pith. machine review for the scientific record. sign in

arxiv: 2511.11147 · v2 · submitted 2025-11-14 · 🌌 astro-ph.CO

Recognition: 1 theorem link

· Lean Theorem

Galactic foreground residue biases in cosmic-microwave-background lensing-convergence reconstruction and delensing of B-mode maps

Kishan Deka , Pawel Bielewicz
Authors on Pith no claims yet

Pith reviewed 2026-05-17 22:37 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords CMB lensingforeground residualsB-mode delensingtensor-to-scalar ratioGalactic emissioncomponent separationcosmic variance
0
0 comments X

The pith

Galactic foreground residuals after cleaning bias CMB lensing reconstruction mostly through their Gaussian components, which remain comparable to cosmic variance and must be corrected for accurate B-mode delensing.

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

This paper uses realistic simulations to show how leftover Galactic emission after multi-frequency cleaning affects reconstruction of the CMB lensing potential and the removal of lensing from B-mode polarization maps. The central finding is that Gaussian parts of the residuals dominate the error budget in lensing reconstruction, while non-Gaussian contributions are three orders of magnitude smaller even in complex foreground models. Component separation reduces the foreground impact by roughly one order of magnitude, yet the remaining Gaussian bias reaches the level of cosmic variance on the lensing power spectrum, requiring explicit correction when constraining the tensor-to-scalar ratio. For next-generation experiments, these residuals become a leading error source once delensing efficiency exceeds about 90 percent.

Core claim

Using realistic simulations of Galactic foregrounds and multi-frequency component separation, the residual foreground contribution to CMB lensing reconstruction errors is dominated by Gaussian components of the residual maps, with errors from non-Gaussian components around three orders of magnitude smaller even for the most complex models considered. Component separation reduces the overall Galactic contribution to lensing errors by one order of magnitude. The bias from the Gaussian residuals is small but comparable to the cosmic variance limit on the lensing power spectrum, so it is corrected when delensing B-mode maps and constraining the tensor-to-scalar ratio. For a simple quadratic delc

What carries the argument

Separation of Gaussian versus non-Gaussian contributions within the residual foreground maps after component separation, used to quantify their separate impacts on the quadratic estimator for lensing convergence.

If this is right

  • Component separation is required to keep Galactic residuals from dominating lensing reconstruction errors.
  • Explicit bias correction for the Gaussian residual term is needed to reach cosmic-variance-limited lensing spectra and unbiased tensor-to-scalar ratio constraints.
  • With quadratic estimators, foreground residuals after cleaning remain two orders of magnitude below leftover lensing uncertainties in current delensing.
  • For experiments achieving 90 percent delensing efficiency, foreground residuals will become one of the dominant error sources in B-mode maps.

Where Pith is reading between the lines

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

  • Improved modeling of Gaussian foreground residuals could further reduce the bias correction step required for future delensing pipelines.
  • The results imply that tests of delensing algorithms on real data should separately quantify Gaussian and non-Gaussian residual contributions rather than treating all residuals as a single contaminant.
  • Similar Gaussian dominance may appear in other small-scale CMB analyses such as patchy reionization or cluster lensing once foreground cleaning reaches comparable depth.

Load-bearing premise

The simulations of Galactic foreground emission and multi-frequency component separation methods accurately represent the small-scale properties and residuals expected in next-generation CMB experiments.

What would settle it

Direct comparison of the measured power spectrum of small-scale non-Gaussianity in real residual maps from a CMB-S4-like experiment against the three-orders-of-magnitude suppression reported in the simulations.

Figures

Figures reproduced from arXiv: 2511.11147 by Kishan Deka, Pawel Bielewicz.

