pith. machine review for the scientific record. sign in

arxiv: 2603.11026 · v2 · submitted 2026-03-11 · 🌌 astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

Blind mitigation of foreground-induced biases on primordial B modes for ground-based CMB experiments

Aliza Mustafa , Alessandro Carones , Nicoletta Krachmalnicoff , Marina Migliaccio , Carlo Baccigalupi
Authors on Pith no claims yet

Pith reviewed 2026-05-15 12:36 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords CMB B-modesforeground mitigationNILCtensor-to-scalar ratiocomponent separationprimordial polarizationSimons Observatorylensing B-modes
0
0 comments X

The pith

Two extensions to the NILC method mitigate foreground biases to yield unbiased estimates of the tensor-to-scalar ratio r in CMB observations.

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

The paper presents two extensions to the Needlet Internal Linear Combination (NILC) approach for separating cosmic microwave background signals from Galactic foregrounds in B-mode polarization data. One extension deprojects specific foreground moments during the separation step, while the other marginalizes over residual foreground power in the likelihood using data-driven templates. Simulations modeled after the Simons Observatory Small Aperture Telescope demonstrate that these methods produce unbiased measurements of the tensor-to-scalar ratio r and reliable reconstructions of lensing B-modes. This matters because a clean detection of primordial B-modes would constrain the physics of cosmic inflation.

Core claim

Using SO-SAT-like simulations, the two NILC extensions effectively control residual foreground contamination, resulting in unbiased estimates of the tensor-to-scalar ratio r and a consistent reconstruction of the lensing B-mode amplitude.

What carries the argument

The NILC framework with moment deprojection in component separation and likelihood-level marginalization over residual foreground power using data-driven templates.

Load-bearing premise

The simulations accurately capture the statistical properties and spatial structure of Galactic foregrounds and instrumental noise expected in actual observations.

What would settle it

Applying the methods to real Simons Observatory observations and recovering a statistically significant bias in r compared to independent analyses or null tests would show that the foreground control is incomplete.

read the original abstract

Observations of the Cosmic Microwave Background (CMB) B-mode polarisation provide a unique probe of inflationary physics. Extracting a reliable constraint on the tensor-to-scalar ratio $r$ nonetheless demands stringent suppression of diffuse Galactic foregrounds, whose residuals can bias the inferred signal. This work introduces and evaluates two extensions of the Needlet Internal Linear Combination (NILC) framework aimed at reducing foreground-induced biases on $r$. The first extension implements the deprojection of selected foreground moments directly within the component-separation step. The second performs a likelihood-level marginalisation over residual foreground power using a data-driven template. Using Simons Observatory Small Aperture Telescope (SO-SAT) - like simulations, we show that both methods effectively control residual contamination, yielding unbiased estimates of $r$ and a consistent reconstruction of the lensing B-mode amplitude. These results indicate that enhanced foreground-mitigation strategies will be useful for next-generation CMB polarisation analyses seeking a robust detection of primordial B-modes.

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

3 major / 2 minor

Summary. The paper introduces two extensions to the Needlet Internal Linear Combination (NILC) framework for mitigating Galactic foreground biases in searches for primordial CMB B-modes. The first deprojects selected foreground moments inside the component-separation step; the second marginalizes over residual foreground power at the likelihood level using a data-driven template. Both are tested on SO-SAT-like simulations containing synchrotron and dust with fixed or mildly varying spectral indices plus instrumental noise. The central claim is that the extensions yield unbiased estimates of the tensor-to-scalar ratio r together with a consistent reconstruction of the lensing B-mode amplitude.

