CMB Lensing Reconstruction on the Full Sky

Takemi Okamoto; Wayne Hu

REVIEW

CMB Lensing Reconstruction on the Full Sky

Takemi Okamoto, Wayne Hu

Astrophysics · Submitted 3 Jan 2003 · 25 references

Pith's verdict: A clean, correct full-sky generalization of the flat-sky lensing quadratic estimators, with the real-space construction and the ΘE cross estimator as the genuinely new pieces; the only real soft spot is that the noise curves are N^(0)-only.

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arXiv IDastro-ph/0301031 (version 2)

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CitationTakemi Okamoto, Wayne Hu. "CMB Lensing Reconstruction on the Full Sky." arXiv:astro-ph/0301031 (2003). https://pith.science/paper/UC4VUMBJ

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Editor's note

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Bottom line: this is a solid methods paper and the derivation holds together. Okamoto and Hu take the flat-sky quadratic estimators of Hu & Okamoto (their ref. [10]) onto the sphere, derive the minimum-variance weights and noise covariance from scratch (Eqs. 29–39), and then show how to build all of them with real-space products plus fast harmonic transforms (Eqs. 44–47). The new content is the full-sky generalization and the polarization cross estimators — ΘE, ΘB, EE, EB — which the paper says plainly in Sect. IV. That claim checks out against the cited prior work, which was flat-sky and, apart from EB, did not have the cross terms. The ΘE estimator is the one that matters for near-term experiments where polarization information comes mostly through the T–E channel, so this is not a cosmetic extension. The approximation is quantified rather than asserted: the non-optimal ΘE weighting differs from minimum variance by under 1% (Fig. 2), and the flat-sky limit is checked explicitly in Appendix B with the residual shown in Fig. 3. The tensor/spin-weight appendix is careful and the conventions are stated, which is more than most papers in this area bother to do. I found no circular step; the self-citations are used as benchmarks, not as load-bearing inputs. On the soft spot: the reader flags the Gaussian variance as unflagged, and that is slightly off — the text immediately before Eq. (35) says they minimize the Gaussian variance, so the assumption is named. What is not done is quantifying the consequence. The noise in Eq. (39) and Fig. 1 is N^(0)-only; the connected lensing trispectrum (what people later called N^(1)) is largest at L ≲ 100, exactly where the paper says the S/N peaks. So the absolute low-L forecast is optimistic and the 'minimum variance' label is minimum-variance-within-the-Gaussian-ansatz. That does not touch the estimator construction, and it does not touch the ΘE-versus-minimum-variance comparison, since both are quadratic in the same fields and a common correction largely cancels in the ratio. So: a real caveat on the forecast, not on the method. Minor: no code shipped, so reproducibility is re-implement-from-equations. Worth a serious referee, and worth citing if you are doing full-sky lensing or delensing.

Referee report

Significance

If correct, this is a foundational methods paper: it supplies the all-sky generalization of the quadratic lensing estimators that had previously been written only in the flat-sky limit, and it does so in a form that can be implemented with fast spherical harmonic transforms. The derivations are self-contained rather than quoted: the minimum-variance weights and noise covariance are obtained from scratch in Eqs. (29)–(39), and the approximations are quantified rather than asserted, with Fig. 2 bounding the ΘE non-optimality below 1% and Fig. 3 bounding the flat-sky/full-sky noise difference below about 1% for L > 200. The paper has no free parameters and introduces no ad hoc entities; its claims are checkable against the flat-sky results of the cited prior work, and Appendix B performs exactly that check. The extension to the ΘE, ΘB and EB channels is a genuine addition, since most of the incremental polarization information for near-term experiments is expected in the temperature-polarization cross-correlation.

