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Weak Gravitational Lensing of the CMB

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Antony Lewis, Anthony Challinor

Astrophysics · Submitted 26 Jan 2006 · 186 references

Pith's verdict: The standard reference for CMB lensing, and it earns that status: careful derivations, honest about approximations, with a few genuinely new results buried in a review.

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arXiv IDastro-ph/0601594 (version 4)
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CitationAnthony Challinor, Antony Lewis. "Weak Gravitational Lensing of the CMB." arXiv:astro-ph/0601594 (2006). https://pith.science/paper/CCWXLA4R
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The paper

The paper claims

Weak lensing of the cosmic microwave background is a calculable, observationally important distortion that modifies CMB spectra, generates B modes and non-Gaussianity, and can be mapped back to constrain cosmology.

The reading

The cosmic microwave background is the light left over from the early universe. As that light travels toward us, it passes through galaxies, clusters, and dark matter. Gravity bends the light, so each patch of the sky appears slightly shifted, by about two arcminutes on average. This is weak gravitational lensing. The shift is small, but it matters because the CMB has sharp acoustic peaks. Lensing blurs those peaks and moves power from large scales to small scales. It also twists the polarization pattern, turning part of the E-mode signal into B modes. Those B modes are a nuisance for anyone trying to detect primordial gravitational waves, because inflation also predicts B modes. The review explains how to calculate all of this. It starts from photon paths in general relativity, defines a lensing potential, and shows how to compute lensed temperature and polarization spectra. It then describes the non-Gaussian signature: lensing makes the four-point function nonzero even when the unlensed CMB is Gaussian. That signature can be used to reconstruct the lensing potential from the observed sky. Once the potential is known, one can delens the map and partly recover the unlensed CMB. The review also covers cluster masses, moving lenses, sky curvature, and simulations. It is a guide to both the physics and the practical analysis pipeline.
Extended reading notes

Core claim

The central claim is that weak gravitational lensing has several important effects on the CMB: it changes the power spectra, induces non-Gaussianities, and generates a B-mode polarization signal that is an important source of confusion for the signal from primordial gravitational waves. The review further claims that these effects can be calculated from general relativity and standard cosmological perturbation theory, and that the lensing potential can be reconstructed from the lensed sky. If correct, lensing is both a required systematic for precision CMB cosmology and a signal that carries information about late-time matter.

Load-bearing premise

The quantitative small-scale predictions, especially the lensed B-mode and trispectrum, assume the lensing potential is a Gaussian field whose power spectrum can be obtained by applying a halofit-style non-linear correction to linear theory. Section 3.3 states that halofit is expected to be accurate at the few percent level for standard LambdaCDM models, and Section 5.4 notes that the B-mode prediction depends on this. If baryonic feedback or non-Gaussian lensing correlations change the small-scale potential, the arcminute-scale lensed spectra and delensing forecasts shift.

Original abstract

Weak gravitational lensing has several important effects on the cosmic microwave background (CMB): it changes the CMB power spectra, induces non-Gaussianities, and generates a B-mode polarization signal that is an important source of confusion for the signal from primordial gravitational waves. The lensing signal can also be used to help constrain cosmological parameters and lensing mass distributions. We review the origin and calculation of these effects. Topics include: lensing in General Relativity, the lensing potential, lensed temperature and polarization power spectra, implications for constraining inflation, non-Gaussian structure, reconstruction of the lensing potential, delensing, sky curvature corrections, simulations, cosmological parameter estimation, cluster mass reconstruction, and moving lenses/dipole lensing.

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, 3,674 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

Weak gravitational lensing has several important effects on the cosmic microwave background (CMB): it changes the CMB power spectra, induces non-Gaussianities, and generates a B-mode polarization signal that is an important source of confusion for the signal from primordial gravitational waves. The lensing signal can also be used to help constrain cosmological parameters and lensing mass distributions. We review the origin and calculation of these effects. Topics include: lensing in General Relativity, the lensing potential, lensed temperature and polarization power spectra, implications for constraining inflation, non-Gaussian structure, reconstruction of the lensing potential, delensing, sky curvature corrections, simulations, cosmological parameter estimation, cluster mass reconstruction, and moving lenses/dipole lensing.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Weak gravitational lensing has several important effects on the cosmic microwave background (CMB): it changes the CMB power spectra, induces non-Gaussianities, and generates a B-mode polarization signal that is an important source of confusion for the signal from primordial gravitational waves. background 1.00 0.04 0.13 asserted 1.00
s1 The lensing signal can also be used to help constrain cosmological parameters and lensing mass distributions. background 0.95 0.06 0.10 hedged possibility 0.64
s2 We review the origin and calculation of these effects. contribution 0.97 0.49 0.04 asserted 0.54
s3 Topics include: lensing in General Relativity, the lensing potential, lensed temperature and polarization power spectra, implications for constraining inflation, non-Gaussian structure, reconstruction of the lensing potential, delensing, sky curvature corrections, simulations, cosmological parameter estimation, cluster mass reconstruction, and moving lenses/dipole lensing. contribution 0.85 0.10 0.04 unclear 0.51

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.93.