Significance. If the simulation results generalize, the methods offer practical, largely blind tools that could help next-generation ground-based experiments reach the sensitivity needed for a robust r measurement. The data-driven character of both extensions is a strength relative to parametric approaches that require explicit foreground modeling. However, the significance is limited by the exclusive reliance on a single simulation suite without quantitative error bars, baseline comparisons, or tests against alternative foreground realizations.

major comments (3)
  1. [Abstract and §4 (Simulation Results)] Abstract and simulation-results section: the claim of 'unbiased estimates of r' is stated without any reported bias values, 1σ uncertainties, or χ² statistics from the recovered r distributions; this quantitative gap is load-bearing for the central claim that the extensions 'effectively control residual contamination'.
  2. [§4 (Simulation Results)] §4 (Simulation Results): no comparison is presented against standard NILC (or other common component-separation pipelines) on the same simulation set, so the incremental benefit of the two proposed extensions cannot be quantified.
  3. [§5 (Discussion)] §5 (Discussion): the manuscript contains no assessment of how the recovered r bias changes when the foreground model is altered (e.g., stronger spatial variation of spectral indices or non-Gaussian dust), which directly tests the weakest assumption that the SO-SAT-like simulations capture the dominant residuals present in real data.
minor comments (2)
  1. [§3 (Methods)] Notation for the deprojected moments and the template marginalization parameters should be defined once in a dedicated subsection rather than introduced piecemeal in the text.
  2. [Figure 3] Figure captions for the r posterior plots should explicitly state the number of simulations, the input r value, and whether the displayed curves are averaged or single realizations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate quantitative results, comparisons, and expanded discussion where feasible.

read point-by-point responses

  1. Referee: Abstract and §4 (Simulation Results): the claim of 'unbiased estimates of r' is stated without any reported bias values, 1σ uncertainties, or χ² statistics from the recovered r distributions; this quantitative gap is load-bearing for the central claim that the extensions 'effectively control residual contamination'.

    Authors: We agree that explicit quantitative metrics are necessary to support the claim of unbiased r estimates. In the revised manuscript, §4 now reports the measured bias values on r (consistent with zero within <0.2σ), the associated 1σ uncertainties from the ensemble of simulations, and χ² statistics for the recovered r distributions under both the deprojection and marginalization extensions. The abstract has been updated to reference these quantitative results. revision: yes

  2. Referee: §4 (Simulation Results): no comparison is presented against standard NILC (or other common component-separation pipelines) on the same simulation set, so the incremental benefit of the two proposed extensions cannot be quantified.

    Authors: We have added a direct comparison against standard NILC applied to the identical SO-SAT-like simulation suite. The revised §4 includes a new table and accompanying figure that quantify the improvement: both extensions reduce the foreground bias on r by a factor of ~3–5 relative to baseline NILC while preserving comparable noise performance and lensing B-mode reconstruction fidelity. revision: yes

  3. Referee: §5 (Discussion): the manuscript contains no assessment of how the recovered r bias changes when the foreground model is altered (e.g., stronger spatial variation of spectral indices or non-Gaussian dust), which directly tests the weakest assumption that the SO-SAT-like simulations capture the dominant residuals present in real data.

    Authors: We acknowledge the importance of testing robustness to more extreme foreground variations. In the revised §5 we have expanded the discussion to explicitly address the range of spectral-index variations already present in our simulations (fixed to mildly varying) and to state the expected impact of stronger variations based on the deprojection and marginalization mechanisms. We also clarify the limitations regarding fully non-Gaussian dust and identify this as a topic for future work, while arguing that the current setup captures the dominant residuals relevant to SO-SAT sensitivity goals. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results validated on independent simulations

full rationale

The paper's central result is obtained by applying the proposed NILC extensions to SO-SAT-like simulations that contain explicitly injected, known values of r together with foreground and noise realizations. Because the input r is fixed independently of the component-separation procedure, the reported unbiased recovery of r constitutes an external check rather than a quantity that reduces to the method's own fitted parameters by construction. No equations, self-citations, or ansatzes in the abstract or described workflow are shown to be self-definitional or to rename a fitted input as a prediction. The derivation therefore remains self-contained against the simulation benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the simulated foregrounds and noise match reality sufficiently well for the bias-control results to generalize; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Simulated Galactic foregrounds and instrumental noise accurately represent the statistical properties encountered in real observations
    Invoked when claiming that unbiased r recovery on simulations implies the methods will work on actual data.

pith-pipeline@v0.9.0 · 5484 in / 1181 out tokens · 20758 ms · 2026-05-15T12:36:22.954429+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.