Major objections (2)

[Sect. III, Eq. (39) and Fig. 1] The noise covariance in Eq. (39), and therefore the curves in Fig. 1, is computed under the Gaussian assumption that the four-point function of the observed fields factorizes into two-point functions. The text immediately preceding Eq. (35) does state this ('we derive the minimum variance estimator by minimizing the Gaussian variance'), so the assumption is named rather than hidden. What is not stated is its consequence: lensing itself generates a connected trispectrum whose contribution to the estimator variance is largest at low L, precisely the regime the paper identifies in Sect. I as the signal-to-noise peak (L ~ 10^2). The low-L curves in Fig. 1 should therefore be described explicitly as Gaussian-ansatz noise, with a sentence noting that the connected (non-Gaussian) contribution is omitted and is expected to be most important where the plotted signal-to-noise is highest. This does not affect the estimator derivation, and it largely cancels in the ratio plotted in Fig. 2, since both estimators compared there are quadratic in the same fields; the request is for an explicit caveat and, if feasible, an order-of-magnitude estimate of the omitted term at L < 100.
[Sect. IV, Eqs. (44)–(47)] Section IV asserts that the real-space estimators of Eq. (45) reproduce the harmonic-space estimators of Eq. (29) with Â^α_L = A^α_L for every channel except ΘE, and that this is 'straightforward to verify'. This equivalence is the practical payload of the section, since it is what licenses the use of fast transforms, so it should be demonstrated rather than asserted. Please add an appendix showing the algebra for at least one pure channel (ΘΘ) and one mixed channel (EE or ΘB), making explicit how the spin-gradient operations of Eq. (A17) and the filtered fields of Eq. (47) combine to reproduce the weights f^α_{l1Ll2} of Table I and the normalization of Eq. (34).

Minor objections (7)

[Eq. (22)] The parity factor is defined as β_{ll'lL} but is used everywhere else, including Eq. (21) and Eq. (28), as β_{ll'L}; the extra l in the subscript of the definition appears to be a typographical error.
[Eqs. (42)–(43)] Equation (42) is followed by a bare '(43)' with no equation attached; the numbering should be corrected.
[Sect. IV, Eq. (48) and Fig. 2] The abstract and Sect. III describe the estimators as minimum variance, while the ΘE estimator actually constructed in Sect. IV uses the non-optimal weights of Eq. (48). It would help to state in the abstract or at the start of Sect. IV that the minimum-variance label refers to the harmonic-space estimators of Sect. III, and that the real-space ΘE estimator is a near-optimal variant whose penalty is bounded by Fig. 2.
[Sect. II, Eq. (21)] The estimator is normalized through Eq. (33) at first order in the deflection, so it is unbiased only to O(φ). A sentence noting that the O(φ^2) correction is neglected, and that it is small for the deflection amplitudes considered, would make the accuracy of the reconstruction explicit.
[Fig. 1] The caption should state which five estimators are plotted (ΘΘ, ΘE, EE, ΘB, EB) and identify the curves, and should note that for a single channel the noise is simply N^{αα}_L = A^α_L, as stated after Eq. (39).
[References [9] and [14]] Reference [14] (Challinor & Chon) is listed as 'Phys. Rev. D in press' with no volume or page and with the year 2000, and reference [9] (Hirata & Seljak) is likewise listed as in press; both should be updated to their final bibliographic data if they have since appeared.
[Sect. II title and throughout] There are numerous typesetting artifacts in the text as submitted, for example 'MUL TIPOLE' in the Sect. II heading, 'Physi cs' in the author affiliation, 'gravitatio nal' and 'reconstruc tion' in the abstract, 'the the change' before Eq. (10), and 'with with' in the Fig. 1 caption; these should be cleaned up in the final version.

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The paper

The paper claims

The paper derives minimum-variance estimators that recover the CMB lensing potential across the whole spherical sky, plus a fast real-space recipe for building them.

The reading

Gravitational lensing by intervening dark matter imprints a small, coherent distortion on the microwave background, and that distortion shows up as correlations between multipole moments that would otherwise be independent. The authors' aim is to write down, for a spherical sky rather than a small flat patch, the best possible estimator of the projected lensing potential built from products of two measured multipoles — the minimum-variance quadratic estimator — together with its expected noise. The motivation is scale: lensing is most detectable at angular scales of a few degrees, so any survey covering more than about a percent of the sky cannot treat the sky as flat without bias. The paper also supplies a practical construction in which the harmonic-space convolutions are replaced by ordinary products of filtered fields in real space, so that fast spherical harmonic transforms do the work. It extends earlier flat-sky treatments to estimators built from the temperature-polarization cross-correlation, which the authors argue will carry most of the extra polarization information for realistic noise levels.

