Mass Reconstruction with CMB Polarization

Takemi Okamoto (U. Chicago); U. Chicago); Wayne Hu (CfCP

REVIEW

Mass Reconstruction with CMB Polarization

Wayne Hu (CfCP, U. Chicago), Takemi Okamoto (U. Chicago)

Astrophysics · Submitted 30 Nov 2001 · 15 references

Pith's verdict: The paper that establishes the EB quadratic estimator as the best of the six pairings, on a physical argument that holds up; the L~1000 reach is the one number to treat as provisional, since it rests on the Gaussian noise term alone.

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arXiv IDastro-ph/0111606 (version 3)

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CitationTakemi Okamoto (U. Chicago), U. Chicago), Wayne Hu (CfCP. "Mass Reconstruction with CMB Polarization." arXiv:astro-ph/0111606 (2001). https://pith.science/paper/HYY6ZDX5

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Editor's note

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This is the paper that puts the EB channel on the map as the best of the six quadratic estimators, and the reason it works is physical rather than numerical: the sin 2φ kernel suppresses the low-L contribution to B, so the EB signal lives at high L where the unlensed B field is zero. That argument is correct, and it is what every arcminute lensing analysis since has built on. Worth your time if you care about CMB lensing or polarization experiment design, and worth a serious referee either way. What is genuinely new here, given Hu 2001a,b for the temperature-only estimator and Hu 2000b and Benabed et al. for statistical EB correlation measures, is the explicit extension to all six pairings with closed-form normalization and noise (Eqs. 10-18), the real-space filtered-field construction in §4 (Eqs. 25-28) that makes the estimator actually implementable, and the quantitative demonstration in Fig. 3 that EB dominates at high L. The identity N_αα = A_α is used correctly, not asserted. The figures are doing real work: the toy lensing picture, the noise-versus-beam and noise-versus-noise-level scan, the EB-versus-ΘΘ reconstruction, and the B-mode decontamination demo in Fig. 9. On soft spots, the one that matters is the L~1000 number. It uses only the disconnected noise N^(0); the connected trispectrum N^(1) is deferred to Okamoto & Hu (in preparation), and §5.1 says so. At L~1000 with the reference experiment the B field in the estimator is lensing-dominated, so N^(1) is not obviously negligible. The authors note the temperature case was minor (Hu 2001a), and the later published follow-up found N^(1)_EB small, so the claim probably survives, but within this paper the specific L~1000 figure is provisional and the abstract runs slightly ahead of the text. The rest is minor: flat-sky throughout with the all-sky generalization deferred, and the foreground-negligibility claim cited to Hu 2000a rather than demonstrated. The dependence on externally known unlensed spectra is stated openly in §4, so it is not a hidden flaw. One correction to the record: the classifier reports no stated limitations, which is wrong. §4 and §5.1 both state limitations explicitly, and the authors deserve credit for that. Recommendation: accept and send out. The derivation is checkable by hand, the central physical argument holds, and the one gap is acknowledged and was closed in the follow-up.

Referee report

Significance

If the signal-to-noise estimates hold, arcminute-resolution polarization experiments become mass-mapping instruments reaching the linear regime over 0.002 < k < 0.2 h/Mpc, roughly an order of magnitude beyond temperature-only reconstruction. The estimator is fully specified with no free parameters, the normalization and noise are derived from the same assumed spectra, and the noise thresholds quoted in the abstract (26 μK-arcmin, 0.3 μK-arcmin, 3') are falsifiable predictions that later experiments can test. Credit is due for the explicit real-space construction in Eqs. (25)–(28) and the simulation in Fig. 5, which make the method reproducible rather than schematic, and for the parity argument that ΘΘ and EB are independent, which underlies the consistency tests proposed in §5.1 and §5.3.

Major objections (2)

[§5.1, Eq. (29), Fig. 6, and abstract sentence s3] The L ~ 1000 reach is computed with the Gaussian (disconnected) noise N_αα = A_α of Eq. (18) only. The connected trispectrum contribution N^(1) is explicitly deferred to Okamoto & Hu (in preparation), and the only quantitative evidence that it is small is for the temperature estimator (Hu 2001a). That argument does not automatically transfer to EB at L ~ 1000: Fig. 2 shows the detector noise below the lensed BB power at l ~ 2000, so the B field entering the EB estimator there is lensing-dominated rather than noise-dominated, which is precisely the regime where the connected term is least suppressed by a small C^BB. The authors acknowledge the Gaussian approximation in §5.1, but the abstract's headline number does not carry the caveat. Either estimate N^(1)_EB at L ~ 1000 for the reference experiment, or state in the abstract that the L ~ 1000 figure is a Gaussian-approximation estimate pending the trispectrum calculation.
[§4 and §5.1] The reach to l ~ 2000 in the B field assumes that the arcminute-scale polarized sky is cosmological, an assumption supported by citation to Hu (2000a) rather than demonstrated here. At the reference experiment's 1 μK-arcmin sensitivity this is a strong assumption, and it is the single external input on which the L ~ 1000 claim depends. The paper should state it explicitly as an assumption in §4 and note that the quoted noise thresholds are foreground-free; this is a framing request, not a demand to reproduce the cited foreground analysis.

