How to measure CMB power spectra without losing information

Max Tegmark

REVIEW

How to measure CMB power spectra without losing information

Max Tegmark

Astrophysics · Submitted 21 Nov 1996 · 27 references

Pith's verdict: The core result holds: a quadratic estimator that provably saturates the Fisher bound and runs in n², with only the ordinary caveats about the fiducial covariance.

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arXiv IDastro-ph/9611174 (version 2)

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CitationMax Tegmark. "How to measure CMB power spectra without losing information." arXiv:astro-ph/9611174 (1996). https://pith.science/paper/IF5TG6UY

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Editor's note

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The thing to know up front: Section III is right. Tegmark takes the best-quadratic-estimator problem, which naively needs an n(n+1)/2-dimensional matrix inversion, and shows the answer collapses to E_l = (1/2) F_ll C^-1 P_l C^-1. He then proves the resulting y has covariance exactly F and mean Fc, so F^-1 y is unbiased with covariance F^-1 — the Cramer-Rao floor. That is a clean piece of algebra, and it is why the paper gets cited. The second real contribution is the interpretation: z = C^-1 x is a high-pass-filtered, edge-tapered map, and the estimator is just a truncated spherical-harmonic expansion of z. That is why it degrades gracefully to Hauser-Peebles at low S/N and approaches the eigenmode method at high S/N without an eigenvalue solve. The n^2 argument (iterative solve for C^-1 Y, F from V = Y^t C^-1 Y without ever forming the full matrix) is sound, and the appendix on pseudo-inverses for monopole and dipole removal is a tidy bonus that also applies to mapmaking. On the stress-test concern: real, but smaller than stated. Yes, the saturation proof is conditional on the C inside E_l being the true covariance — equation 9 defines C that way, so the math is honest about its own hypothesis. Plug in a wrong fiducial and W stops being diagonal, so C_l-hat becomes a windowed mixture; that is what every quadratic estimator since does. The paper does not dwell on it, but Section V.D does say the fit can be iterated, and the COBE fiducial (Q = 18.4 microkelvin, from reference 11) is a mild circularity only because the error bars there are wide anyway. I would not call this a crack in the claim; it is a caveat the field absorbed without trouble. Genuine soft spots, in order: the low-multipole non-Gaussianity is relegated to footnote 2, which matters more than the footnote suggests for anyone feeding these points into a likelihood; and the chi-squared parameter-fitting proposal in V.D is explicitly untested, which the author says plainly. Who should read it: anyone doing CMB power-spectrum estimation, and anyone who wants a worked example of Fisher-matrix reasoning applied to real data. It deserves referees.

Referee report

Significance

If the optimality proof holds, this is a foundational result for CMB data analysis: it establishes that the angular power spectrum is a lossless compression of a Gaussian CMB map, and it supplies a practical algorithm whose map-level steps scale as n^2, making megapixel analyses feasible. The proof is short and self-contained, the COBE application is concrete, and the intuitive picture (high-pass filtering plus edge softening) is a genuine strength. The paper also gives a useful taxonomy of data-presentation choices (raw, band powers, Cholesky, symmetric square root) that has been widely adopted. The result is falsifiable in the sense that the predicted covariance F^{-1} can be checked against Monte-Carlo skies, and the paper's own COBE numbers provide such a check.

Major objections (1)

[Abstract claim (1); Section III.E, Eqs. (26)–(30); Section IV.A] The optimality proof is exact only when the fiducial covariance C used to build E_l = (1/2) C^{-1} P_l C^{-1} equals the true sky covariance. The paper uses a fiducial C taken from a prior brute-force likelihood fit (Section IV.A, Q_rms,ps = 18.4 μK, citing Ref. [11]) but does not analyze the consequence of a wrong fiducial. If C_fid ≠ C_true, then ⟨y⟩ = W(C_fid) c_true with W ≠ F, so c̃ = F^{-1} y is biased and its covariance is not F^{-1}(C_true); the Cramér–Rao bound for the true F is then not saturated. The abstract's unconditional statement that 'no other method can measure C_l with smaller error bars' should be qualified accordingly, and the paper should either discuss iterating the fiducial (as it already suggests for the chi-squared step in Section V.D) or quantify the degradation. A concrete test would be to repeat the COBE analysis with the fiducial spectrum scaled by, say, ±20% and show the shift in Ĉ_l relative to the quoted error bars.

