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Examining statistical isotropy of CMB low multipoles from Planck PR4 data

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Using the latest Planck PR4 temperature map, the paper finds that nine of the first sixty CMB multipoles are intrinsically anisotropic, a deviation from statistical isotropy with a cumulative probability of about 0.3% — close to 3-sigma…

desk verdict A careful PR4 update of the Power-tensor anomalies whose headline 0.3% significance is one borderline multipole away from being a 0.9% result. read the letter →

arxiv 2506.22795 v1 pith:NHVVJOYB submitted 2025-06-28 astro-ph.CO

Pavan Kumar Aluri , Sanjeet Kumar Patel This is my paper · ORCID
classification astro-ph.CO
keywords cosmicmicrowavebackgroundstatisticalisotropyCMBanomalieslowmultipolesPowertensorPlanckPR4multipolealignmentquadrupole-octopole
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 asks whether the long-claimed "anomalies" in the cosmic microwave background's largest angular scales survive in the most recent Planck data release. Applying the Power tensor method to the Planck PR4 Commander temperature map, it finds that nine of the first sixty multipoles ($l=2$ to 61) have anomalously low Power entropy, meaning their power is concentrated along a preferred axis rather than spread isotropically. The chance of finding this many anomalous modes under the hypothesis that the sky is statistically isotropic is about 0.3%, a rejection near the $3\sigma$ level. The same analysis finds no statistically significant planar modes and no preferred collective alignment axis for the sixty multipoles, though the quadrupole-octopole alignment persists.

What carries the argument

The central object is the Power tensor $A_{ij}(l)$, a $3\times 3$ matrix built from each multipole's spherical-harmonic coefficients and angular-momentum matrices; it maps a multipole onto an ellipsoid whose normalized eigenvalues define a Power entropy $S(l)=-\sum_\alpha \lambda_\alpha \log\lambda_\alpha$. Low $S(l)$ means power is concentrated along a single axis (the principal eigenvector, PEV), i.e., the mode is intrinsically anisotropic. The Alignment tensor $X_{ij}=(1/N_l)\sum_l \tilde{e}_i^l \tilde{e}_j^l$, formed by averaging the PEVs of many multipoles, tests for collective preferred directions. The paper also uses a binomial cumulative probability to convert the number of modes falling below a per-multipole $p$-value threshold into an overall significance for the first sixty multipoles.

What would settle it

Run many statistically isotropic CMB realizations through the same dipole-subtraction, downgrade, and inpainting pipeline and count how many yield 9 or more of the 60 multipoles with Power entropy $p \le 0.05$; if that fraction is much larger than 0.3%, the reported significance is an artifact of the mask-inpainting procedure rather than intrinsic anisotropy. A more direct check: for a single isotropic simulation, compare Power entropies and principal eigenvectors computed from the full-sky map and from its masked-and-inpainted version; large differences falsify the preservation assumption.

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Extended reading notes

Core claim

The paper reports that in the Planck PR4 Commander-cleaned temperature map, the multipoles $l=6, 13, 14, 16, 17, 30, 34, 40,$ and $56$ have Power entropy $p$-values $\le 0.05$ against isotropic simulations. Using the binomial distribution with $n=60$ multipoles and a per-multipole threshold $P=0.05$, the cumulative probability of finding $k^*=9$ or more such modes is $P_{\mathrm{cuml}} \approx 0.0028$ (Eq. 7), i.e., statistical isotropy is rejected at close to $3\sigma$. The modes with anomalously small smallest eigenvalues of the Power tensor ($l=13,14,16,30,34,56$) are a subset of these anisotropic modes, so the paper concludes they are axially anisotropic rather than intrinsically planar. The quadrupole and octopole remain aligned at $\theta_{23} \approx 14.8^\circ$ with $p$-values of 0.08 (100 Planck simulations) and 0.035 (5000 ideal realizations), while higher multipoles aligned with the quadrupole are collectively insignificant ($P_{\mathrm{cuml}} \approx 0.34$). The Alignment tensor over $l=2$–61 yields no significant preferred axis or plane ($p \approx 0.21$ and 0.20), and its principal axis lies within about $6^\circ$–$35^\circ$ of the anomalous low-multipole axes, broadly close to the dipole, quadrupole, and octopole directions.

Load-bearing premise

The analysis assumes that the sparse inpainting of the masked galactic region preserves the statistical properties of the low-multipole coefficients, so that Power tensor statistics computed from the inpainted map and from identically processed simulations are unbiased.

