsabs.harvard.edu/abs/2024arXiv241006450H p. arXiv:2410.06450

    2024 doi arXiv

  25. [33]

    D., 2007, @doi [Computing in Science & Engineering] 10.1109/MCSE.2007.55 , 9, 90

    Hunter J. D., 2007, @doi [Computing in Science & Engineering] 10.1109/MCSE.2007.55 , 9, 90

    2007 doi

  26. [34]

    Ivezi\` c Z ., et al., 2019, @doi [ ] 10.3847/1538-4357/ab042c , 873, 111

    2019 doi

  27. [35]

    E., Raftery A

    Kass R. E., Raftery A. E., 1995, Journal of the American Statistical Association, 90, 773

    1995

  28. [36]

    Koposov S., et al., 2023b, joshspeagle/dynesty: v2.1.2, @doi 10.5281/zenodo.7995596 , https://doi.org/10.5281/zenodo.7995596

    doi

  29. [37]

    Koposov S., et al., 2023a, joshspeagle/dynesty: v2.1.3, @doi 10.5281/zenodo.8408702 , https://doi.org/10.5281/zenodo.8408702

    doi

  30. [38]

    Kumar Aluri P., et al., 2023, @doi [Classical and Quantum Gravity] 10.1088/1361-6382/acbefc , https://ui.adsabs.harvard.edu/abs/2023CQGra..40i4001K 40, 094001

    2023 doi

  31. [39]

    Land K., et al., 2008, @doi [ ] 10.1111/j.1365-2966.2008.13490.x , https://ui.adsabs.harvard.edu/abs/2008MNRAS.388.1686L 388, 1686

    2008

  32. [40]

    Maartens R., Clarkson C., Chen S., 2018, @doi [ ] 10.1088/1475-7516/2018/01/013 , https://ui.adsabs.harvard.edu/abs/2018JCAP...01..013M 2018, 013

    2018 doi

  33. [41]

    Marocco F., et al., 2021, @doi [ ] 10.3847/1538-4365/abd805 , https://ui.adsabs.harvard.edu/abs/2021ApJS..253....8M 253, 8

    2021 doi

  34. [42]

    McConnell D., et al., 2020, @doi [ ] 10.1017/pasa.2020.41 , https://ui.adsabs.harvard.edu/abs/2020PASA...37...48M 37, e048

    2020 doi

  35. [43]

    T., Lewis G

    Mittal V., Oayda O. T., Lewis G. F., 2024, @doi [ ] 10.1093/mnras/stad3706 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.527.8497M 527, 8497

    2024 doi

  36. [44]

    T., Lewis G

    Mittal V., Oayda O. T., Lewis G. F., in prep

  37. [45]

    P., et al., 2011, @doi [ ] 10.1071/AS11021 , https://ui.adsabs.harvard.edu/abs/2011PASA...28..215N 28, 215

    Norris R. P., et al., 2011, @doi [ ] 10.1071/AS11021 , https://ui.adsabs.harvard.edu/abs/2011PASA...28..215N 28, 215

    2011 doi

  38. [46]

    P., et al., 2021, @doi [ ] 10.1017/pasa.2021.42 , https://ui.adsabs.harvard.edu/abs/2021PASA...38...46N 38, e046

    Norris R. P., et al., 2021, @doi [ ] 10.1017/pasa.2021.42 , https://ui.adsabs.harvard.edu/abs/2021PASA...38...46N 38, e046

    2021 doi

  39. [47]

    T., Lewis G

    Oayda O. T., Lewis G. F., 2023, @doi [ ] 10.1093/mnras/stad1454 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.523..667O 523, 667

    2023 doi

  40. [48]

    T., Mittal V., Lewis G

    Oayda O. T., Mittal V., Lewis G. F., Murphy T., 2024, @doi [ ] 10.1093/mnras/stae1399 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.531.4545O 531, 4545

    2024 doi

  41. [49]

    Peebles P., 2022, @doi [Annals of Physics] 10.1016/j.aop.2022.169159 , 447, 169159

    2022

  42. [50]

    J., Wilkinson D

    Peebles P. J., Wilkinson D. T., 1968, @doi [Physical Review] 10.1103/PhysRev.174.2168 , https://ui.adsabs.harvard.edu/abs/1968PhRv..174.2168P 174, 2168

    1968 doi

  43. [51]

    Pirani F. A. E., 1965, in , Vol. 1, Lectures on General Relativity. Prentice-Hall, pp 249--373

    1965

  44. [52]

    Planck Collaboration et al., 2020, @doi [ ] 10.1051/0004-6361/201833880 , https://ui.adsabs.harvard.edu/abs/2020A&A...641A...1P 641, A1

    2020 doi

  45. [53]