What was examined, and what was not

sources read
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not examined
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  • 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. Weak lensing geometry for a source (the CMB) at comovi
Figure 2. The power spectrum of the deflection angle (given in te
Figure 3. Cumulative contribution of different redshifts to th
Figure 4. Contributions of different wavenumbers k (in Mpc−1 ) to the power spectrum of the lensing potential for a concordance ΛCDM model. l ≥ 1 the Bessel functions go to zero at the origin, jl(kχ) → 0 as χ → 0, so the l ≥ 1 power spectrum is finite and well defined. The last scattering surface is a long way away, so the lensing potential has contributions out to quite high redshift as show in
Figure 5. Correlation of the lensing potential with the CMB tem
Figure 6. Top: the lensed temperature power spectrum (solid) a
Figure 7. The functions σ 2 (β) ≡ Cgl(0) − Cgl(β) [solid] and Cgl,2(β) [dashed] as a function of angular separation β (in radians) for a typical concordance model. For accuracy the results are calculated using the full-sky generalization, and use the linear power spectrum for C ψ l . Here we defined R Θ ≡ 1 2 h|∇T| 2 i = 1 4π Z dl l l 4C Θ l ∼ 109µK 2 (4.17) similarly to Eq. (4.13) to measure the power in the unlensed CMB temperature gradient. The lensed CMB power is therefore proportional to the unlensed gradient power RΘ, and to the power in the deflection angle l 2C ψ
Figure 8. Fractional difference between the lensed
Figure 9. Contributions to the total blackbody-spectrum temp
Figure 10. The unlensed E-power spectrum l(l + 1)C E l /2π (top, dashed), lensing potential power spec￾trum [l(l + 1)]2C ψ l (dot-dashed), and the contribution to the large-scale lensed B-mode power spec￾trum from each log l given by half their product (thin solid). The dotted line is the large-scale white spectrum l(l + 1)C B l /2π given by Eq. (5.34), which can be compared to the full numerical result (thick solid). The lensed B-mode power spectrum has a close-to-white spectrum at l ≪ 1000. C˜B l ∼ Z d 2 l ′ (2π) 2 l ′4C ψ l ′ C E l ′ sin2 2(φl ′ − φl) = 1 4π Z dl ′ l ′ l ′4C ψ l ′ l ′2C E…
Figure 11. The contributions to the low-l (l ∼ 10) C B l power spectrum from lensing potential with l ≤ lmax. A significant fraction of the power comes from the tail of C ψ l at high l. down to |k ⊥ ψ − kE|. This is the power that will be observed in the B-mode on large scales (and also in the E-mode if it weren’t dominated by the much larger unlensed signal from reionization and last scattering). There are contributions to the large scale B modes from all scales on which there is non-zero E and lensing potential power. Since the relevant E and ψ modes are dominant on small scales (the…
Figure 12. The unlensed T-E correlation and E-mode power spectra compared to the smoother lensed results. C˜E l ≈ C ψ l Z d 2 l ′ (2π) 2 [l ′ · l] 2 C E |l ′ | cos2 2(φl ′ − φl) = 1 2 l 2C ψ l R E (5.35) for the E-polarization and C˜B l ≈ C ψ l Z d 2 l ′ (2π) 2 [l ′ · l] 2 C E |l ′ | sin2 2(φl ′ − φl) = 1 2 l 2C ψ l R E = C˜E l (5.36) for the B-polarization. So on small scales C˜E l ∼ C˜B l , and the power is proportional to the deflection angle power at that scale, as shown in
Figure 13. Top: the lensed E (top solid) and lensed B (bottom solid) power spectra, compared to the unlensed E spectrum (dotted) and the asymptotic result of Eq. (5.35) (dashed). Bottom: the fractional change in the E power spectrum due to lensing. All results are for a fiducial standard ΛCDM cosmology. 5.3.2 Lensed polarization correlation functions As in the case of the temperature spectrum, the series expansion in the deflection angle that we have used in the previous section is not expected to be very accurate on small scales, so for a more accurate calculation of the lensed power spectra…
Figure 14. The 95%-confidence regions for the CMB polarization
Figure 15. A simulated reconstruction of the magnitude of the d
Figure 16. A simulated reconstruction of the lensing converge
Figure 17. The fractional difference between the lensed CMB pow
Figure 18. Effect on lensing potential power spectrum (top) and
Figure 19. Simulated effect of cluster lensing on the CMB temper

Figures reproduced from arXiv:astro-ph/0601594 by the authors.