Editorial extensions

If this is right

  • Statistical isotropy of the CMB temperature sky is rejected at roughly the $3\sigma$ level by the Planck PR4 low multipoles, lending support to earlier WMAP-era and Planck 2018 anomaly claims.
  • The anisotropy is axial rather than planar: every unusually planar mode is also one of the low-entropy anisotropic modes, so explanations that predict planar low-multipole structure are not supported by this data.
  • The quadrupole-octopole alignment, one of the original CMB anomalies, persists in the newest Planck map at marginal significance, suggesting it is not a processing artifact of earlier releases.
  • No significant collective alignment axis exists for the full set of sixty multipoles, so the 'axis of evil' picture is not strengthened as a global preferred direction; whatever is happening is concentrated in specific modes.
  • The collective alignment axis that does emerge lies close to the directions of the dipole, quadrupole, octopole, and the four quadrupole-aligned modes, giving future polarization experiments a concrete axis to test.

Reading between the lines

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

  • Extending beyond the paper: the binomial calculation treats the 60 multipoles as independent, but the galactic mask and inpainting introduce correlations between neighboring $l$, so the effective number of independent modes is smaller; a Monte Carlo that directly counts low-entropy modes per simulated map would give a more faithful significance.
  • Extending beyond the paper: because the same anomaly set was originally identified in WMAP and earlier Planck releases, the threshold and multipole range are not prior-independent; a full look-elsewhere treatment over all exploratory statistics run on these data would reduce the post-trial significance.
  • Extending beyond the paper: the near-coincidence of the collective alignment axis with the CMB kinetic dipole direction hints at a possible Doppler or aberration origin for the axiality; repeating the Power tensor analysis on dipole-removed maps under different assumptions about the solar velocity would test this.
  • Extending beyond the paper: applying the same pipeline to CMB polarization, once signal-to-noise allows, would give an independent check of whether the same multipoles are anisotropic in polarization; if the anomalous set differs, residual foregrounds or temperature-map processing are the likely source.
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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 / 6 minor

Summary. The paper applies the Power tensor and Alignment tensor statistics to the Planck PR4 NPIPE Commander temperature map, restricted to multipoles l=2 to 61, and compares the observed statistics with 100 Planck FFP simulations and 5000 ideal CMB realizations generated from a best-fit theoretical power spectrum. The central result is that 9 of the 60 multipoles have Power entropy p-values at or below 0.05, giving a binomial cumulative probability Pcuml ≈ 0.0028 (claimed as close to 3-sigma significance), which the authors interpret as evidence for a significant number of intrinsically anisotropic low multipoles. Additional results concern planarity (no independently planar modes beyond those already anisotropic), alignment of l=3,44,59,61 with the quadrupole, and the absence of collective alignment in the range l=2-61 according to the Alignment tensor.

Significance. The Power tensor is an established public statistic, and the paper provides a useful update using the latest Planck PR4 map. The analysis pipeline is described in detail, and the use of two simulation ensembles gives a degree of cross-checking. The headline result is falsifiable and the binomial global test is a reasonable way to assess the multiplicity of low-l anomalies. However, the significance is borderline and is sensitive to the choice of reference simulations and to the inpainting/mask treatment, so the robustness of the central claim needs to be demonstrated.