Minor objections (11)

[Eqs. (10)–(12) and (18)] Please verify the powers of L in the normalization chain. As printed, A_α = L² [∫ f_α F_α]^{-1} combined with the explicit L in Eq. (10) gives ⟨d_α⟩ = L³ φ rather than the L φ of Eq. (12), while N_αα = A_α in Eq. (18) does require A_α = L²/∫ f_α F_α. One of the L factors needs checking; a one-line dimensional check would settle it.
[§2 vs §3] The lensing wavevector is defined as L = l − l′ in Eqs. (4)–(5) but as L = l + l′ in Eqs. (9)–(10). The physics is unchanged under l′ → −l′, but the reader will trip on the sign flip; state the convention once and use it throughout.
[§3, first paragraph] The statement that there are '6 estimators corresponding to the 3! pairs of Θ, E, B' has the right count for the wrong reason: 3! counts ordered pairs of distinct fields, whereas the six estimators are the unordered pairings with repetition (ΘΘ, ΘE, ΘB, EE, EB, BB). Suggest 'the six unordered pairings'.
[Abstract, sentence s2] The two noise levels quoted, ~40 μK-arcmin and ~26 μK-arcmin, refer to different quantities: §2 gives Planck ΔP ≈ 40√2 μK-arcmin and ΔT ≈ 27 μK-arcmin, and §3 states that the ΘE+EE combination surpasses ΘΘ below ΔT ≈ 26 μK-arcmin. Say explicitly which of ΔT and ΔP each number denotes.
[Table 1 caption] The caption reads 'Minimum variance filters', but the table lists f_α, the lensing kernels of Eq. (9); the minimum-variance filters F_α are defined in Eq. (13). Retitle the table to avoid confusion between the two.
[§4, Eqs. (25)–(27)] The tensor index structure of E_ijk, B_ij and the Levi-Civita contraction in Eq. (27) is opaque on first reading. A sentence explaining that E_ijk is symmetric in its three indices (4 unique components in 2D) and B_ij in its two (3 unique components), and how the contraction produces the vector G_i, would make the construction checkable without reverse-engineering it.
[§5.3] The cross-reference 'see Hu 2001 for the analogous technique using the temperature field' is ambiguous among Hu 2001a, 2001b and 2001c; specify which.
[References] Seljak & Zaldarriaga 1998 is cited in §5.3 but the reference list contains only the 1999 Phys. Rev. D 60, 043504 entry; Kaiser & Squires 1993 and Zaldarriaga 2000 appear in the list but are not cited in the text; and Hu 2001c and Huterer 2001 are given as 'in press' without volume or page numbers. Please reconcile the citations and complete the entries.
[§5.4] The sentence 'Decontamination is not feasible through the ΘΘ estimator since most of the B-modes on degree scales arise from fluctuations in the deflection field with L ≳ 200' compresses the causal chain; the operative point is that ΘΘ is noise-dominated at L ≳ 200, so it cannot reconstruct the modes that generate degree-scale B-modes. Expanding this by one clause would help.
[Fig. 9] The figure compares the true and reconstructed lensing B field only visually. A quantitative residual, such as the cross-correlation coefficient between the two maps or the residual B power in the low-pass band, would substantiate the claim that the procedure is feasible.
[§2 and §4] The flat-sky approximation is used for the 10° × 10° and 25° × 25° simulations, and the all-sky treatment is deferred to Okamoto & Hu (in preparation). For the L < 100 applications in §5.3 (ISW cross-correlation) curvature is not negligible; a sentence noting where the flat-sky results are expected to be accurate would be useful.

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The paper

The paper claims

The E–B pairing of CMB polarization fields is the most sensitive way to reconstruct the intervening mass, reaching multipoles near 1000 where temperature alone stops at a few hundred.

The reading

Gravitational lensing by intervening matter bends the cosmic microwave background, and that bending leaves a statistical trace: multipoles that should have been independent become correlated in a way that encodes the lens. The paper asks how much of the lensing mass can be recovered from those correlations, and argues the answer depends sharply on which pair of fields you multiply together. The authors build the minimum-variance estimator for each of the six pairings of temperature, E-mode polarization and B-mode polarization, and find that the cross-correlation of E with B is the strongest of them. The reason is structural: with no gravitational waves there is no primary B-mode, so the B field contributes no cosmic variance to the noise, and the E–B pairing draws its signal from small angular scales where the temperature field is already swamped by its own cosmic variance. The payoff claimed is a projected mass map out to multipoles of about $L = 1000$, roughly an order of magnitude finer than temperature alone, once an experiment is sensitive enough to map the lensing B-modes near $l = 2000$.

What carries the argument

The load-bearing object is the minimum-variance quadratic estimator of the deflection field: $d_alpha(L) = A_alpha(L), L int d^2 l_1/(2pi)^2, x(l_1) x'(l_2) F_alpha(l_1,l_2)$ with $l_2 = L - l_1$, normalized so that its ensemble average over CMB realizations returns $d(L) = L varphi(L)$. The optimal filter $F_alpha$ is the solution of a variance-minimization problem and reduces to simple forms when the two fields are the same or when their unlensed cross-spectrum vanishes; for the minimum-variance filter the noise of each estimator collapses to $N_{alphaalpha}(L) = A_alpha(L)$, and the five non-trivial estimators are combined by inverse-covariance weights. The E–B case is then recast as a real-space operation: because $sin 2varphi_{ll'} = 2(hat{l}cdothat{l}')(hat{n}cdot(hat{l}timeshat{l}'))$, the estimator becomes a sum of dot and cross products of four filtered E fields and three filtered B fields, which is what makes it implementable as a map-level product rather than a four-dimensional integral over pairs of multipoles.