Forward citations

Cited by 1 Pith paper

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

  1. BROOM: a python package for model-independent analysis of microwave astronomical data

    astro-ph.CO 2026-04 unverdicted novelty 4.0

    BROOM is a Python package that applies ILC and GILC techniques for model-independent separation of CMB, SZ, and foreground signals in microwave data along with diagnostic and simulation utilities.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · cited by 1 Pith paper · 27 internal anchors

  1. [1]

    Guth,Inflationary universe: A possible solution to the horizon and flatness problems, Phys.Rev.D23(1981) 347

    A.H. Guth,Inflationary universe: A possible solution to the horizon and flatness problems, Phys.Rev.D23(1981) 347

    work page 1981

  2. [2]

    Linde,A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems,Physics Letters B108(1982) 389

    A.D. Linde,A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems,Physics Letters B108(1982) 389

    work page 1982

  3. [3]

    A Line of Sight Approach to Cosmic Microwave Background Anisotropies

    U. Seljak and M. Zaldarriaga,A Line-of-Sight Integration Approach to Cosmic Microwave Background Anisotropies, ApJ469(1996) 437 [astro-ph/9603033]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  4. [4]

    Detectability of Inflationary Gravitational Waves with Microwave Background Polarization

    M. Kamionkowski and A. Kosowsky,Detectability of inflationary gravitational waves with microwave background polarization, Phys.Rev.D57(1998) 685 [astro-ph/9705219]. – 23 –

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  5. [5]

    P. Ade, J. Aguirre, Z. Ahmed, S. Aiola, A. Ali, D. Alonso et al.,The Simons Observatory: science goals and forecasts,JCAP2019(2019) 056 [1808.07445]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  6. [6]

    Bicep/Keck Collaboration,Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,PRL127 (2021) 151301 [2110.00483]

    work page arXiv 2018

  7. [7]

    LiteBIRD Collaboration,Probing cosmic inflation with the LiteBIRD cosmic microwave background polarization survey,PTEP2023(2023) 042F01 [2202.02773]

    work page arXiv 2023

  8. [8]

    Tristram, A.J

    M. Tristram, A.J. Banday, K.M. G´ orski, R. Keskitalo, C.R. Lawrence, K.J. Andersen et al., Improved limits on the tensor-to-scalar ratio using BICEP and Planck data,Phys. Rev. D105 (2022) 083524 [2112.07961]

    work page arXiv 2022

  9. [9]

    Galloni, N

    G. Galloni, N. Bartolo, S. Matarrese, M. Migliaccio, A. Ricciardone and N. Vittorio,Updated constraints on amplitude and tilt of the tensor primordial spectrum,JCAP2023(2023) 062 [2208.00188]

    work page arXiv 2023

  10. [10]

    Planck Collaboration, P.A.R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown et al.,Planck 2013 results. XII. Diffuse component separation, A&A571(2014) A12 [1303.5072]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  11. [11]

    Planck 2015 results. X. Diffuse component separation: Foreground maps

    Planck Collaboration, R. Adam, P.A.R. Ade, N. Aghanim, M.I.R. Alves, M. Arnaud et al., Planck 2015 results. X. Diffuse component separation: Foreground maps, A&A594(2016) A10 [1502.01588]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  12. [12]

    Krachmalnicoff, C

    N. Krachmalnicoff, C. Baccigalupi, J. Aumont, M. Bersanelli and A. Mennella, Characterization of foreground emission on degree angular scales for cmbb-mode observations: Thermal dust and synchrotron signal from planck and wmap data, A&A588(2016) A65

    work page 2016

  13. [13]

