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REVIEW 3 major objections 4 minor 58 references

Towards detecting Primordial non-Gaussianity in the CMB using Spherical Convolutional Neural Networks

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Spherical CNNs trained on full-sky CMB maps can measure the local non-Gaussianity parameter $f_{\rm NL}$ with errors within about ten percent of the optimal bispectrum estimator at low resolution.

desk verdict A careful proof-of-concept that spherical CNNs can estimate fNL from full-sky maps within ~10-27% of KSW at low nside, but the flat-sky 'impossible' result is shrinkage, not a fluke, and the paper should say so. read the letter →

arxiv 2412.12377 v1 pith:PCV2EKNL submitted 2024-12-16 astro-ph.CO

Jorik Melsen , Thomas Flöss , P. Daniel Meerburg This is my paper · ORCID
classification astro-ph.CO PACS 98.70.Vc98.80.Es07.05.Mh
keywords primordialnon-GaussianityCMBsphericalconvolutionalneuralnetworksDeepSphereHEALPixlocalfNLbispectrumestimatorFisherforecast
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether a spherical convolutional neural network can measure the primordial non-Gaussianity parameter $f_{\rm NL}$ directly from full-sky cosmic microwave background maps, bypassing the bispectrum statistics that traditional estimators compute. On simulated maps, the graph-based DeepSphere architecture recovers $f_{\rm NL}$ with Gaussian-test error within roughly ten percent of the optimal KSW bispectrum error at nside 16 to 64, and within about 24 percent at nside 128. Adding noise or a galactic mask degrades this only mildly, and training on a larger set of independent simulations brings nside 64 to parity with the optimal error. A reader should care because the method is a proof of concept that map-level learning can approach the performance of a theoretically optimal estimator, opening the door to non-Gaussian signals for which no efficient optimal estimator exists.

What carries the argument

The load-bearing object is the DeepSphere graph-based spherical convolution layer: a HEALPix map is represented as a graph whose nodes are pixels, and convolution assigns one learned weight per ring of neighbours at a given radius, making the operation rotation-equivariant and avoiding the artificial zero-padding of pixel-based HEALPix CNNs. The network is trained with MSE loss on $f_{\rm NL}$ and compared against the Komatsu–Spergel–Wandelt (KSW) bispectrum estimator, whose Fisher error at $f_{\rm NL}=0$ serves as the optimality floor for the whole comparison.

What would settle it

Expand the flat-sky experiment of Appendix C to a much larger Gaussian test set, for example 100,000 independently simulated maps, and check whether the CNN's RMSE stays below the KSW/Fisher value of 90; if the sub-optimal result persists, the Fisher floor used to judge the spherical models is miscalibrated. A complementary check is to recompute the Fisher errors with an independent bispectrum implementation and compare the two floors directly.

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Extended reading notes

Core claim

The paper's central claim is that a rotation-equivariant graph CNN on HEALPix spheres can learn the mapping from a full-sky temperature map to the local $f_{\rm NL}$ amplitude, and that its error is bounded by the same Fisher/KSW floor that limits the optimal bispectrum estimator. In the clean noiseless case, the best DeepSphere models give Gaussian-test RMSEs of 206, 103, 52 and 28 at nside 16, 32, 64 and 128, against KSW errors of 189, 93, 47 and 22; the masked and noisy versions follow the same pattern, and with 40,500 training maps from 900 independent seeds the nside-64 RMSE falls to 47, matching the bispectrum bound. The paper reads the residual gap at nside 128 and the run-to-run spread in performance as a training-data limitation rather than a fundamental architectural one. It treats the single flat-sky result that lands slightly below the Fisher bound (RMSE 85 against 90) as a finite-test-set fluctuation, not as evidence that the benchmark itself is wrong.

Load-bearing premise

The argument depends on the KSW/Fisher bispectrum error being the true optimal floor, because every 'near-optimal' claim measures distance to that number; the single CNN result that falls below it (flat-sky RMSE 85 versus Fisher 90) is attributed to finite test-set noise, so if that attribution is wrong the benchmark is miscalibrated and the spherical comparisons inherit the error.