References
How later papers used it

451 later papers cite it

Evidence: substantial. 411 of the citing papers share no author with it, and 45 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 727 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

Uses its method or tool 13 of 39 papers with this label · 3 by the same authors

By the same authors

Compares with it 1 of 4 papers with this label

  • Compared its own expression with the one used to compute the lensing in this paper, finding the only differences were exchanging the blurring potential for the lensing one and additional factors l from angular derivatives.

    Lensing anomalies from the epoch of reionisation (2019) · 0.91 · paraphrased: the licence does not allow quoting

How the literature uses it

329 arXiv papers cite this in 543 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 · 253 sentences without a confident label

Builds on it 2 papers
  • Following the derivation of the post-Born correction to temperature anisotropies in [ref] , we first generalize the results of [ref] writing the following relation between the lensed and unlensed temperature anisotropies $$ and polarization tensor $_mn$ of the photon field valid up to fourth order in the deflection angles $ai$ (the superscript $(i)$ denotes the order).

    CMB-lensing beyond the leading order: temperature and polarization anisotropies · Weak lensing corrections beyond leading order in real space · 0.94

  • We can generalize the result of [ref] and write the following relation between the lensed and unlensed temperature fluctuations $$ valid up to fourth order in the deflection angles $ai$ (the superscript $i$ denotes the order).

    CMB-lensing beyond the Born approximation · Weak Lensing Corrections beyond leading order · 0.92

Uses its method or tool 30 papers
  • These lensed CMB observables can be calculated through [ref] C_ℓ^L,TT ≈(1-ℓ^2 R^ϕ) C_ℓ^TT +∫d^2 ℓ^'/(2 π)^2[ℓ^' ·(ℓ-ℓ^')]^2 C_|ℓ-ℓ^'|^ϕϕ C_ℓ^'^TT, C_ℓ^L,EE ≈(1-ℓ^2 R^ϕ) C_ℓ^EE +∫d^2 ℓ^'/(2 π)^2[ℓ^' ·(ℓ-ℓ^')]^2 C_|ℓ-ℓ^'|^ϕϕ C_ℓ^'^EE cos^2 2(ϕ_ℓ^'-ϕ_ℓ), C_ℓ^L,BB ≈∫d^2 ℓ^'/(2 π)^2[ℓ^' ·(ℓ-ℓ^')]^2 C_|ℓ-ℓ^'|^ϕϕ C_ℓ^'^EE sin^2 2(ϕ_ℓ^'-ϕ_ℓ), where $ℓ=(ℓ, ϕ_ℓ)$, the superscript $L$ denotes the lensed field and $R^ϕ$ is half of a total deflection angle power, R^ϕ≡1/2⟨|∇ϕ|^2⟩=1/4 π ∫d ℓ/ℓ ℓ^4 C_ℓ^ϕϕ.

    CMB lensing from early-formed dark matter halos · Lensing potential with EFHs · 0.97

  • To see this and calculate the S/N it is convenient to Fourier transform ([ref]) and get the familiar result [ref] T^SL() = T() - ' ·(- ')δψ(') T(- ') - 1/2 d”/(2π)^2 ' ·[' + ” - ] ' ·” T(') δψ(”) δψ^*('+” - ).

    How Sensitive is the CMB to a Single Lens? · Realistic S/N · 0.81

  • For this calculation, we follow the prescription of Refs. [ref] and use the integral solvers implemented in the package as part of the Boltzmann code .

    The Atacama Cosmology Telescope: Probing new signatures of ultralight axions with gravitational lensing · Gravitational lensing · 1.00

Compares with it 1 paper
  • This allows us to compare this expression with the one used to compute the lensing, namely [ref] [equation] The only differences are exchanging the blurring potential for the lensing one and the appearance of additional factors $l$ due to the angular derivatives.

    Lensing anomalies from the epoch of reionisation · Blurring, lensing and their correlation · 0.91

Cites it as support 3 papers
Cites it as background 210 papers
  • where the total estimated power spectrum $C_L^ϕ̂ϕ̂$ has contributions from the signal of interest itself $C_L^ϕϕ$, from chance Gaussian fluctuations $N^(0)$ [ref] , from additional secondary lensing contractions $N^(1)$ [ref] , and non-Gaussian contributions $N_L^(3/2)$.

    Non-Gaussian deflections in iterative optimal CMB lensing reconstruction · Non-Gaussian deflections effect on the reconstruction · 0.92

  • 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

  • In $$ the deflection potential field $ψ^Λ()$ is statistically isotropic and follows a Gaussian distribution [ref] ⟨ψ_l m^Λ ⟩=0, ⟨ψ^Λ_lm^* ψ^Λ_l'm' ⟩=δ_l l'δ_m m' C^ψ_l, where as usual we expand $ψ^Λ() = ∑_lmψ_lmY_lm()$ and the angle brackets stand for an ensemble average.

    How Sensitive is the CMB to a Single Lens? · Ideal S/N · 0.93

Discussion

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