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REVIEW 2 major objections 7 minor 44 references

Detecting rotation from lensing in the CMB

T0 review · 2 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper asks whether the second-order lensing curl of the CMB, and the polarization rotation it may cause, can be measured by current and planned CMB experiments.

desk verdict Solid, honest forecasting paper; the headline S/N numbers rest on the contested β=-ω relation, which the authors flag and provide ω-only alternatives for. read the letter →

arxiv 2501.04158 v2 pith:DYOJI5ET submitted 2025-01-07 astro-ph.CO

Julien Carron , Enea di Dio , Ruth Durrer This is my paper · ORCID
classification astro-ph.CO
keywords cosmicmicrowavebackgroundCMBlensingpost-BornrotationpolarizationB-modequadraticestimatorslarge-scalestructurecross-correlation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the second-order, “post-Born” curl of CMB lensing—the rotational part of the deflection angle that appears when a photon is deflected more than once—can be measured, and whether that rotation also rotates the polarization of the CMB. The authors build quadratic estimators for the polarization rotation angle $\beta$ and the lensing rotation angle $\omega$, compute their noise and biases for current and planned experiments, and forecast detections from the CMB alone and from cross-correlating the CMB with large-scale structure templates. Under the contested assumption $\beta=-\omega$, they find that SPT-3G after its nominal seven-year survey will detect the combined signal at signal-to-noise $\approx 7$ through LSS cross-correlation, and that CMB-S4 deep will reach $\approx 38$; from the CMB alone, CMB-S4 deep reaches about $3.9\sigma$. If these forecasts hold, the open dispute over whether the lensing curl rotates polarization can be decided with data already being collected.

What carries the argument

The machinery is a pair of optimal quadratic estimators, one for polarization rotation $\beta$ and one for the lensing curl $\omega = -\frac{1}{2}\epsilon^{ij}\nabla_i \alpha_j$, constructed as likelihood gradients of inverse-variance-filtered CMB polarization maps. In the squeezed limit, a long-wavelength rotation of the image and a long-wavelength rotation of the polarization produce the same local EB power, so the estimators are degenerate at the dipole and only partially separated at $L\geq 2$ by shear/B-mode information; the paper quantifies this with response functions $R^{\beta\beta}$, $R^{\omega\omega}$, $R^{\beta\omega}$ and Gaussian noise $N^{(0)}_L$, plus the lensing-induced $N^{(1)}_L$ bias. It then applies iterative internal delensing, which reduces both $N^{(0)}_L$ and $N^{(1)}_L$ by orders of magnitude, and builds LSS rotation templates from the bispectrum $b^{\omega ij}$ of $\omega$ with convergence, galaxy, and CIB tracers, characterized by a correlation coefficient $F_L$ that enters the forecast signal-to-noise.

What would settle it

Measure the cross-spectrum $C^{\beta\omega}_L$ between the reconstructed polarization rotation and lensing rotation maps from the same CMB data. Under $\beta=-\omega$ it must equal $-C^{\omega\omega}_L$, while under no polarization rotation it is zero; equivalently, the SPT-3G-7y template-cross-correlation S/N should fall near 7.1 in the first case and near 4.9 in the second. A measured value between these, with uncertainties excluding both, would falsify the exact relation.

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Extended reading notes

Core claim

At second order in gravitational lensing, the deflection angle acquires a curl component $\omega$, and a separate body of work claims this curl also rotates the polarization by $\beta=-\omega$. The paper's central computational claim is that the two effects can be measured jointly with quadratic estimators acting on the local EB (E-mode/B-mode) polarization cross-power they create, and that the $\beta=-\omega$ combination is much easier to detect than the curl alone because the B-mode signals add coherently. Concretely, the paper forecasts S/N $\approx 7.1$ for SPT-3G-7y, $\approx 38.3$ for CMB-S4 deep, and $\approx 39.4$ for PICO when the CMB rotation estimators are cross-correlated with templates built from CMB lensing convergence, galaxy clustering, and the cosmic infrared background; without polarization rotation the same SPT-3G measurement yields S/N $\approx 4.9$. A purely internal CMB detection remains marginal, at $3.9\sigma$ for CMB-S4 deep even with iterative delensing.

Load-bearing premise

The headline forecasts assume the exact relation $\beta=-\omega$, meaning the polarization rotation and lensing rotation fields have equal power and are perfectly anti-correlated; the paper adopts this contested relation from other work without deriving it and does not show how the forecasts degrade under partial decorrelation.

Editorial extensions

If this is right

  • SPT-3G-7y should detect the combined lensing-rotation signal at S/N $\approx 7.1$ with LSS templates if $\beta=-\omega$, so the rotation question can be addressed with data now being collected.
  • CMB-S4 deep reaches S/N $\approx 38.3$ and PICO $\approx 39.4$ under the same assumption, making the post-Born curl a measurable signal rather than a theoretical correction.
  • A CMB-only measurement stays marginal at $3.9\sigma$ for CMB-S4 deep, so delensing and external templates are required for a high-significance detection.
  • If polarization rotation is absent, SPT-3G-7y still detects the image rotation at S/N $\approx 4.9$ with the same templates, so the curl itself should be detected either way.
  • The $\beta$ and $\omega$ estimators are nearly degenerate on the largest scales and correlated elsewhere, so any joint measurement must include their cross-response to avoid misattributing signal.