What carries the argument

The load-bearing object is the spin-dependent gradient operator $D_i = $ a covariant derivative acting on a spin-$s$ field, which lets the authors write the lensing correction to temperature and polarization harmonics in one unified form (Eq. 21) and then reuse the same operator to build the estimator in real space. Its action on spin-weighted spherical harmonics simply raises and lowers the spin index (Eq. A17), which is why the otherwise awkward harmonic-space convolutions collapse into pointwise products of filtered maps. Around this operator sit two other pieces of machinery: the minimum-variance weight solution (Eq. 35), which for a single field reduces to $f^{*}_{l_1 L l_2} / (2 C^{aa}_{l_1} C^{aa}_{l_2})$ and for uncorrelated fields to $f^{*}_{l_1 L l_2} / (C^{aa}_{l_1} C^{bb}_{l_2})$, and the noise covariance (Eq. 39) that converts those weights into a forecast for how well the projected mass can be mapped.

Extended reading notes

Core claim

The central claim is a complete set of full-sky quadratic estimators for the lensing potential $d^{ab}_{LM}$, formed as a weighted sum over multipole pairs of the observed temperature, $E$-mode, and $B$-mode fields (Eq. 29), with weights $g^{ab}_{l_1 l_2}(L)$ chosen to minimize the estimator variance (Eq. 35) and a Gaussian noise covariance $N^{ab}_L$ given in Eq. 39. The paper tabulates the six quadratic pairs — $TT$, $TE$, $EE$, $TB$, $EB$, $BB$ — with explicit weight functions (Table I), and shows that the $TE$ and $TB$ channels are genuinely new relative to prior flat-sky work. It then shows that the same estimators can be built in angular space from a single operator $P[X,Y] = -D_i[X D_i Y]$ acting on suitably filtered fields (Eqs. 44-47), so that no explicit harmonic-space convolution is needed. The approximate real-space $TE$ estimator is shown to differ from the minimum-variance one by less than 1% in noise, and the full-sky expressions are shown to differ from flat-sky ones by less than about 1% for $L > 200$, with the flat-sky versions slightly overestimating the noise.

Load-bearing premise

The noise forecasts and the claim that the approximate $TE$ estimator is essentially as good as the optimal one both assume the estimator's variance is Gaussian, meaning the four-point function of the observed fields factorizes into products of two-point functions — an assumption that lensing itself violates at exactly the low multipoles where the signal-to-noise is highest.

Editorial extensions

If this is right

A survey covering a large fraction of the sky can reconstruct the projected lensing potential with near-optimal weights rather than the flat-sky approximations that were previously the only option.
The temperature-polarization cross-correlation becomes a usable lensing channel, which matters for experiments whose polarization sensitivity limits what the $EE$ channel alone can deliver.
Lens reconstruction can be implemented with fast spherical harmonic transforms on the sphere or fast Fourier transforms in the flat-sky limit, using the same code structure in both cases.
Full-sky and flat-sky estimators agree to about 1% in noise above $L \approx 200$, so the flat-sky shortcut is adequate at small angular scales but not where lensing signal-to-noise peaks.
The formalism gives a route to removing the lensing contaminant from large-area polarization maps, which is a prerequisite for searches for primordial gravitational waves via $B$-modes.

Reading between the lines

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

If the estimator noise is quoted under a Gaussian four-point assumption, then the quoted signal-to-noise at the low multipoles where lensing peaks is likely optimistic in detail, because the lensed fields have a connected four-point function that the covariance in Eq. 39 does not include.
The real-space construction is the part most likely to outlive the specific weight formulas: once the filtered-field products exist, swapping in improved weights or non-Gaussian corrections is a local change rather than a rewrite.
The same spin-gradient operator that organizes the lensing expansion should also organize delensing: subtracting the estimated deflection from the maps uses the identical $D_i$ structure, so an implementation of the estimator is most of an implementation of the delenser.
Because the $B$-mode estimator is dropped on the assumption that unlensed $B$-power is small, the framework is implicitly tuned to a cosmology without a large primordial tensor contribution; a detection of primordial $B$-modes would require revisiting that omission.

Original abstract

Gravitational lensing of the microwave background by the intervening dark matter mainly arises from large-angle fluctuations in the projected gravitational potential and hence offers a unique opportunity to study the physics of the dark sector at large scales. Studies with surveys that cover greater than a percent of the sky will require techniques that incorporate the curvature of the sky. We lay the groundwork for these studies by deriving the full sky minimum variance quadratic estimators of the lensing potential from the CMB temperature and polarization fields. We also present a general technique for constructing these estimators, with harmonic space convolutions replaced by real space products, that is appropriate for both the full sky limit and the flat sky approximation. This also extends previous treatments to include estimators involving the temperature-polarization cross-correlation and should be useful for next generation experiments in which most of the additional information from polarization comes from this channel due to sensitivity limitations.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,353 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.44 · 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.