Extended reading notes

Core claim

The central claim is that the correlation between E-mode and B-mode polarization induced by lensing yields a higher signal-to-noise reconstruction of the deflection field than any other quadratic combination of the CMB fields, and that this is what pushes mass reconstruction from degree scales down to the arcminute regime. The authors compute, for each pairing $xx'$ of $x = $ temperature, E, B, the response $f_alpha$ of the two-point correlation to a fixed lensing potential, then minimize the estimator variance subject to the normalization that the estimator averages to the true deflection. The resulting noise curves show the best estimator switching from temperature–temperature to E–B as experimental sensitivity improves: polarization begins to help at roughly $40$, $mu$K-arcmin and a $7'$ beam, overtakes temperature near $26$, $mu$K-arcmin and $4'$, and keeps improving until the lensing B-modes are mapped to $l approx 2000$ around $0.3$, $mu$K-arcmin and $3'$. The physical reason E–B wins is that the unlensed B field vanishes in the absence of gravitational waves, so it adds no sample variance, and because a B-mode at wavenumber $l$ cannot be generated by neighbouring modes from a low-$L$ lens, the signal prefers high $l$ where the temperature estimator is cosmic-variance limited. The paper then writes the E–B estimator explicitly as a product of filtered fields on the sky, demonstrates it on a simulated $10^circ times 10^circ$ field, and lays out four uses: measuring the projected linear power spectrum to the sample-variance limit, cross-correlating with cosmic shear to anchor tomography at high redshift, cross-correlating with the temperature field to isolate the integrated Sachs–Wolfe effect and hence dark energy, and reconstructing the lensing B-modes so they can be removed from a primordial gravitational wave signal.

Load-bearing premise

The estimators are optimal and their noise curves correct only if the unlensed temperature and polarization power spectra are known in advance and the lensed fields are Gaussian to the required order, an assumption the authors state explicitly in Section 4 and one that also carries the cited, not demonstrated, premise that polarized foregrounds are negligible at arcminute scales.

Editorial extensions

If this is right

A polarization experiment reaching roughly $0.3$, $mu$K-arcmin with a $3'$ beam can measure the projected deflection power spectrum to the sample-variance limit out to $L approx 1000$, corresponding to wavenumbers $0.002 < k < 0.2$ in $h$/Mpc, the whole linear regime today.
Cross-correlating the reconstructed deflection field with cosmic shear in redshift bands turns CMB lensing into the high-redshift anchor of a tomographic probe of structure growth.
The deflection–temperature cross-correlation at $L < 100$ isolates the integrated Sachs–Wolfe effect, so a detection of any such cross-correlation in a flat universe is effectively a direct detection of dark energy.
Because the small-scale B-modes used for reconstruction are not contaminated by large-angle gravitational wave B-modes, the reconstruction can be used to synthesize and subtract the lensing B-mode foreground from a degree-scale inflationary signal.
Five independent estimators with comparable signal-to-noise at $L < 500$ provide consistency checks that a foreground or instrumental systematic would have to mimic in several distinct quadratic combinations at once.

Reading between the lines

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

The vanishing unlensed B-mode makes the E–B estimator a built-in null test: any arcminute-scale B power that is not lensing-shaped would show up as a disagreement between the deflection map inferred from E–B and the one inferred from temperature or E–E, which is a cheap systematic diagnostic the paper does not spell out.
Since the filters require the unlensed spectra as input, the method's precision is coupled to the same experiment's cosmological parameter fit; a joint estimation of spectra and deflection, rather than the two-step procedure assumed here, is likely needed before the sample-variance limit is actually reached.
Delensing with this estimator is the step that makes a primordial B-mode search limited by detector noise rather than by lensing, so the sensitivity threshold quoted here is in effect also the threshold at which an inflationary gravitational wave measurement becomes possible at all.
The tomographic gain is largest where the cosmic shear survey has the most statistical power, at $L > 100$; a modest overlap area with a deep shear survey may capture most of the tomographic value even if the CMB experiment covers far more sky.

Original abstract

Weak gravitational lensing by the intervening large-scale structure of the Universe induces high-order correlations in the cosmic microwave background (CMB) temperature and polarization fields. We construct minimum variance estimators of the intervening mass distribution out of the six quadratic combinations of the temperature and polarization fields. Polarization begins to assist in the reconstruction when E-mode mapping becomes possible on degree-scale fields, i.e. for an experiment with a noise level of ~40 uK-arcmin and beam of ~7', similar to the Planck experiment; surpasses the temperature reconstruction at ~26 uK-arcmin and 4'; yet continues to improve the reconstruction until the lensing B-modes are mapped to l ~ 2000 at ~0.3 uK-arcmin and 3'. Ultimately, the correlation between the E and B modes can provide a high signal-to-noise mass map out to multipoles of L ~ 1000, extending the range of temperature-based estimators by nearly an order of magnitude. We outline four applications of mass reconstruction: measurement of the linear power spectrum in projection to the cosmic variance limit out to L ~ 1000 (or wavenumbers 0.002 < k < 0.2 in h/Mpc), cross-correlation with cosmic shear surveys to probe the evolution of structure tomographically, cross-correlation of the mass and temperature maps to probe the dark energy, and the separation of lensing and gravitational wave B-modes.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,634 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.32 · human participants 0.01 · controlled comparison 0.18 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.72 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Weak gravitational lensing by the intervening large-scale structure of the Universe induces high-order correlations in the cosmic microwave background (CMB) temperature and polarization fields. We construct minimum variance estimators of the intervening mass distribution out of the six quadratic combinations of the temperature and polarization fields. Polarization begins to assist in the reconstruction when E-mode mapping becomes possible on degree-scale fields, i.e. for an experiment with a noise level of ~40 uK-arcmin and beam of ~7', similar to the Planck experiment; surpasses the temperature reconstruction at ~26 uK-arcmin and 4'; yet continues to improve the reconstruction until the lensing B-modes are mapped to l ~ 2000 at ~0.3 uK-arcmin and 3'. Ultimately, the correlation between the E and B modes can provide a high signal-to-noise mass map out to multipoles of L ~ 1000, extending the range of temperature-based estimators by nearly an order of magnitude. We outline four applications of mass reconstruction: measurement of the linear power spectrum in projection to the cosmic variance limit out to L ~ 1000 (or wavenumbers 0.002 < k < 0.2 in h/Mpc), cross-correlation with cosmic shear surveys to probe the evolution of structure tomographically, cross-correlation of the mass and temperature maps to probe the dark energy, and the separation of lensing and gravitational wave B-modes.