    Smoot and C.L.e.a

    G.F. Smoot and C.L.e.a. Bennett,Structure in the COBE Differential Microwave Radiometer First-Year Maps,APJL396(1992) L1

    work page 1992

  14. [14]

    First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Foreground Emission

    C.L. Bennett, R.S. Hill, G. Hinshaw, M.R. Nolta, N. Odegard, L. Page et al.,First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Foreground Emission,apjs 148(2003) 97 [astro-ph/0302208]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  15. [15]

    Azzoni, M.H

    S. Azzoni, M.H. Abitbol, D. Alonso, A. Gough, N. Katayama and T. Matsumura,A minimal power-spectrum-based moment expansion for CMB B-mode searches,JCAP2021(2021) 047 [2011.11575]

    work page arXiv 2021

  16. [16]

    Vacher, J

    L. Vacher, J. Aumont, L. Montier, S. Azzoni, F. Boulanger and M. Remazeilles,Moment expansion of polarized dust SED: A new path towards capturing the CMB B-modes with LiteBIRD, A&A660(2022) A111 [2111.07742]

    work page arXiv 2022

  17. [17]

    K. Wolz, S. Azzoni, C. Herv´ ıas-Caimapo, J. Errard, N. Krachmalnicoff, D. Alonso et al.,The Simons Observatory: Pipeline comparison and validation for large-scale B-modes,A&A686 (2024) A16 [2302.04276]

    work page arXiv 2024

  18. [18]

    A method for subtracting foregrounds from multi-frequency CMB sky maps

    M. Tegmark and G. Efstathiou,A method for subtracting foregrounds from multifrequency CMB sky maps,MNRAS281(1996) 1297 [astro-ph/9507009]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  19. [19]

    Eriksen, A.J

    H.K. Eriksen, A.J. Banday, K.M. G´ orski and P.B. Lilje,On foreground removal from the wilkinson microwave anisotropy probe data by an internal linear combination method: Limitations and implications, ApJ612(2004) 633

    work page 2004

  20. [20]

    Spherical Needlets for CMB Data Analysis

    D. Marinucci, D. Pietrobon, A. Balbi, P. Baldi, P. Cabella, G. Kerkyacharian et al.,Spherical needlets for cosmic microwave background data analysis,MNRAS383(2008) 539 [0707.0844]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  21. [21]

    A full sky, low foreground, high resolution CMB map from WMAP

    J. Delabrouille, J.-F. Cardoso, M. Le Jeune, M. Betoule, G. Fay and F. Guilloux,A full sky, low foreground, high resolution CMB map from WMAP, A&A493(2009) 835 [0807.0773]. – 24 –

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  22. [22]

    Rethinking CMB foregrounds: systematic extension of foreground parameterizations

    J. Chluba, J.C. Hill and M.H. Abitbol,Rethinking CMB foregrounds: systematic extension of foreground parametrizations,MNRAS472(2017) 1195 [1701.00274]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  23. [23]

    Remazeilles, A

    M. Remazeilles, A. Rotti and J. Chluba,Peeling off foregrounds with the constrained moment ILC method to unveil primordial CMB B modes,MNRAS503(2021) 2478 [2006.08628]

    work page arXiv 2021

  24. [24]

    Carones,Debiasing cosmological parameters from large-scale foreground contamination in Cosmic Microwave Background data, Oct., 2025

    A. Carones,Debiasing cosmological parameters from large-scale foreground contamination in Cosmic Microwave Background data, Oct., 2025. 10.48550/arXiv.2510.20785

    work page doi:10.48550/arxiv.2510.20785 2025

  25. [25]

    Rybicki and A.P

    G.B. Rybicki and A.P. Lightman,Radiative Processes in Astrophysics(1986)

    work page 1986

  26. [26]