major comments (3)
  1. [§5, Eq. (7), Table 1] The cumulative probability Pcuml ≈ 0.0028 depends on counting k* = 9 modes with p ≤ 0.05; this count is not robust because the l=6 mode has p = 0.03 from the 100 PR4 simulations but p = 0.0534 from the 5000 ideal realizations, straddling the threshold. If the latter ensemble is used as the reference, k* becomes 8 and Pcuml increases to roughly 0.01, reducing the claimed significance from ~3σ to ~2.5σ. The paper must either justify why the 100 PR4 simulations are the primary reference or show that the discrepancy for l=6 is within sampling uncertainty.
  2. [§3, Appendix A] The two simulation ensembles are not processed identically: the 100 PR4 simulations are downgraded and inpainted exactly as the data, while the 5000 ideal CMB realizations are generated at Nside=256 with a 40′ beam and no masking or inpainting step is described. The p-values from the ideal set are therefore not a clean cross-check for a masked and inpainted data map. The authors should process the ideal realizations through the same mask/inpainting pipeline, or explicitly restrict the cross-check to statistics that are insensitive to inpainting, and explain why the l=6 p-value differs across ensembles.
  3. [§3, Appendix A] The assertion that iSAP sparse inpainting 'preserves the statistical properties with the rest of the unmasked sky' is not validated for the chosen mask (fsky ≈ 0.92) and multipole range. The mask is selected after testing several options and rejecting those that produce visible discontinuities, which is a data-dependent choice that is not accounted for in the p-values. The authors should validate the inpainting assumption, for example by comparing Power tensor distributions from inpainted ideal simulations with those from full-sky ideal simulations, and should test the sensitivity of the anomalous-mode list to the mask choice.
minor comments (6)
  1. [Abstract, §5] The 'cumulative probability of 0.3%' is a rounded version of 0.0028 ≈ 0.28%; please report the exact value (e.g., 0.0028) alongside the rounded figure.
  2. [§4.1, Table 1] For l=30, the p-value from the 100 PR4 simulations is reported as '<0.01,' which is a coarse upper limit; consider reporting the exact binomial confidence interval or the fraction of simulations below the observed value.
  3. [§4.2] The quadrupole-octopole alignment p-values (0.08 and 0.035) are described as 'marginally anomalous at about 2σ level,' but p=0.08 corresponds to roughly 1.4σ for a one-sided test; the language should be consistent with the quoted p-values.
  4. [Appendix A] The sky fraction of the mask is stated as both fsky ≈ 0.92 and fsky ≈ 0.9; please make the values consistent.
  5. [Fig. 5] The histograms for the Alignment entropy and the smallest eigenvalue use similar colors; add distinguishing line styles or labels for clarity.
  6. [Throughout] The term 'intrinsically anisotropic' is used for multipoles with small Power entropy p-values; this conflates a statistical statement about the data with a physical property. Suggest using 'significantly anisotropic' or 'anomalously anisotropic' instead.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 0.3% significance is a Monte Carlo comparison under the statistical-isotropy null, not a fitted or self-citation-derived quantity.

full rationale

The paper's central claim is that nine of the first sixty multipoles have Power entropy p-values at or below 0.05, with a cumulative binomial probability Pcuml ≈ 0.0028 given by Eq. (7). This is a direct, self-contained comparison of the observed Power tensor statistics against two simulation ensembles: 100 Planck PR4 FFP simulations processed with the same downgrading and inpainting as the data, and 5000 ideal CMB realizations generated from the best-fit theoretical power spectrum. No model parameter or statistic is fitted to the data and then renamed as a prediction; the p-values are empirical tail probabilities from simulations generated under statistical isotropy, and Eq. (7) is an exact binomial calculation using the reported k*, n, and P. The Power tensor and Alignment tensor definitions are given explicitly in Eqs. (2)-(5), so the methodology does not rely on a black-box prior result. The self-citations to Refs. [22, 23, 25] concern earlier applications, slightly altered definitions, and foreground-related bias studies; these are methodologically ancillary and not the load-bearing justification for the main result. The potential fragility of the result due to borderline multipoles such as l=6, whose p-value differs between the two ensembles (0.03 vs. 0.0534), and the assumptions about iSAP inpainting preserving statistical properties, are statistical robustness and correctness concerns rather than circularity: they do not make the data-to-simulation comparison equivalent to its own inputs. The paper is therefore self-contained against external benchmarks for its central claim.

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

The central claim rests on the null model of a statistically isotropic Gaussian CMB, on inpainting preserving low-l statistics, and on the independence assumption in the binomial global test. The analysis choices (threshold, l-range, beam, mask) are chosen by hand and not robustly varied.

free parameters (4)
  • p-value threshold P = 0.05
    Chosen by hand as the 2-sigma significance level; the anomalous mode list and the binomial cumulative probability (Eq. 7) depend on this threshold.
  • multipole range used in the global test = l = 2 to 61 (n = 60)
    The 'first sixty multipoles' range is chosen by hand; the binomial n and the set of tested modes depend on this range.
  • effective Gaussian beam FWHM for downgrading = 40 arcmin
    The PR4 map is downgraded to Nside 256 with 40-arcmin beam; this changes the spherical harmonic coefficients and thus the Power tensor.
  • galactic mask = Extended WMAP 9-year kp8 mask, fsky about 0.92
    Chosen after testing several masks to avoid inpainting discontinuities; no robustness check across masks is reported.
assumptions (4)
  • domain assumption Under statistical isotropy, the CMB is a realization of a Gaussian random field with the Planck 2018 best-fit power spectrum, and the 5000 ideal CMB maps are valid null realizations.
    Section 3 generates the ideal simulations from CAMB using Planck 2018 parameters; all p-values compare the data against these maps.
  • standard math The Power tensor in Eq. (2) has ensemble average delta_ij Cl/3 under isotropy, so the normalized eigenvalues and Power entropy are unbiased isotropy diagnostics.
    Invoked in Section 2.1; follows from the two-point function of a_lm and properties of angular momentum matrices.
  • ad hoc to paper iSAP sparse inpainting preserves the statistical properties of the unmasked sky at low multipoles.
    Stated in Section 3 and Appendix A; the FFP simulations are processed with the same inpainting, but no independent validation of this assumption is provided.
  • domain assumption The 60 per-multipole tests are approximately independent, so the binomial distribution in Eq. (7) is a valid global null statistic.
    Different-l a_lm are theoretically independent, but the nonlinear Power tensor statistics may introduce mild correlations; the paper does not test this.