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	Weak gravitational lensing by the intervening large-scale structure of the Universe induces high-order correlations in the cosmic microwave background (CMB) temperature and polarization fields.	background 1.00	0.06	0.04	asserted 1.00
s1	We construct minimum variance estimators of the intervening mass distribution out of the six quadratic combinations of the temperature and polarization fields.	contribution 0.52	0.95	0.08	asserted 1.00
s2	Polarization begins to assist in the reconstruction when E-mode mapping becomes possible on degree-scale fields, i.e. for an experiment with a noise level of ~40 uK-arcmin and beam of ~7', similar to the Planck experiment; surpasses the temperature reconstruction at ~26 uK-arcmin and 4'; yet continues to improve the reconstruction until the lensing B-modes are mapped to l ~ 2000 at ~0.3 uK-arcmin and 3'.	result 0.96	0.69	0.17	asserted 0.99
s3	Ultimately, the correlation between the E and B modes can provide a high signal-to-noise mass map out to multipoles of L ~ 1000, extending the range of temperature-based estimators by nearly an order of magnitude.	result 0.88	0.72	0.55	asserted 0.92
s4	We outline four applications of mass reconstruction: measurement of the linear power spectrum in projection to the cosmic variance limit out to L ~ 1000 (or wavenumbers 0.002 < k < 0.2 in h/Mpc), cross-correlation with cosmic shear surveys to probe the evolution of structure tomographically, cross-correlation of the mass and temperature maps to probe the dark energy, and the separation of lensing and gravitational wave B-modes.	contribution 0.58	0.43	0.11	asserted 0.95

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from proof or derivation 0.78.

What was examined, and what was not

sources read

arXiv listing abstract v3

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.** — An exaggerated example of the lensing effect on a 10◦ × 10◦ field. Top: (left-to-right) unlensed temperature field, unlensed E-polarization field, spherically symmetric deflection field d(n). Bottom: (left-to-right) lensed temperature field, lensed E-polarization field, lensed B-polarization field. The scale for the polarization and temperature fields differ by a factor of 10. gravitational waves. 2. LENSING Weak lensing by the large-scale structure of the Universe remaps the primary temperature field Θ(nˆ) = ∆T (nˆ)/T and dimensionless Stokes parameters Q(nˆ) and U(nˆ) as…

**Figure 2.** — Power spectra of the CMB temperature and polarization fields compared with the detector noise of the Planck satellite and a nearly perfect experiment with a noise level of ∆T = ∆P / √ 2 = 1µK-arcmin and a beam of σ = 4′ (long dashed lines, thick for polarization, thin for temperature). The Planck experiment has sufficient signal-to-noise to map the Θ field but can only marginally map the E-polarization field; the nearly perfect experiment can map the all three fields to l = 2000. on the E-polarization is similar to that of the temperature and reflects the fact that cos 2ϕl ′ l ≈ 1…

**Figure 3.** — Deflection signal (dd) and noise power spectra of the quadratic estimators and their minimum variance (mv) combination: (a) Planck experiment (b) reference experiment. As the sensitivity of the experiment improves the best quadratic estimator switches from ΘΘ to EB. Only the EB-estimator can reconstruct the mass distribution at L ∼ > 200. For the reference experiment, all 5 estimators have sufficient signal-to-noise to produce maps at L ∼ < 200. The EB estimator has the best signal-to-noise, and allows for mapping to L ∼ < 1000. The reason is that there is no noise variance cont

**Figure 4.** — Deflection signal (dd) and noise power spectra for the minimum variance (mv; solid lines) and ΘΘ (dashed lines) estimators as a function of (a) beam size σ and (b) noise level ∆T = ∆P / √ 2. The noise saturates to its minimum by σ ≈ 2 − 4 ′ and ∆T ≈ 0.1 − 0.3µK-arcmin as the polarization field is mapped to the cosmic variance limit out to l < 2000. The noise variance hd ∗ mv(L)dmv(L ′ )i = (2π) 2 δ(L − L ′ )[C dd L + Nmv(L)] (21) becomes Nmv = 1 P αβ(N−1)αβ . (22) Note that the ΘΘ and EB estimators are independent and those estimators that are correlated have correlation…

**Figure 5.** — Mass reconstruction on a 10◦ × 10◦ field with the reference experiment (∆T = ∆P / √ 2 = 1µK-arcmin and σ = 4′ ): (a) deflection field, (b) ΘΘ-reconstruction, (c) EB-reconstruction. 2' 0.1 1 0.3 3 10 30µK–arcmin 4' 8' 16' 16' 30 10 3 ≤1 8' 4' 2' ΘΘ mv (a) 3µK–arcmin (b) 4' beam 10 10–7 10–8 10–7 10–8 100 L 1000 L(L+1)C L /2 π dd L(L+1)C L /2 π dd dd dd

**Figure 6.** — Statistical errors achievable on the deflection power spectrum with the Planck (fsky = 0.65) and reference experiments (fsky = 1). Boxes represent band averaging width and 1σ errors. The polarization information in the reference experiment allows for a cosmic variance limited measurement of the projected power spectrum out to L ∼ 1000. In this regime, the fluctuations are almost completely linear (dashed lines). dot and cross products Gi(nˆ) = 2X jkm Eijk(nˆ)Bjm(nˆ)ǫkm3 , (27) where ǫijk is the Levi-Civita symbol. The deflection field is then reconstructed as dEB(L) = − AEB(L) L…