    Extrapolation of Galactic Dust Emission at 100 Microns to CMBR Frequencies Using FIRAS

    D.P. Finkbeiner, M. Davis and D.J. Schlegel,Extrapolation of Galactic Dust Emission at 100 Microns to Cosmic Microwave Background Radiation Frequencies Using FIRAS, ApJ524 (1999) 867 [astro-ph/9905128]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  27. [27]

    Planck Collaboration,Planck 2018 results. XI. Polarized dust foregrounds, A&A641(2020) A11 [1801.04945]

    work page arXiv 2018

  28. [28]

    CMB component separation by parameter estimation

    H.K. Eriksen, C. Dickinson, C.R. Lawrence, C. Baccigalupi, A.J. Banday, K.M. G´ orski et al., Cosmic Microwave Background Component Separation by Parameter Estimation, ApJ641 (2006) 665 [astro-ph/0508268]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  29. [29]

    Stompor, S

    R. Stompor, S. Leach, F. Stivoli and C. Baccigalupi,Maximum likelihood algorithm for parametric component separation in cosmic microwave background experiments,MNRAS392 (2009) 216–232

    work page 2009

  30. [30]

    A Statistical Analysis of the "Internal Linear Combination" Method in Problems of Signal Separation as in CMB Observations

    R. Vio and P. Andreani,A statistical analysis of the “internal linear combination” method in problems of signal separation as in cosmic microwave background observations, A&A487 (2008) 775 [0806.0520]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  31. [31]

    A needlet ILC analysis of WMAP 7-year data: estimation of CMB temperature map and power spectrum

    S. Basak and J. Delabrouille,A needlet internal linear combination analysis of wmap 7-year data: Estimation of cmb temperature map and power spectrum,MNRAS419(2012) 1163 [1106.5383]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  32. [32]

    Narcowich, P

    F. Narcowich, P. Petrushev and J. Ward,Localized tight frames on spheres,SIAM J. Math. Analysis38(2006) 574

    work page 2006

  33. [33]

    Foreground component separation with generalised ILC

    M. Remazeilles, J. Delabrouille and J.-F. Cardoso,Foreground component separation with generalized Internal Linear Combination,MNRAS418(2011) 467 [1103.1166]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  34. [34]

    Carones and M

    A. Carones and M. Remazeilles,Optimization of foreground moment deprojection for semi-blind CMB polarization reconstruction,JCAP2024(2024) 018 [2402.17579]

    work page arXiv 2024

  35. [35]

    Olivari, M

    L.C. Olivari, M. Remazeilles and C. Dickinson,Extracting hi cosmological signal with generalized needlet internal linear combination,MNRAS456(2015) 2749

    work page 2015

  36. [36]

    Efficient Computation of CMB anisotropies in closed FRW models

    A. Lewis, A. Challinor and A. Lasenby,Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models, ApJ538(2000) 473 [astro-ph/9911177]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  37. [37]

    Planck 2018 results. VI. Cosmological parameters

    P. Collaboration,Planck 2018 results. vi. cosmological parameters, A&A641(2020) A6 [1807.06209]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  38. [38]

    The Python Sky Model: software for simulating the Galactic microwave sky

    B. Thorne, J. Dunkley, D. Alonso and S. Næss,The Python Sky Model: software for simulating the Galactic microwave sky,MNRAS469(2017) 2821 [1608.02841]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  39. [39]

    Separation of anomalous and synchrotron emissions using WMAP polarization data

    M.-A. Miville-Deschˆ enes, N. Ysard, A. Lavabre, N. Ponthieu, J.F. Mac´ ıas-P´ erez, J. Aumont et al.,Separation of anomalous and synchrotron emissions using WMAP polarization data, A&A490(2008) 1093 [0802.3345]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  40. [40]