Figure 1
Figure 1. Figure 1: Analytical zeroth-order lensing reconstruction bias, N (0) L for dif￾ferent estimators given by Eq. (14). The instrumental specification is considered for a CMB-S4-like experiment with beam resolution 2.5 ar￾cminutes and white noise level in temperature ∆T = 2 µK-arcmin and polarisation ∆P = 2.8 µK-arcmin. The disconnected bias contribution term, N (0) L , is expressed analytically as, N (0)XYX′Y ′ L [Cℓ] … view at source ↗
Figure 2
Figure 2. Figure 2: Masks used in our analysis to mimic CMB-S4 sky coverage shown in Ecliptic coordinates with Galactic emission at 145 GHz in the background. The left and middle panels are considered for wide sky survey corresponding to the LATs configuration: the left panel shows mask used for component separation (fsky = 0.64) and the middle panel shows mask used for lensing reconstruction (fsky = 0.37). The right panel sh… view at source ↗
Figure 3
Figure 3. Figure 3: Beam-deconvolved noise power spectra for temperature (solid) and polarisation (dashed) in LAT configuration. Synchrotron emission arises from relativistic electrons ac￾celerated along the Galactic magnetic field. Synchrotron emis￾sion is modelled with a power law (Gold et al. 2011), I sync ν = I sync ν0 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Angular power spectra for CMB B-mode cleaned maps for the SAT configuration and sky coverage. The grey solid zig-zag lines are BB power spectra of 100 simulations and the average is shown as red stars. The theoretical input power spectra are shown as black solid line. The residual HILC noise and foreground levels are shown as dotted and dashed lines, respectively. The magenta solid line is average delensed… view at source ↗
Figure 6
Figure 6. Figure 6: Relative bias in semi-analytic N (0) L noise due to foreground con￾tamination in MF channels (solid lines) before component separation and residual foreground in HILC maps (dashed lines) after component separation for different models. We define the fractional change in the zeroth-order recon￾struction noise as, ∆N (0) L N (0) L = N (0),FG L − N (0) L N (0) L , (31) where, N (0),FG L is noise power spectru… view at source ↗
Figure 8
Figure 8. Figure 8: The mean value of the tensor-to-scalar ratio, r, and its standard deviation, σ(r), estimated from 100 simulations for the three consid￾ered foreground models and foreground-free (FG-free) maps. Vertical dashed line corresponds to the fiducial value of r. To show the off-diagonal contributions of the covariance ma￾trix, we present the corresponding correlation matrices for our three models in Fig. B.1. The … view at source ↗
Figure 9
Figure 9. Figure 9: as foreground res. temp. gives a quantitative measure of how residual foreground contamination in the B-mode template propagates into the constraint on r. The foreground res. and the noise res. terms arise from residual foreground and instrumental noise, respectively, that is present in the component-separated B￾mode SAT maps. The lensing res. term captures the remaining B￾mode power due the incomplete rem… view at source ↗
read the original abstract

Diffuse contamination from Galactic foreground emission is one of the main concerns for reconstruction of the cosmic microwave background (CMB) lensing potential for next-generation CMB polarisation experiments. Using realistic simulations, we investigated the impact of Galactic foreground residuals from multi-frequency foreground-cleaning methods on CMB lensing reconstruction and the de-lensing of B-mode maps. We also assessed how these residuals affect constraints on the tensor-to-scalar ratio for a CMB-S4--like experiment. We paid special attention to the errors coming from the small scale non-Gaussianity of the foreground residuals. We show that component separation is essential for the lensing reconstruction that reduces Galactic emission contribution to the lensing reconstruction errors by one order of magnitude. The residual foreground contribution is dominated by terms coming from Gaussian components of the residual maps. Errors coming from non-Gaussian components are around three orders of magnitude smaller than the Gaussian one, even for recent and the most complex models of the Galactic emission considered in this work. Although the bias in the reconstruction errors due to the Gaussian component of the residuals being small, it is comparable to the cosmic variance limit for the lensing power spectrum. For this reason, we corrected for this bias in the de-lensing of B-mode maps and constraining the tensor-to-scalar ratio. We also show that for the de-lensed B-mode maps with a simple quadratic estimator, that is, residuals of the Galactic emission after component separation, errors are two orders of magnitude smaller than uncertainties from leftover of the lensing signal. However, for high-sensitivity CMB experiments and more efficient de-lensing algorithms that remove up to 90% of the lensing signal, the foreground residuals will become one of the main sources of errors.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript uses realistic simulations of Galactic foregrounds (including recent complex models) and multi-frequency component separation to quantify residual biases in CMB lensing-convergence reconstruction and quadratic-estimator delensing of B-mode maps. It reports that component separation reduces foreground contamination to lensing errors by roughly one order of magnitude, that Gaussian components of the residuals dominate the error budget, and that non-Gaussian contributions are suppressed by three orders of magnitude even at small scales. The Gaussian residual bias, though small, is comparable to the cosmic-variance limit on the lensing power spectrum and is therefore corrected before delensing; after correction, foreground residuals remain two orders of magnitude below lensing residuals for a simple quadratic estimator but become a leading systematic for high-sensitivity experiments that achieve ~90 % delensing efficiency. Implications for the tensor-to-scalar ratio constraint in a CMB-S4-like survey are also assessed.