**Figure 8.** — Statistical errors on the cross correlation of CMB deflections and the temperature field for the Planck (fsky = 0.65) and reference (fsky = 1) experiments. The five estimators of the deflection field obtainable with polarization information enables five nearly independent, cosmic variance limited detections of the cross correlation for L < 100 (shown offset slightly for clarity). The cross correlation is extremely sensitive to the properties of the dark energy. to scales in the matter power spectrum of 0.002 ∼ < k ∼ < 0.2 in h/Mpc representing the whole linear regime today.…

**Figure 9.** — Large-angle (l < 100) lensing B-polarization field (top) and the reconstructed B-polarization field from the small angle EB deflection estimator and the observed E-field. Detector noise appropriate for the reference experiment has been added to this 25◦ ×25◦ patch. Reconstruction techniques can help separate the gravitational wave and lensing B-modes. and 1000 deg2 of overlap with a cosmic shear survey out to median source redshift z = 1, divided into three redshift bands zs < 1, 1 < zs < 1.5, zs > 1.5. Errors on the cosmic shear side assume n = 56 gal/arcmin2 , and an intrinsic…

Showing 8 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/0111606 by the authors.

References

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Bias to CMB lensing from lensed foregrounds
astro-ph.CO 2019-08 conditional novelty 7.0 of 10

Lensed extragalactic foregrounds create a percent-level bias in CMB lensing estimators that is significant for upcoming Simons Observatory measurements and can be reduced by modified estimators.
Maximum a posteriori CMB lensing reconstruction
astro-ph.CO 2017-04 accept novelty 7.0 of 10

An iterative maximum-posterior estimator recovers the CMB lensing map with no significant likelihood approximations, and in CMB-S4 simulations it improves the tensor-to-scalar ratio error bar by up to a factor of abou...
Planck 2013 results. XVII. Gravitational lensing by large-scale structure
astro-ph.CO 2013-03 accept novelty 7.0 of 10

Planck detects CMB gravitational lensing at over 25 sigma, maps the lensing potential over roughly 70% of the sky, and measures its power spectrum to 4% in amplitude, sharpening CMB-only constraints on curvature and o...
The shape of the CMB lensing bispectrum
astro-ph.CO 2011-01 accept novelty 7.0 of 10

Lensing of the cosmic microwave background produces a bispectrum in temperature and, newly identified, in polarization, and its bias on local non-Gaussianity estimates is small and correctable.
Constraining Dynamical Dark Energy with Tomographic Galaxy and CMB Weak Lensing Through Higher-Order Statistics
astro-ph.CO 2026-10 accept novelty 6.0 of 10

Higher-order lensing statistics, added to tomographic galaxy and CMB lensing power spectra, are forecast to reduce dark-energy parameter uncertainties by about a quarter in idealized Rubin and Simons Observatory setups.
Faster CMB lensing with control variates
astro-ph.CO 2026-05 unverdicted novelty 6.0 of 10

A control variate technique using differenced estimates from realistic masked and isotropic simulations reduces the computational cost of CMB lensing bias calculations by a factor of three to five.
Wavelet Flow For Extragalactic Foreground Simulations
astro-ph.CO 2025-05 conditional novelty 6.0 of 10

A Wavelet Flow generative model jointly produces CMB lensing convergence and cosmic infrared background maps whose power spectra and Minkowski functionals match the training simulation within a few percent.
Impact of Galactic non-Gaussian foregrounds on CMB lensing measurements
astro-ph.CO 2025-05 conditional novelty 6.0 of 10

Galactic dust bias on CMB lensing amplitude is below 0.3σ for ACT but can exceed 0.3σ for Simons Observatory with wide masks, depending on the dust model.
Towards a multi-tracer neutrino mass measurement with line-intensity mapping
astro-ph.CO 2024-12 conditional novelty 6.0 of 10

AtLAST [CII] line-intensity mapping, combined with CMB-S4 and DESI BAO forecasts, could reach sigma(sum m_nu) ~ 18 meV, and ~11.5 meV with a 21-cm tau prior.
The Cosmological Preference for Negative Neutrino Mass
astro-ph.CO 2024-07 conditional novelty 6.0 of 10

The cosmological preference for negative neutrino mass is driven by CMB lensing a few percent stronger than the standard model predicts, and it survives swaps of the BAO dataset, the Planck likelihood, and the dark en...
No $\nu$s is Good News
astro-ph.CO 2024-05 conditional novelty 6.0 of 10

DESI plus CMB data, reanalyzed without a positivity prior, prefer a sum of neutrino masses of -160 +/- 90 meV, excluding the minimum 58 meV sum at about 3 sigma.
Optimal filtering for CMB lensing reconstruction
astro-ph.CO 2019-09 accept novelty 6.0 of 10

Optimal anisotropic CMB filtering plus a second-stage filter on the reconstructed lensing convergence map reduces lensing power spectrum variance by factor 2-5 relative to isotropic filtering, with up to 30% additiona...
KSZ tomography and the bispectrum
astro-ph.CO 2018-10 conditional novelty 6.0 of 10

The pair-sum, kSZ-template, velocity matched-filter, velocity-growth, and velocity-reconstruction statistics are all algebraically the same optimal <ggT> bispectrum estimator, and that estimator measures Pge(kS) and P...
Planck 2018 results. VIII. Gravitational lensing
astro-ph.CO 2018-07 accept novelty 6.0 of 10

Planck's final CMB lensing reconstruction detects lensing at 40 sigma, constrains sigma_8 to about 1 percent when combined with the power spectra, and demonstrates delensing of the acoustic peaks.
CMB Delensing Beyond the B Modes
astro-ph.CO 2016-09 conditional novelty 6.0 of 10

An all-orders calculation of delensed CMB temperature and E-mode spectra, including residual covariances, shows that map-level delensing sharpens acoustic peaks and improves Stage IV constraints on the acoustic scale,...
Impact of post-Born lensing on the CMB
astro-ph.CO 2016-05 accept novelty 6.0 of 10