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Pith. "Pith review of Examining statistical isotropy of CMB low multipoles from Planck PR4 data." pith.science (2026). https://pith.science/paper/NHVVJOYB

@misc{pith2026250622795,
  author       = {Pith},
  title        = {Pith review of: Examining statistical isotropy of CMB low multipoles from Planck PR4 data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NHVVJOYB}},
  note         = {Machine review of arXiv:2506.22795}
}
abstract

Low multipoles ($l$) in cosmic microwave background (CMB) temperature anisotropies have shown some `peculiarities' when examined since the release of the full sky CMB maps, using a variety of tests. In this paper, we concern ourselves with the very first peculiarities seen in CMB data viz., a breakdown of statistical isotropy in the form of axiality and planarity of these low-$l$ modes, and preferred alignments among them. We scrutinize the latest CMB data from ESA's Planck mission, PR4, to evaluate the current status of these deviations. We employ the Power tensor method which allows an invariant characterization of the distribution of power in a given multipole, and apply it to probe the first sixty multipoles i.e., $l=2$ to 61. We find that there are significant number of modes that are intrinsically anisotropic with a cumulative probability of $0.3\%$. However since the planarity study reveals that those modes that are unusually planar are subset of these anisotropic modes, we conclude that they may not be intrinsically planar. The quadrupole is still well aligned with the octopole. Besides, $l=3$, higher multipoles aligned with quadrupole are found to be insignificant. Interestingly, the collective alignment axis of the first sixty multipoles is found to be broadly closer to the axis of dipole, quadrupole, octopole and other modes aligned with $l=2$.

Figures

Figures reproduced from arXiv: 2506.22795 by the authors.

Figure 1
Figure 1. Eigenvalues of Power tensor for each ‘l’ in the CMB map from Planck PR4 data obtained using Commander method. The horizontal red line is the expectation value of the eigenvalues when isotropy holds viz., ⟨λi⟩ = 1/3, and the two dashed lines denote the ±20% deviation around this expectation i.e., ±1/5 th × 1/3 = ±1/15. 4 Analysis and results 4.1 Probing axial and planar modes with Power tensor Power entropy, S(l), al… view at source ↗
Figure 2
Figure 2. Top : Power entropy with 95% confidence level derived from 100 Planck PR4 Commander simulations and 5000 ideal CMB realizations based on the theoretical power spec￾trum from Planck 2018 data release. Bottom : p-value plot of the Power entropy statistic obtained by comparing the data statistic from Planck PR4 Commander CMB map with the same from simulations. Here also the two curves correspond to p-values obtained us… view at source ↗
Figure 3
Figure 3. p-value plot of the smallest eigenvalue of Power tensor, λ obs 1 , for the first sixty multi￾poles of the CMB map derived using the Planck NPIPE processing pipeline (PR4) and cleaned using the Commander method. The 2σ (95%) level is also indicated by a gray horizontal line. The curves correspond to the two different simulation sets used viz., 100 Planck like simulations provided by Planck collaboration and 5000 mock… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Top : Alignment angle between CMB quadrupole (l = 2) and higher multipoles upto lmax = 61 in Planck PR4 Commander map. Bottom : p-values of the observed alignment statistic “1 − cos θ2l” characterizing the alignment between quadrupole and rest of the multipoles l = 3 t…
Figure 5
Figure 5. Figure 5: Left : Histogram of the Alignment entropy, SX, as obtained from 100 Planck PR4 Commander processed simulations and 5000 ideal CMB realizations generated based on best fit theoretical power spectrum from Planck 2018 data. The observed Alignment entropy, S obs X , as see…
Figure 6
Figure 6. Figure 6: Various anisotropy axes (PEVs) corresponding to different multipoles obtained using [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: First row : Commander cleaned CMB temperature map from Planck’s public release 4. Second row : Galactic mask used in the present study with a sky fraction of fsky ≈ 0.92 is shown to the left and the same mask with additional azimuthal cut to fit out the dipole in the C…