Significance. If the simulation results are robust, the work supplies a concrete, quantitative basis for foreground-mitigation strategies in next-generation CMB polarization surveys. It isolates the Gaussian versus non-Gaussian residual contributions, demonstrates that the latter are negligible, and shows that a straightforward bias correction suffices to keep foregrounds sub-dominant for current delensing algorithms. These findings directly inform survey design and analysis pipelines for CMB-S4 and similar experiments.

major comments (2)
  1. [§3] §3 (simulation and component-separation pipeline): the central claim that non-Gaussian residual contributions are three orders of magnitude smaller than Gaussian ones rests on the fidelity of the adopted Galactic emission models at ℓ ≳ 1000. No explicit validation against observed small-scale non-Gaussian statistics (e.g., dust or synchrotron bispectrum or kurtosis measurements) is presented; if the simulations under-represent the amplitude or scale dependence of non-Gaussianity, the reported suppression factor would not hold for real data.
  2. [§4.2] §4.2 (lensing-reconstruction error budget): the statement that the Gaussian residual bias is “comparable to the cosmic variance limit” and therefore requires correction is load-bearing for the subsequent delensing analysis. The precise numerical factor by which the bias exceeds or equals the cosmic-variance floor should be shown explicitly (e.g., via a table or figure comparing the two contributions as a function of multipole) rather than asserted qualitatively.
minor comments (2)
  1. [Figure 5] Figure 5 (or equivalent): the caption should explicitly state the multipole range over which the power spectra are averaged and whether the plotted curves include the full covariance or only diagonal errors.
  2. Notation: the symbol for the residual foreground map after component separation is introduced without a clear definition; a single consistent symbol (e.g., f_res) should be used throughout the text and equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the clarity and robustness of our presentation. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (simulation and component-separation pipeline): the central claim that non-Gaussian residual contributions are three orders of magnitude smaller than Gaussian ones rests on the fidelity of the adopted Galactic emission models at ℓ ≳ 1000. No explicit validation against observed small-scale non-Gaussian statistics (e.g., dust or synchrotron bispectrum or kurtosis measurements) is presented; if the simulations under-represent the amplitude or scale dependence of non-Gaussianity, the reported suppression factor would not hold for real data.

    Authors: We agree that the fidelity of the foreground models at small scales is central to the robustness of the reported suppression of non-Gaussian residuals. The simulations employ the most recent complex Galactic emission models available, which incorporate non-Gaussian features calibrated to existing multi-frequency observations. Direct high-ℓ bispectrum or kurtosis measurements remain limited in the literature, so our results are framed as the impact under current best models. In the revised manuscript we have expanded the discussion in §3 to cite additional validation studies of these models against available small-scale statistics and to explicitly note the reliance on model assumptions. revision: partial

  2. Referee: [§4.2] §4.2 (lensing-reconstruction error budget): the statement that the Gaussian residual bias is “comparable to the cosmic variance limit” and therefore requires correction is load-bearing for the subsequent delensing analysis. The precise numerical factor by which the bias exceeds or equals the cosmic-variance floor should be shown explicitly (e.g., via a table or figure comparing the two contributions as a function of multipole) rather than asserted qualitatively.