Multiple deflections along the line of sight change the CMB lensing bispectrum by an amount comparable to the large-scale-structure bispectrum, while leaving the lensing power spectrum almost untouched.
LSS constraints with controlled theoretical uncertainties
astro-ph.CO 2016-02 conditional novelty 6.0 of 10

Treating theoretical model error as correlated noise in Fisher forecasts shrinks the usable range of scales and weakens predicted large-scale-structure constraints on neutrino mass and equilateral non-Gaussianity by f...
Phases of New Physics in the CMB
astro-ph.CO 2015-08 accept novelty 6.0 of 10

Planck 2015 temperature and polarization data separate free-streaming from non-free-streaming dark radiation, giving Neff = 2.80 +/- 0.24 and Nfluid < 0.67 at 2 sigma, with the separation driven by a phase shift in th...
Measurement of the Cosmic Microwave Background Polarization Lensing Power Spectrum with the POLARBEAR experiment
astro-ph.CO 2013-12 accept novelty 6.0 of 10

Four-point correlations of CMB E- and B-mode polarization over 30 square degrees show gravitational lensing at 4.2 sigma, the first such detection from CMB polarization data alone.
Weak Lensing of the Cosmic Microwave Background by Foreground Gravitational Waves
astro-ph 2006-04 accept novelty 6.0 of 10

Lensing of the cosmic microwave background by a primordial gravitational wave background is calculated and found to be below cosmic variance, making it undetectable in power spectra but relevant as a systematic at sma...
A Global Overview of Starobinsky Inflation beyond slow-roll with CMB-BAO data
astro-ph.CO 2026-09 conditional novelty 5.0 of 10

A global MCMC analysis of Starobinsky inflation without the slow-roll approximation yields parameter constraints that are stronger than those from slow-roll and shows that DESI BAO data shift N* to values around 70, r...
Simultaneous inference of isotropic and anisotropic polarization rotation angles for next-generation cosmic microwave background experiments
astro-ph.CO 2026-07 conditional novelty 5.0 of 10

A Bayesian power-spectrum framework simultaneously infers isotropic and anisotropic cosmic birefringence angles, instrumental miscalibration angles, and foreground parameters from simulated multi-frequency next-genera...
unimpeded: A Public Grid of Nested Sampling Chains for Cosmological Model Comparison and Tension Analysis
astro-ph.CO 2025-11 conditional novelty 5.0 of 10

A public library of pre-computed nested-sampling chains for 8 cosmological models and a large dataset grid enables fast Bayesian model comparison; combined analyses most often prefer ΛCDM, with DES–Planck and SH0ES–Pl...
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.
Observable CMB B-modes from Cosmological Phase Transitions
astro-ph.CO 2024-10 unverdicted novelty 5.0 of 10

Phase transitions in dark sectors can generate CMB B-modes with amplitudes competitive with inflation but peaking at smaller angular scales.
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.
Cosmological constraints from unWISE and Planck CMB lensing tomography
astro-ph.CO 2021-05 accept novelty 5.0 of 10

Combining unWISE galaxy clustering with Planck CMB lensing yields S8 = 0.784 ± 0.015, 2.4σ below the Planck primary-CMB value and consistent with other low-redshift lensing measurements.
Measuring the spectrum of primordial gravitational waves with CMB, PTA and Laser Interferometers
astro-ph.CO 2020-07 conditional novelty 5.0 of 10

A unified forecast shows that combining CMB and space-based interferometer data could map the primordial gravitational-wave spectrum across many frequency decades, provided astrophysical foregrounds can be cleaned as assumed.
The Simons Observatory: Science goals and forecasts
astro-ph.CO 2018-08 conditional novelty 5.0 of 10

The paper specifies the Simons Observatory's telescopes, frequency bands, noise model and foreground treatment, and forecasts uncertainties on the tensor-to-scalar ratio, relativistic species, neutrino mass, lensing, ...
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...
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...
Probing cosmological parameters with the CMB: Forecasts from full Monte Carlo simulations
astro-ph 2006-06 conditional novelty 5.0 of 10

Running the full Markov Chain Monte Carlo pipeline on simulated Planck data shows the standard Fisher matrix forecast overestimates the errors on neutrino mass fraction and dark matter density by up to a factor of two.
Microwave Spectro-Polarimetry of Matter and Radiation across Space and Time
astro-ph.CO 2019-09 unverdicted novelty 4.0 of 10

This white paper proposes a large space mission combining a polarized imager, a filter-bank spectrometer, and absolute spectrometers to map the microwave sky from 10 to 2000 GHz and probe cosmology across cosmic time.
Towards a cosmological neutrino mass detection
astro-ph.CO 2015-09 accept novelty 4.0 of 10

Combining a Stage-4 CMB lensing experiment with DESI BAO should reach a 1-sigma neutrino mass sum uncertainty of 15 to 19 meV, enough for a 4-sigma detection of the 60 meV minimum, provided large-scale CMB polarizatio...
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...
COrE (Cosmic Origins Explorer) A White Paper
astro-ph.CO 2011-02 conditional novelty 4.0 of 10

A 15-band full-sky microwave polarimeter with 10 to 30 times Planck's sensitivity would detect primordial tensor modes at r = 0.001 at 3 sigma and measure the CMB lensing deflection to the cosmic variance limit, provi...
CMB-S4 Science Book, First Edition
astro-ph.CO 2016-10 accept novelty 3.0 of 10

A community science book sets CMB-S4 targets: detect or bound primordial gravitational waves at r < 0.002, measure the neutrino mass sum near 15 meV, and pin the light-relic density to sigma(Neff) around 0.027.
The Quest for B Modes from Inflationary Gravitational Waves
astro-ph.CO 2015-10 accept novelty 2.0 of 10

Inflationary gravitational waves should imprint a curl (B-mode) pattern in CMB polarization, and this review lays out the theory, the competing foregrounds, and the experimental strategies for detecting it.
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.
Update on the Physics of the Cosmic Microwave Background (2015-2025)
astro-ph.CO 2026-07 conditional

A status review of CMB science 2015-2025: the six-parameter ΛCDM model still fits the power spectra, primordial B-modes remain undetected with r < 0.036, and the Hubble tension persists at over 5σ.