Minor objections (8)

[Abstract; Acknowledgments; Appendix D; Introduction, point 3; Section III.C; Section V.A.6] There are several typographical errors: 'arbitraty' and 'futute' in the abstract, 'wishers to thank' in the acknowledgments, 'measurents' in Appendix D, 'arguable necessary' in Introduction point 3, and 'chosing' in Sections III.C and V.A.6. The spelling of 'diagonalization'/'diagonalisation' and 'Cramer-Rao'/'Cramér–Rao' should also be made consistent throughout.
[Reference [12]] The citation 'preprint astro-ph/961113' appears to be a malformed arXiv identifier (astro-ph/961113 has only six digits after the slash); please check and correct it.
[Section III.D, Eq. (48)] The compressed notation F = (1/2) Σ_{m,m'} (V_λλ')^2 would be clearer if the sums over λ and λ' were written explicitly and the factor of 1/2 were shown to follow from Eq. (11); as written, a reader may stumble over which indices are being contracted.
[Figure 5 caption] The caption refers to 'power tilting' without defining it; the maps shown are z = C^{-1} x, and the caption should say so explicitly so that the figure is self-contained.
[Section IV.A] The statement that 'we make no attempts to subtract galactic contamination outside this cut' is fine, but a sentence on the expected level of residual foreground contamination and its effect on the lowest multipoles would help the reader judge the COBE results.
[Section V.A.6; Figure 2 bottom; Table 1] The paper states that the symmetric square root M = F^{1/2} is the author's preferred choice and is used in the bottom panel of Figure 2, but the Table 1 caption should repeat which decorrelation was used so that the binned values can be interpreted without cross-referencing the text.
[Section IV; throughout] The paper uses both D_l and C_l extensively; a short glossary at the start of Section IV defining D_l ≡ l(l+1)C_l/2π, the beam factor B_l^2, and the tilde notation would make the worked example easier to follow.
[Footnote 2] The suggestion to supplement the power-spectrum estimates with a_lm coefficients for l ≲ 4 to retain all information is well taken, but the paper should state whether the COBE results in Figure 2 and Table 1 include such a supplement or not.

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The paper

The paper claims

A quadratic estimator reaches the smallest error bars the CMB allows

The reading

Tegmark asks what it takes to turn a microwave sky map into an angular power spectrum — the curve cosmologists fit models against — without discarding any cosmological information on the way. His answer is a specific quadratic estimator: each multipole is measured by a fixed matrix sandwiched between the pixel vector, $x^{t}E_ Brach x - b_ Brach$, with $E_ Brach = F_{ Brach Brach}C^{-1}P_ Brach C^{-1}/2$, where $C$ is the pixel covariance and $P_ Brach$ encodes the Legendre-polynomial correlation pattern of multipole $ Brach$. He shows that this estimator's covariance matrix comes out exactly equal to the inverse of the power-spectrum Fisher matrix, which is the floor set by the Cramér–Rao inequality, so no unbiased method — including the much slower maximum-likelihood search — can produce smaller error bars. If that holds, a map of millions of pixels can be compressed to a few thousand power coefficients while keeping everything a Gaussian model could tell you, and the compression costs $n^2$ operations rather than $n^3$. The paper also gives the method a physical reading: it is the old truncated-spherical-harmonic expansion applied not to the raw map but to $z = C^{-1}x$, a map that has been high-pass filtered and had its edges softened.

What carries the argument

The load-bearing object is the power-spectrum Fisher matrix $F_{ Brach Brach'} = 2^{-1} Brach[C^{-1}P_ Brach C^{-1}P_{ Brach'}]$ and the fact that it plays three roles at once: it is the window-function matrix of the estimator, it is the covariance matrix of the raw estimates $y_ Brach$, and its inverse is the covariance of the deconvolved estimates. Because a single matrix supplies all three, the proof of optimality is a one-line substitution rather than a variational argument. A second piece of machinery makes the method usable in practice: the monopole and dipole are projected out, leaving a singular covariance, and the paper shows that the pseudo-inverse $M = Brach[C + Brach ZZ^{t}]^{-1} Brach^{t}$ is independent of the regulator $ Brach$ and reproduces the same optimality proof, so the singular case costs nothing. The computational claim rides on rewriting $ Brach P_ Brach z$ as a sum over $m$ of squared spherical-harmonic projections of $z$, reducing the work to one linear solve plus matrix-vector products.