    Authors: We thank the referee for highlighting this point. We have added a new figure (Figure 5 in the revised manuscript) that directly compares the Gaussian residual bias to the cosmic-variance limit on the lensing power spectrum as a function of multipole. The figure shows that the bias lies within a factor of approximately 1–2 of the cosmic-variance floor over the multipole range relevant for delensing, thereby justifying the bias correction applied in the subsequent analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct simulation comparisons

full rationale

The paper derives its central claims (Gaussian residuals dominate lensing errors by three orders of magnitude over non-Gaussian terms; bias correction applied to delensing) exclusively from comparisons of simulated foreground residuals against cosmic-variance limits. No equations, fitted parameters, or self-citations are shown to reduce any reported prediction or bias to the input simulations by construction. The analysis remains self-contained against external simulation benchmarks with no self-definitional, fitted-input, or uniqueness-imported steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on the assumption that the chosen Galactic emission models and component-separation pipelines produce residuals representative of real observations; no explicit free parameters are stated in the abstract.

axioms (1)
  • domain assumption Simulations of Galactic foregrounds and multi-frequency cleaning accurately capture the Gaussian and non-Gaussian properties relevant to lensing reconstruction at small scales.
    Invoked when concluding that non-Gaussian errors remain negligible even for complex models.

pith-pipeline@v0.9.0 · 5623 in / 1299 out tokens · 32293 ms · 2026-05-17T22:37:11.743560+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages · 3 internal anchors

  1. [1]

    CMB-S4 Science Case, Reference Design, and Project Plan

    Abazajian, K., Addison, G., et al. 2019 [arXiv:1907.04473]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  2. [2]

    Abbott, L. F. & Wise, M. B. 1984, Nucl. Phys. B, 244, 541

    work page 1984

  3. [3]

    H., Adachi, S., Ade, P., et al

    Abitbol, M. H., Adachi, S., Ade, P., et al. 2019 [arXiv:1907.08284]

    work page arXiv 2019

  4. [4]

    J., Sherwin, B

    Abril-Cabezas, I., Qu, F. J., Sherwin, B. D., et al. 2025, Phys. Rev. D, 112, 023522

    work page 2025

  5. [5]

    Ade, P. A. R., Ahmed, Z., Amiri, M., et al. 2021, Phys. Rev. D, 103, 022004

    work page 2021

  6. [6]

    2025, The Open Journal of Astrophysics, 8 Baleato Lizancos, A., Challinor, A., & Carron, J

    Alonso, D. 2025, The Open Journal of Astrophysics, 8 Baleato Lizancos, A., Challinor, A., & Carron, J. 2021a, Journal of Cosmology and Astroparticle Physics, 2021, 016 Baleato Lizancos, A., Challinor, A., & Carron, J. 2021b, Physical Review D, 103 Baleato Lizancos, A., Challinor, A., Sherwin, B. D., & Namikawa, T. 2022, Monthly Notices of the Royal Astron...

    work page 2025

  7. [7]

    2020, Journal of Cosmology and Astroparti- cle Physics, 2020, 030–030

    Beck, D., Errard, J., & Stompor, R. 2020, Journal of Cosmology and Astroparti- cle Physics, 2020, 030–030

    work page 2020

  8. [8]

    2024, The Astrophysical Journal, 964, 148

    Belkner, S., Carron, J., Legrand, L., et al. 2024, The Astrophysical Journal, 964, 148

    work page 2024

  9. [9]

    L., Larson, D., Weiland, J

    Bennett, C. L., Larson, D., Weiland, J. L., et al. 2013, The Astrophysical Journal Supplement Series, 208, 20 Benoit-Lévy, A., Déchelette, T., Benabed, K., et al. 2013, A&A, 555, A37 BICEP/KeckXIII. 2021, Physical Review Letters, 127 BICEP/KeckXII. 2021, Physical Review D, 103

    work page 2013

  10. [10]