How later papers used it

245 later papers cite it

Evidence: substantial. 240 of the citing papers share no author with it, and 106 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 416 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

26 later papers by other authors are listed below: 26 used its method or tool.

Uses its method or tool 26 of 98 papers with this label

Reconstructed lensing convergence fields from CMB temperature maps using this paper's minimum variance quadratic estimator.

First Measurement of the Cross-Correlation of CMB Lensing and Galaxy Lensing (2013) · 1.00 · paraphrased: the licence does not allow quoting
Used correlations between Fourier modes of CMB temperature and polarization maps to estimate the lensing potential with pairs of filtered maps using this paper's quadratic estimator.

A Measurement of the Cosmic Microwave Background Lensing Potential and Power Spectrum from 500 deg$^2$ of SPTpol Temperature and Polarization Data (2019) · 1.00 · paraphrased: the licence does not allow quoting
We used these correlations to build a lensing potential with the minimum variance estimator from [ref] .

A measurement of cluster masses using Planck and SPT-SZ CMB lensing (2024) · Concise overview · 1.00
Computed the power spectrum of the convergence field using this paper's optimal quadratic estimator.

Detection of the Power Spectrum of Cosmic Microwave Background Lensing by the Atacama Cosmology Telescope (2011) · 1.00 · paraphrased: the licence does not allow quoting
Estimated the error on the reconstructed gravitational potential following this paper and its public implementation.

Cosmology with Rayleigh Scattering of the Cosmic Microwave Background (2020) · 1.00 · paraphrased: the licence does not allow quoting
Implemented this paper's TE estimator to measure the bias to TE reconstructions.

The impact of extragalactic foregrounds on internal delensing of CMB B-mode polarization (2022) · 1.00 · paraphrased: the licence does not allow quoting
Computed the noise for CMB lensing reconstruction using this paper's quadratic estimator technique.

Can CMB Lensing Help Cosmic Shear Surveys? (2013) · 1.00 · paraphrased: the licence does not allow quoting
CMB lensing maps are obtained using the quadratic estimator [ref] .

Cosmology from Cross-Correlation of ACT-DR4 CMB Lensing and DES-Y3 Cosmic Shear (2023) · ACT CMB lensing data · 1.00
Used the same modified form of this paper's estimator as in another work to estimate the lensing potential from correlations between Fourier modes in pairs of CMB maps.

Optimal CMB Lensing Reconstruction and Parameter Estimation with SPTpol Data (2020) · 1.00 · paraphrased: the licence does not allow quoting
Reconstructed the lensing convergence field from its CMB maps with this paper's minimum variance quadratic estimator, exploiting the mode couplings induced by lensing.

The Atacama Cosmology Telescope: A CMB lensing mass map over 2100 square degrees of sky and its cross-correlation with BOSS-CMASS galaxies (2020) · 1.00 · paraphrased: the licence does not allow quoting
Then, we looked for a lensing signal at the cluster position using a quadratic estimator [ref] that extracts the lensing potential from the correlations between Fourier modes in the CMB.

Measuring cluster masses with Planck and SPT-SZ CMB lensing (2024) · Pipeline · 1.00
Reconstructed the lensing potential using flat-sky implementations of this paper's quadratic estimators from the quicklens package.

Exploring cosmic origins with CORE: gravitational lensing of the CMB (2017) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's EB quadratic estimator method, implemented by the quicklens software package, for the CMB lensing reconstruction noise.

Searching for Signal of Primordial Black Hole from CMB Lensing and $\gamma$-ray Emissions (2022) · 1.00 · paraphrased: the licence does not allow quoting
Calculated the lensing reconstruction noise using this paper's minimum variance quadratic estimator, including the improvement from iterative delensing with another work's EB estimator.

Aspects of Dark Matter Annihilation in Cosmology (2018) · 1.00 · paraphrased: the licence does not allow quoting
Estimated the reconstruction noise as detailed in this paper, assuming a reconstruction process based on quadratic combinations of the lensed CMB maps from another work.

The Observational Future of Cosmological Scalar-Tensor Theories (2016) · 1.00 · paraphrased: the licence does not allow quoting
Estimated the lensing spectrum from the ACT temperature maps using this paper's optimal quadratic estimator.

The Atacama Cosmology Telescope: Cosmological parameters from three seasons of data (2013) · 1.00 · paraphrased: the licence does not allow quoting
Implemented and investigated this paper's quadratic estimator for estimating the CMB lensing signal in its lensing estimation pipeline.

CMB lensing reconstruction in the presence of diffuse polarized foregrounds (2012) · 0.98 · paraphrased: the licence does not allow quoting
Followed this paper's recipe to express the estimator and determine the weight functions by minimizing the noise contribution with the Lagrange-multiplier technique.

Three-dimensional weak gravitational lensing of the 21-cm radiation background (2022) · 0.98 · paraphrased: the licence does not allow quoting
Following this paper, chose the normalization so that the estimator is unbiased and determined the filter by minimizing the variance at lowest order in the lensing expansion.

Gravitational Lensing of the CMB: a Feynman Diagram Approach (2014) · 0.88 · paraphrased: the licence does not allow quoting
Applied this paper's quadratic lensing estimator to a combined CMB temperature map to estimate the CMB convergence.