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Works this paper leans on

34 extracted references · 32 canonical work pages

  1. [1]

    Schwarz, Craig J

    Dominik J. Schwarz, Craig J. Copi, Dragan Huterer, and Glenn D. Starkman. CMB anomalies after Planck. Classical and Quantum Gravity, 33(18):184001, September 2016

    work page 2016

  2. [2]

    Beyond ΛCDM: Problems, solutions, and the road ahead

    Philip Bull et al. Beyond ΛCDM: Problems, solutions, and the road ahead. Physics of the Dark Universe, 12:56–99, June 2016

    work page 2016

  3. [3]

    Covariance of CMB anomalies

    Jessica Muir, Saroj Adhikari, and Dragan Huterer. Covariance of CMB anomalies. Phys. Rev. D, 98(2):023521, July 2018

    work page 2018

  4. [4]

    C. L. Bennett et al. Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Obser- vations: Are There Cosmic Microwave Background Anomalies? ApJS, 192(2):17, February 2011

    work page 2011

  5. [5]

    C. L. Bennett et al. Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Obser- vations: Final Maps and Results. ApJS, 208(2):20, October 2013

    work page 2013

  6. [6]

    Planck Collaboration, P. A. R. Ade, et al. Planck 2013 results. XXIII. Isotropy and statistics of the CMB. A&A, 571:A23, November 2014

    work page 2013

  7. [7]

    Planck Collaboration, P. A. R. Ade, et al. Planck 2015 results. XVI. Isotropy and statistics of the CMB. A&A, 594:A16, September 2016. 15

    work page 2015

  8. [8]

    Akrami, et al

    Planck Collaboration, Y. Akrami, et al. Planck 2018 results. VII. Isotropy and statistics of the CMB. A&A, 641:A7, September 2020

    work page 2018

Show all 34 references
  1. [9]

    Significance of the largest scale CMB fluctuations in WMAP

    Ang´ elica de Oliveira-Costa, Max Tegmark, Matias Zaldarriaga, and Andrew Hamilton. Significance of the largest scale CMB fluctuations in WMAP. Phys. Rev. D, 69(6):063516, March 2004

    2004

  2. [10]

    Schwarz, Glenn D

    Dominik J. Schwarz, Glenn D. Starkman, Dragan Huterer, and Craig J. Copi. Is the Low- ℓ Microwave Background Cosmic? Phys. Rev. Lett., 93(22):221301, November 2004

    2004

  3. [11]

    Copi, Dragan Huterer, and Glenn D

    Craig J. Copi, Dragan Huterer, and Glenn D. Starkman. Multipole vectors: A new rep- resentation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2⩽l⩽8. Phys. Rev. D, 70(4):043515, August 2004

    2004

  4. [12]

    Ralston and Pankaj Jain

    John P. Ralston and Pankaj Jain. The Virgo Alignment Puzzle in Propagation of Radiation on Cosmological Scales. International Journal of Modern Physics D, 13(9):1857–1877, January 2004

    2004

  5. [13]

    Assessing the effects of foregrounds and sky removal in WMAP

    Anˇ ze Slosar and Uroˇ s Seljak. Assessing the effects of foregrounds and sky removal in WMAP. Phys. Rev. D, 70(8):083002, October 2004

    2004

  6. [14]

    Bielewicz, K

    P. Bielewicz, K. M. G´ orski, and A. J. Banday. Low-order multipole maps of cosmic mi- crowave background anisotropy derived from WMAP. MNRAS, 355(4):1283–1302, Decem- ber 2004

    2004

  7. [15]

    Bielewicz, H

    P. Bielewicz, H. K. Eriksen, A. J. Banday, K. M. G´ orski, and P. B. Lilje. Multipole Vector Anomalies in the First-Year WMAP Data: A Cut-Sky Analysis. ApJ, 635(2):750–760, December 2005

    2005

  8. [16]

    Examination of Evidence for a Preferred Axis in the Cosmic Radiation Anisotropy

    Kate Land and Jo˜ ao Magueijo. Examination of Evidence for a Preferred Axis in the Cosmic Radiation Anisotropy. Phys. Rev. Lett., 95(7):071301, August 2005