    E., Delabrouille, J., et al

    Borrill, J., Clark, S. E., Delabrouille, J., et al. 2025 [arXiv:2502.20452]

    work page arXiv 2025

  11. [11]

    F., Zaldarriaga, M., Tegmark, M., & de Oliveira-Costa, A

    Bunn, E. F., Zaldarriaga, M., Tegmark, M., & de Oliveira-Costa, A. 2003, Phys- ical Review D, 67

    work page 2003

  12. [12]

    & Lewis, A

    Carron, J. & Lewis, A. 2017, Physical Review D, 96

    work page 2017

  13. [13]

    2017, Journal of Cosmology and Astropar- ticle Physics, 2017, 035–035

    Carron, J., Lewis, A., & Challinor, A. 2017, Journal of Cosmology and Astropar- ticle Physics, 2017, 035–035

    work page 2017

  14. [14]

    2022, Journal of Cosmology and As- troparticle Physics, 2022, 039

    Carron, J., Mirmelstein, M., & Lewis, A. 2022, Journal of Cosmology and As- troparticle Physics, 2022, 039

    work page 2022

  15. [15]

    & Lewis, A

    Challinor, A. & Lewis, A. 2005, Physical Review D, 71 DESI Collaboration. 2016, The DESI Experiment Part I: Science,Targeting, and Survey Design

    work page 2005

  16. [16]

    K., Banday, A

    Eriksen, H. K., Banday, A. J., Gorski, K. M., & Lilje, P. B. 2004, The Astrophys- ical Journal, 612, 633–646

    work page 2004

  17. [17]

    & Pollock, M

    Fabbri, R. & Pollock, M. 1983, Physics Letters B, 125, 445

    work page 1983

  18. [18]

    2012, Journal of Cosmology and Astroparticle Physics, 2012, 017–017

    Fantaye, Y ., Baccigalupi, C., Leach, S., & Yadav, A. 2012, Journal of Cosmology and Astroparticle Physics, 2012, 017–017

    work page 2012

  19. [19]

    2012, Astroparticle Physics, 36, 57–63

    Fauvet, L., Macías-Pérez, J., & Désert, F. 2012, Astroparticle Physics, 36, 57–63

    work page 2012

  20. [20]

    & Hill, J

    Ferraro, S. & Hill, J. C. 2018, Phys. Rev. D, 97, 023512

    work page 2018

  21. [21]

    J., Kogut, A., Levin, S., et al

    Fixsen, D. J., Kogut, A., Levin, S., et al. 2011, The Astrophysical Journal, 734, 5

    work page 2011

  22. [22]

    L., et al

    Gold, B., Odegard, N., Weiland, J. L., et al. 2011, The Astrophysical Journal Supplement Series, 192, 15 Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, The Astrophysical Journal, 622, 759–771

    work page 2011

  23. [23]

    2009, Phys

    Grain, J., Tristram, M., & Stompor, R. 2009, Phys. Rev. D, 79, 123515

    work page 2009

  24. [24]

    Guth, A. H. & Pi, S. Y . 1982, Phys. Rev. Lett., 49, 1110

    work page 1982

  25. [25]

    & Lewis, A

    Hamimeche, S. & Lewis, A. 2008, Phys. Rev. D, 77, 103013

    work page 2008

  26. [26]

    2020 [arXiv:2007.14405]

    Han, D., Sehgal, N., MacInnis, A., et al. 2020 [arXiv:2007.14405]

    work page arXiv 2020

  27. [27]

    2011, Physical Review D, 83

    Hanson, D., Challinor, A., Efstathiou, G., & Bielewicz, P. 2011, Physical Review D, 83

    work page 2011

  28. [28]

    & Lewis, A

    Hanson, D. & Lewis, A. 2009, Phys. Rev. D, 80, 063004

    work page 2009

  29. [29]

    2010, Physical Review D, 81

    Hanson, D., Lewis, A., & Challinor, A. 2010, Physical Review D, 81

    work page 2010

  30. [30]