Editorial extensions

If this is right

  • With more independent training data, DeepSphere reaches the optimal bispectrum error at nside 64 and improves at nside 128, which indicates the avenue to full optimality is data volume rather than architecture.
  • CNN errors stay roughly flat across the full $f_{\rm NL}\in[-1000,1000]$ range, while KSW degrades away from $f_{\rm NL}=0$, so map-level learning is the more robust estimator when the signal is large or the target shape is not captured by a separable bispectrum.
  • Noise and masking cause only a mild performance drop, and for noisy nside-64 maps the Gaussian-test error is on par with the optimal value, which matters for eventual application to real survey data.
  • DeepSphere outperforms the pixel-based HEALPix CNN in every tested scenario, suggesting that rotation-equivariant graph convolutions are the right inductive bias for full-sky CMB analysis.
  • Because the models were trained only on local non-Gaussianity in temperature maps, the direct corollary is that the same approach should be tried on other shapes and on polarization data, where the paper notes constraints improve for non-squeezed shapes.

Reading between the lines

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

  • A sharp test of the paper's benchmark logic is the flat-sky anomaly: if an RMSE below the Fisher value of 90 reproduces on a much larger test set, then the KSW/Fisher floor used to certify the spherical models is itself in question, a possibility the paper mentions only as test-set noise.
  • The 400-seed training set with random rotations means the quoted errors benefit from shared large-scale structure across maps; a more demanding and realistic test would generate independent transfer-function realisations for each training map, likely raising the error and lowering the current near-optimality estimates.
  • If near-optimality survives higher resolution and independent simulations, the natural next step is to train one network to estimate several non-Gaussian shapes at once, since a CNN does not need a separable template and could in principle separate local, equilateral and orthogonal contributions from one map.
  • The biggest payoff would come at the trispectrum level or for shapes with no efficient estimator; a concrete pilot would be to train on maps with a specific beyond-bispectrum signal and compare the CNN's recovery of that amplitude against whatever suboptimal estimator currently exists.
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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

3 major / 4 minor

Summary. The paper trains two spherical CNN architectures (DeepSphere and a pixel-based HEALPix CNN) to estimate the local primordial non-Gaussianity parameter fNL directly from simulated full-sky CMB temperature maps, at HEALPix resolutions nside 16, 32, 64, and 128. The models are trained with an MSE loss on maps with fNL drawn uniformly from [-1000,1000], and evaluated on an independent Gaussian test set (fNL=0) as well as on a full-range test set. The main quantitative claim is that the best DeepSphere models achieve Gaussian-test RMSEs within 10% of the KSW/bispectrum Fisher error for nside 16-64 and within about 24-34% at nside 128, with similar behavior for noisy and masked maps. An additional flat-sky experiment reports a CNN RMSE (85) slightly below the bispectrum Fisher error (90), which the paper dismisses as a finite-test-set artifact. The paper concludes that spherical CNNs are a promising complement to traditional bispectrum estimators.

Significance. If the near-optimality claim were established, this would be a valuable proof of concept for using map-level machine-learning estimators for primordial non-Gaussianity, with potential advantages for shapes beyond the bispectrum and for scaling to larger datasets. The paper has several strengths: the CNN is compared against an external estimator (KSW) applied to the same maps, training and test sets use disjoint underlying seeds, multiple training runs are reported in the appendices, and the authors explicitly discuss bias and attempt debiasing. However, the central quantitative claim rests on comparing the CNN's MSE to the unbiased Fisher/KSW bound, and this comparison is not valid for a shrinkage-type estimator trained with an MSE loss on a broad uniform prior. The flat-sky result in Appendix C is, in fact, the expected signature of shrinkage toward the prior mean, not an impossibility. The paper's own discussion therefore highlights a load-bearing methodological gap that must be addressed before the headline claim can be accepted.