Gravitational lensing of the microwave background by the intervening dark matter mainly arises from large-angle fluctuations in the projected gravitational potential and hence offers a unique opportunity to study the physics of the dark sector at large scales. Studies with surveys that cover greater than a percent of the sky will require techniques that incorporate the curvature of the sky. We lay the groundwork for these studies by deriving the full sky minimum variance quadratic estimators of the lensing potential from the CMB temperature and polarization fields. We also present a general technique for constructing these estimators, with harmonic space convolutions replaced by real space products, that is appropriate for both the full sky limit and the flat sky approximation. This also extends previous treatments to include estimators involving the temperature-polarization cross-correlation and should be useful for next generation experiments in which most of the additional information from polarization comes from this channel due to sensitivity limitations.

background contribution method result interpretation main claim

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s1	Studies with surveys that cover greater than a percent of the sky will require techniques that incorporate the curvature of the sky.	background 0.81	0.05	0.04	asserted 0.88
s2	We lay the groundwork for these studies by deriving the full sky minimum variance quadratic estimators of the lensing potential from the CMB temperature and polarization fields.	contribution 0.95	0.97	0.41	asserted 1.00
s3	We also present a general technique for constructing these estimators, with harmonic space convolutions replaced by real space products, that is appropriate for both the full sky limit and the flat sky approximation.	method 0.75	0.93	0.17	asserted 1.00
s4	This also extends previous treatments to include estimators involving the temperature-polarization cross-correlation and should be useful for next generation experiments in which most of the additional information from polarization comes from this channel due to sensitivity limitations.	contribution 0.65	0.74	0.12	asserted 0.73

What kind of paper the classifier reports this is: method or tool 0.87 · evidence from proof or derivation 1.00.

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full text
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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: Deflection and noise power spectra for the quadratic a

**Figure 2.** FIG. 2: Fractional difference between the approximate and th

**Figure 3.** FIG. 3: Fractional differences

Figures reproduced from arXiv:astro-ph/0301031 by the authors.

References

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W. Hu, Phys. Rev. D 64, 083005 (2001)

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W. Hu, Phys. Rev. D 62, 043007 (2000)

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CMB Lensing Reconstruction Using Two Years of Temperature Data from the SPT-3G Summer Survey
astro-ph.CO 2026-07 conditional novelty 5.0 of 10

First CMB lensing reconstruction from two years of SPT-3G Summer temperature data yields A_comb = 1.015 ± 0.053 over 50 < L < 2000, consistent with Planck 2018 ΛCDM.
Searching for primordial features with radio surveys: synergy between the power spectrum and bispectrum
astro-ph.CO 2026-06 unverdicted novelty 5.0 of 10

Joint power spectrum and bispectrum analysis from future HI intensity mapping surveys improves constraints on primordial feature amplitudes by 30-40% and achieves percent-level precision on oscillation frequencies whe...
Detecting rotation from lensing in the CMB
astro-ph.CO 2025-01 accept novelty 5.0 of 10

If lensing field rotation also rotates CMB polarization, SPT-3G plus galaxy-survey templates should detect the effect at SNR~7, and CMB-S4 deep could detect it internally at 3.9σ.
Stage IV CMB forecasts for warm inflation
astro-ph.CO 2024-12 conditional novelty 5.0 of 10

Future CMB surveys are forecast to tighten the dissipation parameter of quartic warm inflation by up to an order of magnitude and to exclude the cubic dissipation model under a null tensor-to-scalar ratio.
CMB lensing from Planck PR4 maps
astro-ph.CO 2022-06 accept novelty 5.0 of 10

Planck PR4 maps with optimal filtering yield CMB lensing amplitude 1.004 ± 0.024 and σ8 Ωm^0.25 = 0.599 ± 0.016, the tightest lensing constraint yet.
The Atacama Cosmology Telescope: Constraints on Cosmic Birefringence
astro-ph.CO 2020-01 accept novelty 5.0 of 10