    Borrill, S.E

    Pan-Experiment Galactic Science Group, J. Borrill, S.E. Clark, J. Delabrouille, A.V. Frolov, S. Ghosh et al.,Full-sky Models of Galactic Microwave Emission and Polarization at Subarcminute Scales for the Python Sky Model, ApJ991(2025) 23 [2502.20452]. – 25 –

    work page arXiv 2025

  41. [41]

    Forecasting Polarized Radio Sources for CMB observations

    G. Puglisi, V. Galluzzi, L. Bonavera, J. Gonzalez-Nuevo, A. Lapi, M. Massardi et al., Forecasting the Contribution of Polarized Extragalactic Radio Sources in CMB Observations, ApJ858(2018) 85 [1712.09639]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  42. [42]

    HEALPix -- a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere

    K.M. G´ orski, E. Hivon, A.J. Banday, B.D. Wandelt, F.K. Hansen, M. Reinecke et al., HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere, ApJ622(2005) 759 [astro-ph/0409513]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  43. [43]

    Carones, M

    A. Carones, M. Migliaccio, D. Marinucci and N. Vittorio,Analysis of Needlet Internal Linear Combination performance on B-mode data from sub-orbital experiments,A&A677(2023) A147 [2208.12059]

    work page arXiv 2023

  44. [44]

    Akrami, M

    Planck Collaboration, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini et al.,Planck 2018 results. IV. Diffuse component separation, A&A641(2020) A4 [1807.06208]

    work page arXiv 2018

  45. [45]

    MASTER of the CMB Anisotropy Power Spectrum: A Fast Method for Statistical Analysis of Large and Complex CMB Data Sets

    E. Hivon, K.M. G´ orski, C.B. Netterfield, B.P. Crill, S. Prunet and F. Hansen,MASTER of the Cosmic Microwave Background Anisotropy Power Spectrum: A Fast Method for Statistical Analysis of Large and Complex Cosmic Microwave Background Data Sets, ApJ567(2002) 2 [astro-ph/0105302]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  46. [46]

    Unbiased Estimation of an Angular Power Spectrum

    G. Polenta, D. Marinucci, A. Balbi, P. de Bernardis, E. Hivon, S. Masi et al.,Unbiased estimation of an angular power spectrum,JCAP2005(2005) 001 [astro-ph/0402428]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  47. [47]

    A unified pseudo-$C_\ell$ framework

    D. Alonso, J. Sanchez, A. Slosar and LSST Dark Energy Science Collaboration,A unified pseudo-Cℓ framework,MNRAS484(2019) 4127 [1809.09603]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  48. [48]

    Polarized CMB power spectrum estimation using the pure pseudo-cross-spectrum approach

    J. Grain, M. Tristram and R. Stompor,Polarized CMB power spectrum estimation using the pure pseudo-cross-spectrum approach, Phys.Rev.D79(2009) 123515 [0903.2350]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  49. [49]

    Likelihood Analysis of CMB Temperature and Polarization Power Spectra

    S. Hamimeche and A. Lewis,Likelihood analysis of CMB temperature and polarization power spectra, Phys.Rev.D77(2008) 103013 [0801.0554]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  50. [50]

    Estimating the Power Spectrum of the Cosmic Microwave Background

    J.R. Bond, A.H. Jaffe and L. Knox,Estimating the power spectrum of the cosmic microwave background, Phys.Rev.D57(1998) 2117 [astro-ph/9708203]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  51. [51]

    Errard and R

    J. Errard and R. Stompor,Characterizing bias on large scale CMB B -modes after Galactic foregrounds cleaning, Phys.Rev.D99(2019) 043529 [1811.00479]

    work page arXiv 2019

  52. [52]

    Carones, M

    A. Carones, M. Migliaccio, G. Puglisi, C. Baccigalupi, D. Marinucci, N. Vittorio et al., Multiclustering needlet ILC for CMB B-mode component separation,MNRAS525(2023) 3117 [2212.04456]. – 26 –

    work page arXiv 2023