major comments (3)
  1. [Section 5, Tables 1-3; Section 4, Eq. (2)] This is the load-bearing issue: without this correction, the headline 'near-optimal' claim conflates an unbiased error bound with the MSE of a shrinkage estimator.
  2. [Appendix C, Table C1] This comment is a concrete test: the authors can compute the expected RMSE of the posterior-mean estimator under the uniform prior for the flat-sky model and compare it to 85.
  3. [Section 6 and Appendix A] This is a presentation issue that affects the reliability of the headline numbers.
minor comments (4)
  1. [Appendix A, Table A8 caption] The caption of Table A8 says 'Results of Deepsphere on the noisy CMB maps after debiasing,' but the table corresponds to the masked dataset (the values match Table 3 and Table A3). The caption should read 'masked' instead of 'noisy'.
  2. [Section 6, paragraph on training data] The text says 'the data only contains 10000 test maps,' but in context this should be 'training maps' (Section 4 states 10000 training maps). This typo makes the argument about insufficient training data confusing.
  3. [Appendix A, debiasing procedure] The debiasing in Appendix A subtracts the mean prediction on the Gaussian test set from the model outputs. Because the bias is estimated on the same test set used to compute the RMSE, this slightly optimistic procedure should either be performed on a separate validation set or explicitly justified as a negligible correction.
  4. [Section 3, Eq. (8)] The notation a^T_L;lm and a^T_NL;lm in Eq. (8) is clear, but the transfer function alpha^X_l(r) in Eq. (7) is written with a superscript X that is never explicitly defined for the temperature case; a brief definition would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CNN predictions are compared against an independently computed KSW/Fisher benchmark on disjoint test maps, and no fitted parameter is renamed as a prediction.

full rationale

The paper's central claim is an empirical benchmark comparison: spherical CNNs are trained on simulated CMB maps and their RMSE is compared with the KSW bispectrum estimator or the bispectrum Fisher error. The training pipeline (Section 4) draws f_NL uniformly from [-1000,1000], applies equation (8) to inject non-Gaussianity, and trains with MSE loss; the test maps use disjoint seeds from the training and validation seeds and are held out. The KSW error in Tables 1-3 is obtained from the external implementation of Duivenvoorden (2020) applied to the same test maps, so the benchmark is not constructed from the CNN outputs and the CNN is not fitted to the benchmark. The Fisher forecast is a bound derived from the bispectrum signal, not from the network. The Appendix C flat-sky result below the Fisher value is attributed to finite test-set noise; whether that attribution is correct is a statistical question about the benchmark comparison, not a case of the prediction being equivalent to its input by construction. Self-references (e.g., Flöss & Meerburg 2024, Kalaja et al. 2021) appear only as background citations and do not carry the load of the derivation. The shrinkage or prior concern raised by a skeptical reader concerns the interpretation of RMSE against an unbiased bound, but it does not make the derivation circular: the CNN is not defined in terms of KSW values, and no equation reduces the claimed prediction to the input benchmark.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central benchmark rests on standard cosmology assumptions: the local fNL parametrization, the correctness of the published Elsner and Wandelt map-generation pipeline, the optimality of the bispectrum at fNL=0, and the correctness of the KSW implementation. The hand-chosen items are ML hyperparameters and the ad hoc noise model. No new physical entities are introduced.

free parameters (3)
  • fNL training range = [-1000, 1000]; later [-200, 200] for nside 128
    Chosen by hand. It sets the regression prior, produces the edge effects near |fNL|=1000, and affects the Gaussian-test error through regression to the mean.
  • Noise power spectrum parameters N_inst^2, l_knee, alpha_knee = 8e4 (uK arcmin)^2, 100, 3
    Chosen ad hoc and explicitly stated as 'not based on any real data'. These parameters determine the noisy-dataset benchmark and the claim of near-optimal performance under noise.
  • Network hyperparameters = weight decay 0.1, dropout 0.1/0.3, 32 feedforward nodes, DeepSphere k=2, leaky ReLU slope 0.3
    Chosen by hand and not optimized; the paper reports up to 13% variation in test error across runs of the same architecture, indicating sensitivity to these choices.
assumptions (5)
  • domain assumption The local fNL parametrization Phi = Phi_L + fNL (Phi_L^2 - <Phi_L^2>) captures the non-Gaussian signal (Eqs. 4 and 5).
    Standard local-shape model; the proof of concept is limited to this shape and the paper does not test other shapes.
  • domain assumption The Elsner and Wandelt (2009) map-generation method produces maps correct at all statistical orders.
    Invoked in Sec. 3 to justify using the 1000 public a_lm pairs. The trustworthiness of training labels depends on this simulation being unbiased.
  • standard math At fNL=0 the bispectrum (KSW) estimator is optimal, so its Fisher error is the benchmark floor.
    Standard Cramer-Rao/Fisher argument; all CNN errors in Tables 1 to 3 and B1 are judged against this bound.
  • domain assumption The KSW implementation (Duivenvoorden 2020) is correct and unbiased at fNL=0.
    The comparison is only as good as the reference estimator; consistency between KSW errors and Fisher forecasts is shown only indirectly.
  • domain assumption Simulated maps from one fiducial pipeline are representative of the fNL estimation problem.
    All training, validation, and test maps come from the same simulation family with the same power spectrum; transfer to real data is not tested.