Extended reading notes

Core claim

The central claim is that the best unbiased estimator of the CMB angular power spectrum is quadratic and has a closed form. Starting from the Gaussian likelihood, Tegmark writes the Fisher matrix for the power spectrum as $F_{ Brach Brach'} = 2^{-1} Brach[C^{-1}P_ Brach C^{-1}P_{ Brach'}]$ and then minimizes the variance of a quadratic estimator $x^{t}E_ Brach x$ subject to the normalization $ Brach P_ Brach E_ Brach = 1$. The Lagrange multiplier solution collapses to $CE_ Brach C = Brach P_ Brach$, i.e. $E_ Brach = F_{ Brach Brach}C^{-1}P_ Brach C^{-1}/2$ — an $n$-dimensional matrix inversion, not the $n^2$-dimensional one a naive pair-product treatment would demand. Rescaling to $y_ Brach = x^{t}E_ Brach x$ gives $ Brach y_ Brach = Fc$ and covariance exactly $F$, so $F^{-1}y$ is unbiased with covariance $F^{-1}$. That is precisely the Cramér–Rao bound, so the estimator is optimal among unbiased estimators and the power spectrum is a lossless compression of a Gaussian map. The paper illustrates the machinery on the 4-year COBE/DMR data ($n = 3881$ pixels after the custom galactic cut), showing window functions whose width scales as $ Brach Brach Brach Brach Brach Brach Brach Brach Brach$, explaining the high-pass and edge-tapering behaviour of $z = C^{-1}x$, and describing how to rotate the estimates into uncorrelated band powers via a Cholesky or symmetric square root of $F$ before a simple chi-squared fit for cosmological parameters.

Load-bearing premise

The estimator is built from a fiducial pixel covariance, and the proof that its error bars reach the theoretical floor assumes that fiducial matches the true sky; the paper fixes it from a prior likelihood fit and does not work out how much is lost when the guess is wrong.

Editorial extensions

If this is right

A megapixel CMB map can be reduced to a few thousand power-spectrum numbers before any parameter fitting, with no loss of information for Gaussian models — turning an infeasible $n^3$ likelihood analysis into a feasible $n^2$ pipeline.
Because the estimates are quadratic, their joint distribution is computable in closed form, so the final cosmological-parameter step can be a chi-squared fit rather than a numerically delicate nonlinear search.
The method is defined for any mask, any survey boundary, and any noise behaviour, since all of that enters only through the pixel covariance matrix $C$.
In the zero-signal-to-noise limit it reduces to the classical truncated-spherical-harmonic expansion, and in the high-signal-to-noise limit it approaches the resolution of the eigenmode method while remaining strictly lossless and avoiding an eigenvalue problem.
Presenting the spectrum as uncorrelated band powers is possible by factoring $F$ — for instance as its symmetric square root — giving non-negative, narrow, symmetric window functions.

Reading between the lines

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

The optimality proof is exact only when the fiducial covariance used to build $E_ Brach$ equals the true sky covariance; in practice one would iterate — estimate the spectrum, rebuild $C$, re-estimate — and the paper's own remark that the procedure can be repeated if the best fit strays far from the fiducial is the seed of that loop.
The same template, a quadratic form built from $C^{-1}$ and a derivative matrix, applies to any parameter that enters the covariance linearly, so the construction plausibly extends to calibration errors, beam uncertainties, or foreground amplitudes treated as extra spectra.
The lossless-compression guarantee is a statement about Gaussian fields; any non-Gaussianity in the map — point sources, lensing, unresolved foregrounds — sits outside the Fisher-matrix argument and would need to be carried separately.
Because the estimator saturates the bound in a single step, it can be read as one exact Newton step of the maximum-likelihood solution, which suggests using it as an initializer for any subsequent nonlinear refinement.

Original abstract

A new method for estimating the angular power spectrum C_l from cosmic microwave background (CMB) maps is presented, which has the following desirable properties: (1) It is unbeatable in the sense that no other method can measure C_l with smaller error bars. (2) It is quadratic, which makes the statistical properties of the measurements easy to compute and use for estimation of cosmological parameters. (3) It is computationally faster than rival high-precision methods such as the nonlinear maximum-likelihood technique, with the crucial steps scaling as n^2 rather than n^3, where n is the number of map pixels. (4) It is applicable to any survey geometry whatsoever, with arbitraty regions masked out and arbitrary noise behaviour. (5) It is not a "black-box" method, but quite simple to understand intuitively: it corresponds to a high-pass filtering and edge softening of the original map followed by a straight expansion in truncated spherical-harmonics. It is argued that this method is computationally feasible even for futute high-resolution CMB experiments with n=10^6-10^7. It is also shown that C_l computed with this method is useful not merely for graphical presentation purposes, but also as an intermediate (and arguably necessary) step in the data analysis pipeline, reducing the data set to a more manageable size before the final step of constraining Gaussian cosmological models and parameters - while retaining all the cosmological information that was present in the original map.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,878 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.58 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