Cosmological lensing ratios with DES Y1, SPT and Planck (2018) · 0.98 · paraphrased: the licence does not allow quoting
We use throughout the quadratic estimator [ref] , using the curved-sky calculations implemented in falafel[<https://github.com/simonsobs/falafel>], and the normalization and reconstruction noise calculations implemented in tempura[<https://github.com/simonsobs/tempura>].

The Atacama Cosmology Telescope: Mitigating the impact of extragalactic foregrounds for the DR6 CMB lensing analysis (2023) · Quadratic Estimator · 0.99
Followed the formalism developed for lens reconstruction in this paper to reconstruct E modes from appropriately weighted combinations of the observed fields.

Reconstructing CMB fluctuations and the mean reionization optical depth (2017) · 0.98 · paraphrased: the licence does not allow quoting
Estimated the unnormalized lensing potential at L by calculating the weighted sum of inverse-variance filtered lensing modes separated by L, following this paper.

A Measurement of Gravitational Lensing of the Cosmic Microwave Background Using SPT-3G 2018 Data (2023) · 0.99 · paraphrased: the licence does not allow quoting
We estimate the noise power of the CMB lensing maps following [ref] .

Prospects for studying the mass and gas in protoclusters with future CMB observations (2023) · Mock observations · 0.96
Used this paper's standard quadratic estimator filter, which minimizes the variance of the estimator subject to the constraint that it is unbiased once applied to the maps.

Probing cosmic birefringence with polarized Sunyaev-Zel'dovich tomography (2022) · 0.93 · paraphrased: the licence does not allow quoting
Performed lensing reconstruction on beam-convolved, noisy temperature maps using this paper's quadratic estimator, replacing unlensed with lensed CMB temperature power spectra in the filters.

CMB Lensing Beyond the Power Spectrum: Cosmological Constraints from the One-Point PDF and Peak Counts (2016) · 0.90 · paraphrased: the licence does not allow quoting

How the literature uses it

163 arXiv papers cite this in 283 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 · 173 sentences without a confident label

Contests it 2 papers

Our most important point is that the well-known Hu and Okamoto [ref] (hereafter, HO02) estimator is not the optimal quadratic estimator that can be constructed from temperature and polarization maps, even if generalized to the full sky, and even when using non-perturbative response functions [ref] .

Quadratic estimators for CMB weak lensing · Introduction · contest 0.89
Recently, [ref] showed that the well-known Hu and Okamoto [ref] estimator is not the most optimal quadratic estimator that can be constructed out of the temperature and polarization maps as was previously thought.

Joint analyses of 2D CMB lensing and 3D galaxy clustering in the spherical Fourier-Bessel basis · CMB lensing reconstruction noise · contest 0.75

Uses its method or tool 55 papers

The estimators are normalized by $A_EE(L)$ and $A_EB(L)$ so that they recover the input deflection power spectrum [ref] .

Measurement of the Cosmic Microwave Background Polarization Lensing Power Spectrum with the POLARBEAR experiment · Introduction · 0.91
The lensing convergence fields are reconstructed from the CMB temperature maps using the minimum variance quadratic estimator of [ref] following the procedure used in [ref] .

First Measurement of the Cross-Correlation of CMB Lensing and Galaxy Lensing · ACT CMB Lensing Data · 1.00
We used these correlations to build a lensing potential with the minimum variance estimator from [ref] .

A measurement of cluster masses using Planck and SPT-SZ CMB lensing · Concise overview · 1.00

Compares with it 2 papers

Including polarizations, this estimator, in principle, suppresses the noise contribution for the reconstructed potentials, but numerically cost, compared to the estimator in Ref. [ref] .

Full-sky lensing reconstruction of gradient and curl modes from CMB maps · Introduction · 0.94
This is in contrast to many lensing estimators in the literature [ref] , which build a separate quadratic estimator $Φ_LM^(XY)$ from each pair of fields ($X,Y$), then combine via $Φ_LM = ∑_X,Yw_XYΦ_LM^(XY)$ for minimum-variance weights $w_XY$.

Searching for Inflationary Physics with the CMB Trispectrum: 1. Primordial Theory & Optimal Estimators · Gravitational Lensing · 0.94

Cites it as support 1 paper

As the sensitivity and resolution of CMB experiments continue to improve, the least noisy CMB lensing reconstructions will come from polarization [ref] .

Real space lensing reconstruction using cosmic microwave background polarization · Discussion · 0.97

Cites it as background 33 papers

In this section we will review the construction of the quadratic estimator for CMB lensing [ref] and for patchy reionization [ref] .

Reconstructing Patchy Reionization with Deep Learning · Quadratic Estimator · 0.82
[equation] Here, $C_L^ξ_jξ_j$ is the target signal we aim to measure, $N_L^0=R^-1$ represents the Gaussian reconstruction noise [ref] , and $N_L^1,ξ_jξ_j$ captures the secondary correlations involving the distortion field itself, of order $𝒪(ξ_j^2)$ [ref] .[$N_0$ has a zero power of the power spectrum $C_L^ξ_jξ_j$. $N_1$ has one power.

Optimal joint reconstruction from CMB observations: application to cosmic birefringence, patchy reionization and CMB lensing · CMB Distortion physics extracted with quadratic estimators · 0.81
Here, we focus on the polarized Sunyaev-Zel'dovich (pSZ) effect induced by the scattering of CMB photons off energetic electrons in the large-scale structure [ref] and evaluate the prospects to probe parity-violation due to an axionlike dark energy (or dark matter) which could change the remote-quadrupole signal that can be inferred from tomographic measurements of pSZ [ref] .

Probing cosmic birefringence with polarized Sunyaev-Zel'dovich tomography · Introduction · 0.92

Discussion

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