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Cite this review

Pith. "Pith review of Towards detecting Primordial non-Gaussianity in the CMB using Spherical Convolutional Neural Networks." pith.science (2026). https://pith.science/paper/PCV2EKNL

@misc{pith2026241212377,
  author       = {Pith},
  title        = {Pith review of: Towards detecting Primordial non-Gaussianity in the CMB using Spherical Convolutional Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCV2EKNL}},
  note         = {Machine review of arXiv:2412.12377}
}
read the original abstract

This paper explores a novel application of spherical convolutional neural networks (CNNs) to detect primordial non-Gaussianity in the cosmic microwave background (CMB), a key probe of inflationary dynamics. While effective, traditional estimators encounter computational challenges, especially when considering summary statistics beyond the bispectrum. We propose spherical CNNs as an alternative, directly analysing full-sky CMB maps to overcome limitations in previous machine learning (ML) approaches that relied on data summaries. By training on simulated CMB maps with varying amplitudes of non-Gaussianity, our spherical CNN models show promising alignment with optimal error bounds of traditional methods, albeit at lower-resolution maps. While we explore several different architectures, results from DeepSphere CNNs most closely match the Fisher forecast for Gaussian test sets under noisy and masked conditions. Our study suggests that spherical CNNs could complement existing methods of non-Gaussianity detection in future datasets, provided additional training data and parameter tuning are applied. We discuss the potential for CNN-based techniques to scale with larger data volumes, paving the way for applications to future CMB data sets.

Figures

Figures reproduced from arXiv: 2412.12377 by the authors.

Figure 1
Figure 1. Scatter plots of the predicted 𝑓NL values of the best DeepSphere model on the full range test set (which contains 𝑓NL values uniformly distributed from −1000 to 1000) versus the correct 𝑓NL values. The red lines here represent a hypothetical flawless estimator. slightly worse, for the best pixel-based models on nside 16 to 64, all Gaussian test errors were within 14% of the optimal error, and for nside 128, all erro… view at source ↗

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Works this paper leans on

58 extracted references · 48 canonical work pages

  1. [1]

    N., et al., 2016, arXiv preprint arXiv:1610.02743

    Abazajian K. N., et al., 2016, arXiv preprint arXiv:1610.02743

    arXiv 2016

  2. [2]

    Ach \'u carro A., et al., 2022, arXiv preprint arXiv:2203.08128

    arXiv 2022

  3. [3]

    A., et al., 2014, Astronomy & Astrophysics, 571, A24

    Ade P. A., et al., 2014, Astronomy & Astrophysics, 571, A24

    work page 2014

  4. [4]

    A., et al., 2016, Astronomy & Astrophysics, 594, A17

    Ade P. A., et al., 2016, Astronomy & Astrophysics, 594, A17

    work page 2016

  5. [5]

    Ade P., et al., 2019, @doi [JCAP] 10.1088/1475-7516/2019/02/056 , 02, 056

    doi:10.1088/1475-7516/2019/02/056 2019

  6. [6]

    J., 1982, Physical Review Letters, 48, 1220

    Albrecht A., Steinhardt P. J., 1982, Physical Review Letters, 48, 1220

    work page 1982

  7. [7]

    M., Steinhardt P

    Bardeen J. M., Steinhardt P. J., Turner M. S., 1983, Physical Review D, 28, 679

    work page 1983

  8. [8]

    Barron D., et al., 2018, Journal of Cosmology and Astroparticle Physics, 2018, 009

    work page 2018

Show all 58 references
  1. [9]

    H., 2015, Working note available on the website http://people

    Beentjes C. H., 2015, Working note available on the website http://people. maths. ox. ac. uk/beentjes/Essays

    2015

  2. [10]

    J., Steiner F., 2017, Classical and Quantum Gravity, 34, 094002

    Buchert T., France M. J., Steiner F., 2017, Classical and Quantum Gravity, 34, 094002

    2017

  3. [11]

    M., Philcox O

    Cabass G., Ivanov M. M., Philcox O. H., Simonovi \'c M., Zaldarriaga M., 2022a, Physical Review D, 106, 043506

  4. [12]