    2005

  9. [17]

    CMB multipole measurements in the pres- ence of foregrounds

    Ang´ elica de Oliveira-Costa and Max Tegmark. CMB multipole measurements in the pres- ence of foregrounds. Phys. Rev. D, 74(2):023005, July 2006

    2006

  10. [18]

    H. K. Eriksen, I. J. O’Dwyer, J. B. Jewell, B. D. Wandelt, D. L. Larson, K. M. G´ orski, S. Levin, A. J. Banday, and P. B. Lilje. Power Spectrum Estimation from High-Resolution Maps by Gibbs Sampling. ApJS, 155(2):227–241, December 2004

    2004

  11. [19]

    H. K. Eriksen, J. B. Jewell, C. Dickinson, A. J. Banday, K. M. G´ orski, and C. R. Lawrence. Joint Bayesian Component Separation and CMB Power Spectrum Estimation. ApJ, 676(1):10–32, March 2008

    2008

  12. [20]

    Akrami, et al

    Planck Collaboration, Y. Akrami, et al. Planck intermediate results. L VII. Joint Planck LFI and HFI data processing. A&A, 643:A42, November 2020

    2020

  13. [21]

    Pramoda Kumar Samal, Rajib Saha, Pankaj Jain, and John P. Ralston. Testing isotropy of cosmic microwave background radiation. MNRAS, 385(4):1718–1728, April 2008

    2008

  14. [22]

    Aluri, John P

    Pavan K. Aluri, John P. Ralston, and Amanda Weltman. Alignments of parity even/odd- only multipoles in CMB. MNRAS, 472(2):2410–2421, December 2017

    2017

  15. [23]

    Rath, Pramoda Kumar Samal, Srikanta Panda, Debesh D

    Pranati K. Rath, Pramoda Kumar Samal, Srikanta Panda, Debesh D. Mishra, and Pavan K. Aluri. Testing statistical isotropy in cosmic microwave background polarization maps. MNRAS, 475(4):4357–4366, April 2018

    2018

  16. [24]

    D. J. Fixsen. The Temperature of the Cosmic Microwave Background. ApJ, 707(2):916– 920, December 2009. 16

    2009

  17. [25]

    Aluri, Pramoda K

    Pavan K. Aluri, Pramoda K. Samal, Pankaj Jain, and John P. Ralston. Effect of foregrounds on the cosmic microwave background radiation multipole alignment. MNRAS, 414(2):1032– 1046, June 2011

    2011

  18. [26]

    Aghanim, et al

    Planck Collaboration, N. Aghanim, et al. Planck 2018 results. VI. Cosmological parameters. A&A, 641:A6, September 2020

    2018

  19. [27]

    Hinshaw et al

    G. Hinshaw et al. Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observa- tions: Temperature Analysis. ApJS, 170:288–334, June 2007

    2007

  20. [28]

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

    Kevork Abazajian et al. CMB-S4 Science Case, Reference Design, and Project Plan. arXiv e-prints, page arXiv:1907.04473, July 2019

    1907 arXiv

  21. [29]

    The Simons Observatory: science goals and forecasts

    Peter Ade et al. The Simons Observatory: science goals and forecasts. J. Cosmology Astropart. Phys., 2019(2):056, February 2019

    2019

  22. [30]

    Allys, et al

    LiteBIRD Collaboration, E. Allys, et al. Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey.arXiv e-prints, page arXiv:2202.02773, February 2022

    2022 arXiv

  23. [31]

    K. M. G´ orski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, and M. Bartelmann. HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere. ApJ, 622(2):759–771, April 2005

    2005

  24. [32]

    Efficient computation of CMB anisotropies in closed FR W models

    Antony Lewis, Anthony Challinor, and Anthony Lasenby. Efficient computation of CMB anisotropies in closed FR W models. ApJ, 538:473–476, 2000

    2000

  25. [33]

    CMB power spectrum parameter degeneracies in the era of precision cosmology

    Cullan Howlett, Antony Lewis, Alex Hall, and Anthony Challinor. CMB power spectrum parameter degeneracies in the era of precision cosmology. J. Cosmology Astropart. Phys., 1204:027, 2012

    2012

  26. [34]

    J. L. Starck, M. J. Fadili, and A. Rassat. Low- ℓ CMB analysis and inpainting. A&A, 550:A15, February 2013. 17

    2013

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