    J., 2013, @doi [ ] 10.1051/0004-6361/201321215 , 555, A117

    Rubart M., Schwarz D. J., 2013, @doi [ ] 10.1051/0004-6361/201321215 , 555, A117

    2013 doi

  46. [54]

    Santos M., et al., 2015, in Advancing Astrophysics with the Square Kilometre Array (AASKA14). p. 19 ( @eprint arXiv 1501.03989 ), @doi 10.22323/1.215.0019

    2015 arXiv

  47. [55]

    J., Starkman G

    Schwarz D. J., Starkman G. D., Huterer D., Copi C. J., 2004, @doi [ ] 10.1103/PhysRevLett.93.221301 , https://ui.adsabs.harvard.edu/abs/2004PhRvL..93v1301S 93, 221301

    2004 doi

  48. [56]

    J., von Hausegger S., Rameez M., Mohayaee R., Sarkar S., Colin J., 2021, @doi [ ] 10.3847/2041-8213/abdd40 , https://ui.adsabs.harvard.edu/abs/2021ApJ...908L..51S 908, L51

    Secrest N. J., von Hausegger S., Rameez M., Mohayaee R., Sarkar S., Colin J., 2021, @doi [ ] 10.3847/2041-8213/abdd40 , https://ui.adsabs.harvard.edu/abs/2021ApJ...908L..51S 908, L51

    2021 doi

  49. [57]

    J., von Hausegger S., Rameez M., Mohayaee R., Sarkar S., 2022, @doi [ ] 10.3847/2041-8213/ac88c0 , https://ui.adsabs.harvard.edu/abs/2022ApJ...937L..31S 937, L31

    Secrest N. J., von Hausegger S., Rameez M., Mohayaee R., Sarkar S., 2022, @doi [ ] 10.3847/2041-8213/ac88c0 , https://ui.adsabs.harvard.edu/abs/2022ApJ...937L..31S 937, L31

    2022 doi

  50. [58]

    M., Schmidt-Rubart M., Schwarz D

    Siewert T. M., Schmidt-Rubart M., Schwarz D. J., 2021, @doi [ ] 10.1051/0004-6361/202039840 , 653, A9

    2021 doi

  51. [59]

    K., 2023, @doi [ ] 10.1093/mnras/stad2161 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.524.3636S 524, 3636

    Singal A. K., 2023, @doi [ ] 10.1093/mnras/stad2161 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.524.3636S 524, 3636

    2023 doi

  52. [60]

    V., eds, American Institute of Physics Conference Series Vol

    Skilling J., 2004, in Fischer R., Preuss R., Toussaint U. V., eds, American Institute of Physics Conference Series Vol. 735, Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 24th International Workshop on Bayesian Inference and Maximum Entropy Methods...

    2004 doi

  53. [61]

    Skilling J., 2006, @doi [Bayesian Analysis] 10.1214/06-BA127 , 1, 833

    2006 doi

  54. [62]

    Virtanen P., et al., 2020, @doi [Nature Methods] 10.1038/s41592-019-0686-2 , https://rdcu.be/b08Wh 17, 261

    2020 doi

  55. [63]

    D., Kl\"ockner H.-R., Schwarz D

    Wagenveld J. D., Kl\"ockner H.-R., Schwarz D. J., 2023, @doi [A&A] 10.1051/0004-6361/202346210 , 675, A72

    2023 doi

  56. [64]

    D., et al., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2408.16619 , https://ui.adsabs.harvard.edu/abs/2024arXiv240816619W p

    Wagenveld J. D., et al., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2408.16619 , https://ui.adsabs.harvard.edu/abs/2024arXiv240816619W p. arXiv:2408.16619

    2024 doi arXiv

  57. [65]

    R., 2004, @doi [arXiv e-prints] 10.48550/arXiv.astro-ph/0412231 , https://ui.adsabs.harvard.edu/abs/2004astro.ph.12231W pp astro--ph/0412231

    Weeks J. R., 2004, @doi [arXiv e-prints] 10.48550/arXiv.astro-ph/0412231 , https://ui.adsabs.harvard.edu/abs/2004astro.ph.12231W pp astro--ph/0412231

    2004 doi arXiv

  58. [66]

    Yoon M., Huterer D., 2015, @doi [ ] 10.1088/2041-8205/813/1/L18 , https://ui.adsabs.harvard.edu/abs/2015ApJ...813L..18Y 813, L18

    2015 doi

  59. [67]

    Zonca A., Singer L., Lenz D., Reinecke M., Rosset C., Hivon E., Gorski K., 2019, @doi [Journal of Open Source Software] 10.21105/joss.01298 , 4, 1298

    2019 doi

  60. [68]

    von Hausegger S., 2024, @doi [ ] 10.1093/mnrasl/slae092 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.535L..49V 535, L49

    2024 doi

Pith tools

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