    M., Philcox O

    Cabass G., Ivanov M. M., Philcox O. H., Simonovi \'c M., Zaldarriaga M., 2022b, Physical Review Letters, 129, 021301

  5. [13]

    Cabella P., Hansen F., Liguori M., Marinucci D., Matarrese S., Moscardini L., Vittorio N., 2006, Monthly Notices of the Royal Astronomical Society, 369, 819

    2006

  6. [14]

    B., Hobson M., Mart \' nez-Gonz \'a lez E., 2011, Monthly Notices of the Royal Astronomical Society, 416, 457

    Casaponsa B., Bridges M., Curto A., Barreiro R. B., Hobson M., Mart \' nez-Gonz \'a lez E., 2011, Monthly Notices of the Royal Astronomical Society, 416, 457

    2011

  7. [15]

    Castorina E., et al., 2019, Journal of Cosmology and Astroparticle Physics, 2019, 010

    2019

  8. [16]

    Collaboration P., et al., 2020, Astronomy & Astrophysics, 641, 1

    2020

  9. [17]

    D'Amico G., Lewandowski M., Senatore L., Zhang P., 2022, arXiv preprint arXiv:2201.11518

    2022 arXiv

  10. [18]

    R., Healy D

    Driscoll J. R., Healy D. M., 1994, Advances in applied mathematics, 15, 202

    1994

  11. [19]

    Ducout A., Bouchet F., Colombi S., Pogosyan D., Prunet S., 2013, Monthly Notices of the Royal Astronomical Society, 429, 2104

    2013

  12. [20]

    Duivenvoorden A., 2020, ksw, https://github.com/AdriJD/ksw

    2020

  13. [21]

    D., 2009, The Astrophysical Journal Supplement Series, 184, 264

    Elsner F., Wandelt B. D., 2009, The Astrophysical Journal Supplement Series, 184, 264

    2009

  14. [22]

    D., 2024, Journal of Cosmology and Astroparticle Physics, 2024, 031

    Fl \"o ss T., Meerburg P. D., 2024, Journal of Cosmology and Astroparticle Physics, 2024, 031

    2024

  15. [23]

    D., 2023, Physical Review D, 107, 023528

    Fl \"o ss T., Biagetti M., Meerburg P. D., 2023, Physical Review D, 107, 023528

    2023

  16. [24]

    Gleyzes J., De Putter R., Green D., Dor \'e O., 2017, Journal of Cosmology and Astroparticle Physics, 2017, 002

    2017

  17. [25]

    Goodfellow I., Bengio Y., Courville A., 2016, Deep learning

    2016

  18. [26]

    M., Hivon E., Banday A

    Gorski K. M., Hivon E., Banday A. J., Wandelt B. D., Hansen F. K., Reinecke M., Bartelmann M., 2005, The Astrophysical Journal, 622, 759

    2005

  19. [27]

    H., 1981, Physical Review D, 23, 347

    Guth A. H., 1981, Physical Review D, 23, 347

    1981

  20. [28]

    Hikage C., Komatsu E., Matsubara T., 2006, The Astrophysical Journal, 653, 11

    2006

  21. [29]

    K., Matarrese S., 2008, Monthly Notices of the Royal Astronomical Society, 389, 1439

    Hikage C., Matsubara T., Coles P., Liguori M., Hansen F. K., Matarrese S., 2008, Monthly Notices of the Royal Astronomical Society, 389, 1439

    2008

  22. [30]

    Ho S., et al., 2015, Journal of Cosmology and Astroparticle Physics, 2015, 040

    2015

  23. [31]

    D., Pimentel G

    Kalaja A., Meerburg P. D., Pimentel G. L., Coulton W. R., 2021, @doi [JCAP] 10.1088/1475-7516/2021/04/050 , 04, 050

    2021 doi

  24. [32]

    P., 2014, arXiv preprint arXiv:1412.6980

    Kingma D. P., 2014, arXiv preprint arXiv:1412.6980

    2014 arXiv

  25. [33]

    N., 2001, Physical Review D, 63, 063002

    Komatsu E., Spergel D. N., 2001, Physical Review D, 63, 063002

    2001

  26. [34]

    D., Spergel D

    Komatsu E., Wandelt B. D., Spergel D. N., Banday A. J., Gorski K. M., 2002, The Astrophysical Journal, 566, 19

    2002

  27. [35]