    2009, Monthly Notices of the Royal Astro- nomical Society, 400, 2169–2173

    Hanson, D., Rocha, G., & Górski, K. 2009, Monthly Notices of the Royal Astro- nomical Society, 400, 2169–2173

    work page 2009

  31. [31]

    Haslam, C. G. T., Salter, C. J., Stoffel, H., & Wilson, W. E. 1982, A&AS, 47, 1

    work page 1982

  32. [32]

    Hazumi, M., Ade, P. A. R., Akiba, Y ., et al. 2019, Journal of Low Temperature Physics, 194, 443

    work page 2019

  33. [33]

    2024, Physical Review D, 110

    Hertig, E., Wolz, K., Namikawa, T., et al. 2024, Physical Review D, 110

    work page 2024

  34. [34]

    Hildebrand, R. H. 1988, QJRAS, 29, 327

    work page 1988

  35. [35]

    M., Netterfield, C

    Hivon, E., Gorski, K. M., Netterfield, C. B., et al. 2002, The Astrophysical Jour- nal, 567, 2–17

    work page 2002

  36. [36]

    & Okamoto, T

    Hu, W. & Okamoto, T. 2002, The Astrophysical Journal, 574, 566–574

    work page 2002

  37. [37]

    1997, Phys

    Kamionkowski, M., Kosowsky, A., & Stebbins, A. 1997, Phys. Rev. D, 55, 7368

    work page 1997

  38. [38]

    2002, Physical Review Letters, 89

    Kesden, M., Cooray, A., & Kamionkowski, M. 2002, Physical Review Letters, 89

    work page 2002

  39. [39]

    2003, Physical Review D, 67

    Kesden, M., Cooray, A., & Kamionkowski, M. 2003, Physical Review D, 67

    work page 2003

  40. [40]

    Kim, J., Naselsky, P., & Christensen, P. R. 2009, Physical Review D, 79

    work page 2009

  41. [41]

    & Challinor, A

    Lewis, A. & Challinor, A. 2006, Physics Reports, 429, 1–65

    work page 2006

  42. [42]

    2000, The Astrophysical Journal, 538, 473–476

    Lewis, A., Challinor, A., & Lasenby, A. 2000, The Astrophysical Journal, 538, 473–476

    work page 2000

  43. [43]

    2001, Physical Review D, 65

    Lewis, A., Challinor, A., & Turok, N. 2001, Physical Review D, 65

    work page 2001

  44. [44]

    Linde, A. D. 1983, Phys. Lett. B, 129, 177 LSST Science Collaboration. 2009 [arXiv:0912.0201]

    work page internal anchor Pith review Pith/arXiv arXiv 1983

  45. [45]

    2018, Physical Review D, 97

    Manzotti, A. 2018, Physical Review D, 97

    work page 2018

  46. [46]

    T., Wu, W

    Manzotti, A., Story, K. T., Wu, W. L. K., et al. 2017, The Astrophysical Journal, 846, 45

    work page 2017

  47. [47]

    2022, Journal of Cosmology and Astroparticle Physics, 2022, 003 Martínez-Solaeche, G., Karakci, A., & Delabrouille, J

    Martire, F., Barreiro, R., & Martínez-González, E. 2022, Journal of Cosmology and Astroparticle Physics, 2022, 003 Martínez-Solaeche, G., Karakci, A., & Delabrouille, J. 2018, Monthly Notices of the Royal Astronomical Society, 476, 1310–1330

    work page 2022

  48. [48]

    2017, Physical Review D, 95

    Namikawa, T. 2017, Physical Review D, 95

    work page 2017

  49. [49]

    2022, Physical Review D, 105

    Namikawa, T., Baleato Lizancos, A., Robertson, N., et al. 2022, Physical Review D, 105

    work page 2022

  50. [50]