A new method for estimating the angular power spectrum C_l from cosmic microwave background (CMB) maps is presented, which has the following desirable properties: (1) It is unbeatable in the sense that no other method can measure C_l with smaller error bars. (2) It is quadratic, which makes the statistical properties of the measurements easy to compute and use for estimation of cosmological parameters. (3) It is computationally faster than rival high-precision methods such as the nonlinear maximum-likelihood technique, with the crucial steps scaling as n^2 rather than n^3, where n is the number of map pixels. (4) It is applicable to any survey geometry whatsoever, with arbitraty regions masked out and arbitrary noise behaviour. (5) It is not a "black-box" method, but quite simple to understand intuitively: it corresponds to a high-pass filtering and edge softening of the original map followed by a straight expansion in truncated spherical-harmonics. It is argued that this method is computationally feasible even for futute high-resolution CMB experiments with n=10^6-10^7. It is also shown that C_l computed with this method is useful not merely for graphical presentation purposes, but also as an intermediate (and arguably necessary) step in the data analysis pipeline, reducing the data set to a more manageable size before the final step of constraining Gaussian cosmological models and parameters - while retaining all the cosmological information that was present in the original map.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A new method for estimating the angular power spectrum C_l from cosmic microwave background (CMB) maps is presented, which has the following desirable properties: (1) It is unbeatable in the sense that no other method can measure C_l with smaller error bars.	contribution 0.98	0.78	0.49	asserted 1.00
s1	(2) It is quadratic, which makes the statistical properties of the measurements easy to compute and use for estimation of cosmological parameters.	contribution 0.55	0.67	0.07	asserted 1.00
s2	(3) It is computationally faster than rival high-precision methods such as the nonlinear maximum-likelihood technique, with the crucial steps scaling as n^2 rather than n^3, where n is the number of map pixels.	contribution 0.66	0.74	0.09	asserted 1.00
s3	(4) It is applicable to any survey geometry whatsoever, with arbitraty regions masked out and arbitrary noise behaviour.	contribution 0.75	0.68	0.07	asserted 1.00
s4	(5) It is not a "black-box" method, but quite simple to understand intuitively: it corresponds to a high-pass filtering and edge softening of the original map followed by a straight expansion in truncated spherical-harmonics.	method 0.69	0.58	0.08	asserted 1.00
s5	It is argued that this method is computationally feasible even for futute high-resolution CMB experiments with n=10^6-10^7.	interpretation 0.52	0.62	0.09	hedged possibility 0.59
s6	It is also shown that C_l computed with this method is useful not merely for graphical presentation purposes, but also as an intermediate (and arguably necessary) step in the data analysis pipeline, reducing the data set to a more manageable size before the final step of constraining Gaussian cosmological models and parameters - while retaining all the cosmological information that was present in the original map.	interpretation 0.73	0.83	0.40	asserted 0.98

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from argument 0.63.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Power spectrum estimation as an intermediate data-c

**Figure 2.** FIG. 2. The power spectrum estimated from the COBE/DMR 4 year

**Figure 3.** FIG. 3. How the window functions depend on the target multipo

**Figure 4.** FIG. 4. How the window functions depend on the sky coverage.

**Figure 5.** FIG. 5. Sky maps before and after power tilting.

**Figure 6.** FIG. 6. Window functions before and after power tilting.

**Figure 8.** FIG. 8. The power spectrum observed by COBE/DMR binned into 8

Showing 7 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9611174 by the authors.

References

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astro-ph.CO 2019-06 accept novelty 5.0 of 10

Approximate narrow-kernel covariance formulas, implemented in the public code NaMaster, reproduce the Gaussian covariance of masked spin-0 and spin-2 power spectra and the resulting cosmological posteriors to a few pe...
Blind foreground subtraction for intensity mapping experiments
astro-ph.CO 2014-09 conditional novelty 5.0 of 10

Polynomial fitting, PCA and ICA remove Galactic and extragalactic radio foregrounds from simulated SKA intensity maps about equally well, leaving a power-spectrum bias near 20 percent of the statistical error.
Correlation of CMB with large-scale structure: II. Weak lensing
astro-ph 2008-01 conditional novelty 5.0 of 10

Cross-correlating a WMAP-derived CMB lensing map with SDSS LRGs, SDSS quasars and NVSS radio sources yields a 2.5-sigma detection of CMB lensing with amplitude 1.06 +/- 0.42 times the WMAP-cosmology prediction.
Results from the Wilkinson Microwave Anisotropy Probe
astro-ph.CO 2014-04 conditional novelty 1.0 of 10

WMAP's nine-year temperature and polarization maps pin the flat Lambda-CDM model to percent-level precision and pass a suite of inflation tests: flat geometry, adiabaticity, super-horizon fluctuations, a slight red ti...