    Komatsu E., et al., 2003, The Astrophysical Journal Supplement Series, 148, 119

    2003

  28. [36]

    N., Wandelt B

    Komatsu E., Spergel D. N., Wandelt B. D., 2005, The Astrophysical Journal, 634, 14

    2005

  29. [37]

    Krachmalnicoff N., Tomasi M., 2019, Astronomy & Astrophysics, 628, A129

    2019

  30. [38]

    V., Roth N., 2014, Physical Review Letters, 113, 221301

    Leistedt B., Peiris H. V., Roth N., 2014, Physical Review Letters, 113, 221301

    2014

  31. [39]

    Liguori M., Matarrese S., Moscardini L., 2003, The Astrophysical Journal, 597, 57

    2003

  32. [40]

    K., Komatsu E., Matarrese S., Wandelt B., 2007, Physical Review D, 76, 105016

    Liguori M., Yadav A., Hansen F. K., Komatsu E., Matarrese S., Wandelt B., 2007, Physical Review D, 76, 105016

    2007

  33. [41]

    D., 1982, Physics Letters B, 108, 389

    Linde A. D., 1982, Physics Letters B, 108, 389

    1982

  34. [42]

    D., Wiaux Y., 2011, IEEE Transactions on Signal Processing, 59, 5876

    McEwen J. D., Wiaux Y., 2011, IEEE Transactions on Signal Processing, 59, 5876

    2011

  35. [43]

    D., et al., 2019, arXiv preprint arXiv:1903.04409

    Meerburg P. D., et al., 2019, arXiv preprint arXiv:1903.04409

    2019 arXiv

  36. [44]

    Mueller E.-M., et al., 2021, arXiv preprint arXiv:2106.13725

    2021 arXiv

  37. [45]

    G., Ma Y.-Z., 2024, Monthly Notices of the Royal Astronomical Society, 529, 3289

    Nagarajappa C. G., Ma Y.-Z., 2024, Monthly Notices of the Royal Astronomical Society, 529, 3289

    2024

  38. [46]

    Novaes C., Bernui A., Ferreira I., Wuensche C., 2015, Journal of Cosmology and Astroparticle Physics, 2015, 064

    2015

  39. [47]

    Oppizzi F., Liguori M., Renzi A., Arroja F., Bartolo N., 2018, Journal of Cosmology and Astroparticle Physics, 2018, 045

    2018

  40. [48]

    Perraudin N., Defferrard M., Kacprzak T., Sgier R., 2019, Astronomy and Computing, 27, 130

    2019

  41. [49]

    Planck Collaboration 2014, Planck map: HFI Mask GalPlane-apo2 2048 R2.00, https://irsa.ipac.caltech.edu/data/Planck/release_2/ancillary-data/previews/HFI_Mask_GalPlane-apo2_2048_R2.00/index.html

    2014

  42. [50]

    J., et al., 2013, Monthly Notices of the Royal Astronomical Society, 428, 1116

    Ross A. J., et al., 2013, Monthly Notices of the Royal Astronomical Society, 428, 1116

    2013

  43. [51]

    Slosar A., Hirata C., Seljak U., Ho S., Padmanabhan N., 2008, Journal of Cosmology and Astroparticle Physics, 2008, 031

    2008

  44. [52]

    M., Zaldarriaga M., 2011, Monthly Notices of the Royal Astronomical Society, 417, 2

    Smith K. M., Zaldarriaga M., 2011, Monthly Notices of the Royal Astronomical Society, 417, 2

    2011

  45. [53]

    Starobinskii A., 1979, JETP Letters, 30, 682

    1979

  46. [54]

    A., Padilla L

    V \'a zquez J. A., Padilla L. E., Matos T., 2018, arXiv preprint arXiv:1810.09934

    2018 arXiv

  47. [55]

    F., Kamionkowski M., 2000, Monthly Notices of the Royal Astronomical Society, 313, 141

    Verde L., Wang L., Heavens A. F., Kamionkowski M., 2000, Monthly Notices of the Royal Astronomical Society, 313, 141

    2000

  48. [56]

    P., Wandelt B

    Yadav A. P., Wandelt B. D., 2008, Physical Review Letters, 100, 181301

    2008

  49. [57]

    P., Wandelt B

    Yadav A. P., Wandelt B. D., 2010, Advances in Astronomy, 2010, 565248

    2010

  50. [58]

    write newline

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