ACTPol polarization data over 456 square degrees give a birefringence power spectrum consistent with zero, improving upper limits on scale-invariant anisotropic birefringence by a factor of 2 to 3.
The promising future of a robust cosmological neutrino mass measurement
astro-ph.CO 2018-08 conditional novelty 5.0 of 10

A uniform grid of 35 mock-likelihood forecasts shows future CMB plus galaxy-survey data should detect a 0.06 eV neutrino mass at 3 to 10 sigma, with the error roughly doubling when dark energy is allowed to vary in time.
Reconstruction of the remote dipole and quadrupole fields from the kinetic Sunyaev Zel'dovich and polarized Sunyaev Zel'dovich effects
astro-ph.CO 2017-07 conditional novelty 5.0 of 10

Minimum-variance quadratic estimators can recover the remote CMB dipole and quadrupole fields from CMB temperature and polarization cross-correlated with galaxy surveys, with high signal-to-noise on the largest scales...
Exploring Cosmic Origins with CORE: Inflation
astro-ph.CO 2016-12 conditional novelty 5.0 of 10

Forecasts indicate a CMB polarization satellite with 1.7 microkelvin-arcmin sensitivity and 5 arcmin resolution would measure the tensor-to-scalar ratio to about 2e-4 and the scalar tilt to 0.0014, enough to separate ...
Exploring Cosmic Origins with CORE: Cosmological Parameters
astro-ph.CO 2016-11 conditional novelty 5.0 of 10

Simulated forecasts for the proposed CORE-M5 CMB satellite project 3 to 8 fold improvements over Planck 2015 on neutrino mass, relativistic species, curvature and dark energy, conditional on foreground residuals sitti...
Future CMB tests of dark matter: ultra-light axions and massive neutrinos
astro-ph.CO 2016-07 conditional novelty 5.0 of 10

CMB-S4 should constrain the ultra-light axion fraction roughly ten times better than Planck over 10^-32 to 10^-23 eV, with lensing breaking the axion-neutrino degeneracy and pushing the CMB mass bound from 10^-25 to 1...
Robust forecasts on fundamental physics from the foreground-obscured, gravitationally-lensed CMB polarization
astro-ph.CO 2015-09 accept novelty 5.0 of 10

An end-to-end Fisher pipeline combining parametric foreground cleaning and three delensing methods finds that a Stage-IV ground survey plus a space mission could reach sigma(r) near 1.3e-4 and sigma(M_nu) near 31 meV.
Constraints on Large-Scale Dark Acoustic Oscillations from Cosmology
astro-ph.CO 2013-10 conditional novelty 5.0 of 10

At most about five percent of the dark matter can have been strongly coupled to a bath of dark radiation, otherwise the resulting dark acoustic oscillations would have left visible imprints on the CMB, CMB lensing, an...
Dark energy with non-adiabatic sound speed: initial conditions and detectability
astro-ph.CO 2010-04 accept novelty 5.0 of 10

Planck cross-correlated with an LSST-like survey can only exclude dark energy sound speeds near zero, and Planck alone cannot detect dark energy clustering at all.
Constraints on Cosmic Strings from the Curl-Mode CMB Lensing Power Spectrum measured by ACT DR6
astro-ph.CO 2026-06 unverdicted novelty 4.0 of 10

ACT DR6 curl lensing yields 2σ bounds Gμ P^{-1} ≤ 3.5×10^{-5} (small-P) and Gμ ≤ 5.0×10^{-5} (P=1), tightened further with Planck 2013 data.
Towards a measurement of the spectral runnings
astro-ph.CO 2016-11 conditional novelty 4.0 of 10

Fisher forecasts give a Stage-4 CMB experiment sigma(alpha_s) = 0.0025 and sigma(beta_s) = 0.0045, enough to test primordial-black-hole scenarios but not the slow-roll predictions of 10^-3 and 10^-5.
Planck 2015 results. XXI. The integrated Sachs-Wolfe effect
astro-ph.CO 2015-02 conditional novelty 4.0 of 10

Cross-correlating Planck 2015 CMB maps with NVSS, SDSS, WISE and the Planck lensing map yields a 4-sigma detection of the integrated Sachs-Wolfe effect, consistent with Lambda-CDM expectations.
A Guide to Designing Future Ground-based CMB Experiments
astro-ph.CO 2014-02 conditional novelty 4.0 of 10