How later papers used it

125 later papers cite it

Evidence: substantial. 119 of the citing papers share no author with it, and 46 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 202 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

6 later papers by other authors are listed below: 6 used its method or tool.

Uses its method or tool 6 of 47 papers with this label

Used a correlation algorithm based on this paper's quadratic estimator technique to mitigate polyphase filter bank related losses.

A VLBI Software Correlator for Fast Radio Transients (2024) · 1.00 · paraphrased: the licence does not allow quoting
Used a quadratic maximum likelihood estimator from this paper, as implemented in another work, to estimate full-sky reconstruction multipoles in a search for an intrinsic dipole.

Constraints on cosmology beyond $\Lambda$CDM with kinetic Sunyaev Zel'dovich velocity reconstruction (2024) · 1.00 · paraphrased: the licence does not allow quoting
In this Letter, we review the TF correction through the quadratic estimator formalism [ref] .

A quadratic estimator view of the transfer function correction in intensity mapping surveys (2025) · Introduction · 0.93
Thus, in order to measure the galaxy-SGWB cross-correlation, we implement a quadratic minimum variance estimator, as described in [ref] .

Tomographic constraints on the production rate of gravitational waves from astrophysical sources (2024) · Galaxy-SGWB cross-correlation · 0.99
Followed this paper's QML approach to estimate the power spectrum.

A map-based method for eliminating systematic modes from galaxy clustering power spectra with application to BOSS (2018) · 0.96 · paraphrased: the licence does not allow quoting
Computed the Fisher information matrix as in this paper to obtain the error bar on each power spectrum wavenumber bin.

Measuring the spectrum of primordial gravitational waves with CMB, PTA and Laser Interferometers (2020) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

68 arXiv papers cite this in 118 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 83 sentences without a confident label

Builds on it 1 paper

In this section we review the power spectrum estimation formalism introduced by [ref] , and adapt it for $21 cm$ tomography.

A Method for 21cm Power Spectrum Estimation in the Presence of Foregrounds · The Mathematical Framework · 0.85

Uses its method or tool 20 papers

[ref] , and are based on optimal quadratic estimators [ref] .

Optimal constraints on Primordial non-Gaussianity with the eBOSS DR16 quasars in Fourier space · Introduction and main results · 0.93
Following Ref. [ref] , we assume a Gaussian likelihood for the data, and denote the unknown systematics as $u$, so that one can express the likelihood function for a data vector $d$ as [equation] where $r(θ,𝐮) ≡d - W·m(θ) -u$, $m(θ)$ is the emulator prediction for the data vector $d$, $W$ is the Window function (which accounts for the point spreading in Fourier k space [ref] ), and $Γ≡Σ^-1$ where $Σ$ in the covariance matrix.

HERA bound on x-ray luminosity when accounting for population III stars · The data Likelihood · 1.00
We estimate the $c_i$ by forming quadratic combinations of the data [ref] , [equation] These are related to the estimated $ĉ_i$ by the response matrix $𝐅$, [equation] where [equation] If $C^gT_l ≪√(C^gg_l C^TT_l)$, then the $ĉ_i$ are good approximations to the maximum likelihood estimates of the $c_i$.

Correlation of CMB with large-scale structure: I. ISW Tomography and Cosmological Implications · Methodology · 0.99

Cites it as background 13 papers

First, efficient estimation of (windowed) polyspectra has been considered in a number of works, including [ref] for the power spectrum, [ref] for the bispectrum and [ref] for the trispectrum, primarily for the amplitudes of specific separable shapes.

Optimal Estimation of the Binned Mask-Free Power Spectrum, Bispectrum, and Trispectrum on the Full Sky: Scalar Edition · Introduction · 0.85
There is a large literature of unbiased power spectrum estimators that are optimal, have well-understood statistical properties, and robustly meet common observational challenges such as partial sky coverage, linear sky contamination, and inhomogeneous noise [ref] .

Catalog-based pseudo-$C_\ell$s · Introduction · 0.86
This is the same problem that occurs in the context of the CMB power spectrum and many works have been devoted to the problem [ref] .

How to detect gravitational waves through the cross-correlation of the galaxy distribution with the CMB polarization · Sky Cuts · 0.98

Discussion

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