    Okamoto, T. & Hu, W. 2003, Physical Review D, 67 Planck Collaboration I. 2020, Astronomy & Astrophysics, 641, A1 Planck Collaboration int. XXII. 2015, Astronomy &; Astrophysics, 576, A107 Planck Collaboration IV. 2020, Astronomy &; Astrophysics, 641, A4 Planck Collaboration V. 2020, Astronomy &; Astrophysics, 641, A5 Planck Collaboration VI. 2020, Astrono...

    work page 2003

  51. [51]

    Polnarev, A. G. 1985, Soviet Ast., 29, 607

    work page 1985

  52. [52]

    2023, Astronomy &; Astrophysics, 678, A165

    Reinecke, M., Belkner, S., & Carron, J. 2023, Astronomy &; Astrophysics, 678, A165

    work page 2023

  53. [53]

    An improved source-subtracted and destriped 408 MHz all-sky map

    Remazeilles, M., Dickinson, C., Banday, A. J., Bigot-Sazy, M. A., & Ghosh, T. 2015 [arXiv:1411.3628]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  54. [54]

    2023, Physical Review D, 107

    Sailer, N., Ferraro, S., & Schaan, E. 2023, Physical Review D, 107

    work page 2023

  55. [55]

    2021, Phys

    Sailer, N., Schaan, E., Ferraro, S., Darwish, O., & Sherwin, B. 2021, Phys. Rev. D, 104, 123514

    work page 2021

  56. [56]

    T., Reichardt, C

    Sayre, J. T., Reichardt, C. L., Henning, J. W., et al. 2020, Physical Review D, 101

    work page 2020

  57. [57]

    & Ferraro, S

    Schaan, E. & Ferraro, S. 2019, Phys. Rev. Lett., 122, 181301

    work page 2019

  58. [58]

    & Hirata, C

    Seljak, U. & Hirata, C. M. 2004, Physical Review D, 69

    work page 2004

  59. [59]

    Seljak, U. b. u. & Zaldarriaga, M. 1997, Phys. Rev. Lett., 78, 2054

    work page 1997

  60. [60]

    Sherwin, B. D. & Schmittfull, M. 2015, Physical Review D, 92

    work page 2015

  61. [61]

    2015, The Astrophysical Journal, 807, 166

    Simard, G., Hanson, D., & Holder, G. 2015, The Astrophysical Journal, 807, 166

    work page 2015

  62. [62]

    M., Hanson, D., LoVerde, M., Hirata, C

    Smith, K. M., Hanson, D., LoVerde, M., Hirata, C. M., & Zahn, O. 2012, Journal of Cosmology and Astroparticle Physics, 2012, 014–014

    work page 2012

  63. [63]

    Starobinskii, A. A. 1979, ZhETF Pisma Redaktsiiu, 30, 719

    work page 1979

  64. [64]

    Tegmark, M., de Oliveira-Costa, A., & Hamilton, A. J. S. 2003, Physical Review D, 68

    work page 2003

  65. [65]

    2017, Monthly Notices of the Royal Astronomical Society, 469, 2821–2833

    Thorne, B., Dunkley, J., Alonso, D., & Næss, S. 2017, Monthly Notices of the Royal Astronomical Society, 469, 2821–2833

    work page 2017

  66. [66]

    & Seljak, U

    Zaldarriaga, M. & Seljak, U. 1997, Physical Review D, 55, 1830–1840

    work page 1997

  67. [67]

    & Seljak, U

    Zaldarriaga, M. & Seljak, U. 1998, Physical Review D, 58

    work page 1998

  68. [68]

    2021, Journal of Open Source Software, 6, 3783 Article number, page 12 of 15

    Zonca, A., Thorne, B., Krachmalnicoff, N., & Borrill, J. 2021, Journal of Open Source Software, 6, 3783 Article number, page 12 of 15

    work page 2021

  69. [69]

    Deka, K. & Bielewicz, P.: Galactic foreground biases in CMB lensing reconstruction and delensing of B-modes Appendix A: HILC method implementation Assuming multi-frequency foreground contaminated CMB ob- servations in theN c frequency bands and frequency maps con- volved with an instrumental beam function, the observed signal for thei th frequency band, w...

    work page 2003