Across a grid of detector counts, beam sizes, and sky fractions, a Stage-IV CMB experiment could reach 1-sigma errors of 15 meV on neutrino mass, 0.0011 on the scalar tilt, and 0.00009 on the tensor-to-scalar ratio, w...
Weak lensing of the CMB
astro-ph.CO 2009-11 accept novelty 1.0 of 10

The review lays out how intervening matter distorts the cosmic microwave background, how that distortion shows up in power spectra and bispectra, and how a quadratic maximum-likelihood estimator can recover the deflec...
Massive neutrinos and cosmology
astro-ph 2006-03 accept novelty 1.0 of 10

A detailed review of how cosmological observations bound the sum of neutrino masses through the free-streaming suppression of small-scale structure, and of future sensitivity forecasts.

How later papers used it

215 later papers cite it

Evidence: substantial. 207 of the citing papers share no author with it, and 106 of those carry a confident label from Pith's citation classifier.

Pith read 31 of the 380 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

13 later papers by other authors are listed below: 13 used its method or tool.

Uses its method or tool 13 of 106 papers with this label

Assumed lensing reconstruction is performed by adopting the method based on the quadratic estimator.

Constraining Light Gravitino Mass from Cosmic Microwave Background (2009) · 1.00 · paraphrased: the licence does not allow quoting
Used a curved-sky quadratic estimator technique to reconstruct the lensing potential from SPT-3G Summer survey temperature data.

CMB Lensing Reconstruction Using Two Years of Temperature Data from the SPT-3G Summer Survey (2026) · 1.00 · paraphrased: the licence does not allow quoting
Adopted the curved sky quadratic estimators developed by Okamoto and Hu.

Forecasts on CMB lensing observations with AliCPT-1 (2022) · 1.00 · paraphrased: the licence does not allow quoting
The lensing deflection field $d̂$ is reconstructed via the curved-sky quadratic estimator (QE) of Okamoto and Hu [ref] , in the implementation described in [ref] .

Constraints on Cosmic Strings from the Curl-Mode CMB Lensing Power Spectrum measured by ACT DR6 (2026) · ACT DR6 curl-mode bandpowers · 1.00
Reconstructed the lensing potential using a quadratic estimator approach, with the estimator for lensed E and B maps given by this paper.

Impact of Anisotropic Birefringence on Measuring Cosmic Microwave Background Lensing (2022) · 0.95 · paraphrased: the licence does not allow quoting
We use the public CLASS delens package [ref] [<https://github.com/selimhotinli/class_delens>] to calculate $N_L^κκ$ for the H&O estimator [ref] .[We cross checked that the CLASS delens package returns the same $N_L^κκ$ as the public tempura package (<https://github.com/simonsobs/tempura>).

Cosmological Parameter Forecasts for a CMB-HD Survey (2023) · CMB Lensing Power Spectrum Noise · 0.97
Determined the optimal combination of the minimum variance estimator by the same analogy as in this paper.

Full-sky lensing reconstruction of gradient and curl modes from CMB maps (2011) · 0.86 · paraphrased: the licence does not allow quoting
Used a general quadratic estimator form for the lensing potential inspired by this paper.

Full-Sky Lensing Reconstruction of 21 cm Intensity Maps (2019) · 0.90 · paraphrased: the licence does not allow quoting
Used the quadratic estimator method of Hu & Okamoto, which provides an algorithm for estimating the noise spectrum from observed CMB primary anisotropy and noise power spectra.

Constraining Fundamental Physics with Future CMB Experiments (2010) · 1.00 · paraphrased: the licence does not allow quoting
Estimated the noise spectrum of the lensing potential following the technique developed in this paper, optimally combining temperature and polarization fields.

Application of cross correlations between CMB and large scale structure to constraints on the primordial non-Gaussianity (2011) · 1.00 · paraphrased: the licence does not allow quoting
Estimated the noise of the CMB lensing potential as the statistical error of lensing reconstruction based on the optimal quadratic estimator.

Probing Neutrinos from Planck and Forthcoming Galaxy Redshift Surveys (2013) · 1.00 · paraphrased: the licence does not allow quoting
Used the quadratic estimator method of Hu & Okamoto, which provides an algorithm for estimating the noise spectrum from observed CMB primary anisotropy and noise power spectra.

Exploring Cosmic Origins with CORE: Cosmological Parameters (2016) · 1.00 · paraphrased: the licence does not allow quoting
Obtained the deflection power spectra through the quadratic estimator of the lensing deflection field presented in this paper.

Future constraints on variations of the fine structure constant from combined CMB and weak lensing measurements (2012) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

174 arXiv papers cite this in 312 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 · 167 sentences without a confident label

Contests it 2 papers

This choice leads to five quadratic estimators, with $ab=TT,TE,EE,EB,TB$; the $BB$ case is excluded because the method of Ref. [ref] is only valid when the lensing contribution is negligible compared to the primary anisotropy, assumption that fails for the B modes in the case of Planck.

Future CMB cosmological constraints in a dark coupled universe · Future CMB data analysis · contest 0.77
This choice leads to five quadratic estimators, with $ab=TT,TE,EE,EB,TB$; the $BB$ case is excluded because the method of Ref. [ref] is only valid when the lensing contribution is negligible compared to the primary anisotropy, assumption that fails for the B modes in the case of Planck.

Future weak lensing constraints in a dark coupled universe · CMB data · contest 0.74

Builds on it 1 paper

In section [ref], we extend the quadratic estimator of Ref. [ref] to the case including both the gradient and curl modes of deflection angle.

Full-sky lensing reconstruction of gradient and curl modes from CMB maps · Introduction · 1.00

Uses its method or tool 83 papers

In harmonic space, the Okamoto & Hu [ref] estimator becomes

CMB temperature lensing power reconstruction · Power spectrum biases · 0.91
We assume lensing reconstruction is performed by adopting the method based on quadratic estimator [ref] , and the expected noise in lensing potential is calculated by the publicly available FuturCMB2 code developed by the authors of [ref] .

Constraining Light Gravitino Mass from Cosmic Microwave Background · Constraints on light gravitino mass · 1.00
In the flat-sky approximation, a minimum-variance estimator for the weak-lensing deflection field, $d(L⃗) ≡√(L(L+1))ϕ(L⃗)$ can be built out of off-diagonal correlations of the CMB temperature fluctuations, and is [ref] [equation] where [equation] The lensing potential power spectrum is estimated from the expectation value of the estimator in Eq. [ref] [equation] where $πM/L$ is the angular coordinate of $L⃗$ and $B(L)$ are the standard Gaussian and non-Gaussian contributions to the full four-point correlation [ref] .

Baryons still trace dark matter: probing CMB lensing maps for hidden isocurvature · CIPs and the CMB · 0.91

Compares with it 6 papers

Note that, if the curl mode is absent, $=0$, the resultant form of the weight function for $$ exactly coincides with the one obtained in Ref. [ref] .

Full-sky lensing reconstruction of gradient and curl modes from CMB maps · Estimator · 0.95
This is in contrast to the first-order formula used to derive the quadratic estimators of Okamoto and Hu [ref] , which contains an unlensed spectrum.

CMB temperature lensing power reconstruction · Discussion · 0.82
This derivation recovers for example the minimum-variance (MV), temperature-only and polarization-only lensing estimators as obtained first by Ref. [ref] (for the gradient term), with identical normalization.

Position-space curved-sky anisotropy quadratic estimation · Derivation of optimal QE weights · 0.93

Cites it as background 24 papers

As is well known, in the spatially homogeneous cosmology, inhomogeneities of the matter (galaxy) distribution modify the sky pattern of CMB polarisation by gravitational lensing and produce B-modes from E-modes [ref] [ref] [ref] [ref] [ref] .

The Gravitational Lensing Effect on the CMB Polarisation Anisotropy in the {\Lambda}-LTB Model · Introduction · 0.83
As discussed, estimates of the lensing-potential power spectrum, $C_L^ϕϕ$, are built out of the (non-Gaussian) connected part of the CMB trispectrum [ref] .

Probing spatial variation of the fine-structure constant using the CMB · Contribution of fine-structure constant fluctuations to CMB lensing estimators · 0.85
The trispectrum, or connected 4-point function, of the CMB is very sensitive to lensing [ref] and is the basis for all current estimators of the lensing potential power spectrum [ref] .

Extragalactic Foreground Contamination in Temperature-based CMB Lens Reconstruction · Introduction · 0.84

Discussion

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