Reading between the lines

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

  • If the true $\beta$ and $\omega$ are only partially anti-correlated, the forecast S/N lies between the $\beta=0$ and $\beta=-\omega$ endpoints (4.9 and 7.1 for SPT-3G-7y); the paper does not compute that interpolation, so its headline numbers represent the maximum under the contested relation.
  • A null $\beta$-$\omega$ cross-correlation would not refute the post-Born curl; it would refute the claim that the curl rotates polarization, effectively deciding the dispute in favor of the no-rotation calculations.
  • The same estimator formalism can separate lensing rotation from other rotation sources such as cosmic birefringence or Faraday rotation using the frequency and redshift dependence of the latter, extending the method beyond the lensing question.
  • Using redshift-resolved galaxy samples or deeper convergence maps could push the LSS template correlation above the $\sim 0.8$ large-scale value the paper finds for CMB-S4 deep, strengthening the forecast further.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. This paper studies the detectability of the second-order (post-Born) lensing curl omega and the associated polarization rotation beta. It constructs minimum-variance quadratic estimators for beta, omega, and kappa, derives their reconstruction noise N^(0) and the lensing-induced N^(1) bias, validates the noise predictions against full-sky simulations, and forecasts detection significances for LiteBIRD, SO, SPT-3G-7y, PICO, and CMB-S4, both internally and in cross-correlation with LSS templates built from CMB lensing convergence, galaxy clustering, and CIB. The central numerical results are conditional on the contested relation beta=-omega: SPT-3G-7y is forecast to reach S/N about 7.1 and CMB-S4-deep about 38 with the combined kappa+g+CIB template, while the omega-only case gives 4.9 for SPT-3G-7y.

Significance. The paper is significant because it gives a self-contained, quantitative path to detecting a genuinely second-order lensing effect with currently planned experiments, and it frames the beta=-omega controversy as a falsifiable observational question. The estimator formalism is careful, the N^(1) treatment is checked with simulations, and the forecasts are transparent about the assumptions on LSS tracers, delensing residuals, and the beta=-omega hypothesis. If the forecasts are correct, the post-Born lensing curl becomes detectable with SPT-3G in cross-correlation, which would be a first. The main weakness is that the headline significance for the polarization-rotation component rests entirely on an exact relation that the paper does not itself establish, and no intermediate partial-coherence scenario is quantified.

major comments (2)
  1. [Section 1; Tables 3 and 4] The headline S/N values (SPT-3G-7y: 7.1; CMB-S4-deep: 38.3 in Table 4) assume beta=-omega exactly, i.e., C_beta beta = C_omega omega and C_beta omega = -C_omega omega. The paper explicitly declines to settle this controversy and provides no intermediate case. Because the omega-only column in Table 3 gives SPT-3G-7y S/N=4.9, the additional discriminating power of the claimed detection is supplied entirely by the contested relation. I recommend adding a one-parameter interpolation beta = -r omega (equivalently C_beta beta = r^2 C_omega omega and C_beta omega = -r C_omega omega) with r in [0,1] to Tables 3-4 and Fig. 7, so that the degradation under partial decorrelation is visible. This is needed to support the conclusion that the polarization-rotation component will be observed at high significance soon.
  2. [Section 3.2; Appendix C; Fig. 4] There is an apparent sign inconsistency in the treatment of beta and omega. Equation (C.1) gives delta B = -2 beta E for polarization rotation, while Eq. (C.8) gives delta B = -2 omega E for the lensing rotation contribution. Under the stated beta=-omega relation, these two rotation-induced B contributions cancel exactly, leaving only the shear part of the curl signal. Yet Section 4.1 and Fig. 4 describe the beta=-omega cross-term as boosting the B-mode power, and Table 1 reports a 3.9 sigma internal detection for this case. The authors should reconcile the sign conventions, for example by explicitly showing how the shear contribution and the partial decorrelation of the large-scale B modes (the reported -0.8 cross-correlation) carry the signal, or correct the equations and figure caption. As written, the reader cannot determine whether the beta=-omega forecasts correspond to the same physical scenario illustrated in Fig. 1.
minor comments (7)
  1. [Introduction] The first paragraph contains a typo: 'It has let to' should read 'It has led to'.
  2. [Throughout] The word 'degenaracy'/'degenaracies' appears in several places (e.g., Section 3.2 and Section 5) and should be corrected to 'degeneracy'/'degeneracies'.
  3. [Section 2 and Fig. 1] The text states that positive image rotation by omega is equivalent to polarization rotation by beta=-omega, while the Fig. 1 caption describes a counter-clockwise polarization rotation by the same angle beta; since positive beta is defined as counter-clockwise, these statements appear to conflict unless 'same angle' means same magnitude rather than same signed angle. Please harmonize the wording.
  4. [Table 1 caption] The caption says all numbers include internal iterative delensing self-consistently, but the surrounding text emphasizes the delensing procedure mainly for the deep configuration; please state explicitly which experimental configurations are delensed and which are not.
  5. [Eq. (4.1)] The signal-to-noise sum starts at L>=30 with no stated Lmax; please specify the maximum multipole used for each configuration.
  6. [Section 3.2] The full-sky simulation validation of the N^(1) predictions is mentioned but not shown; a figure or a short table comparing simulated and predicted spectra would make the claim easier to verify.
  7. [Fig. 2 caption] The caption refers to 'the kappa spectrum and our fiducial post-Born lensing curl spectrum' but does not identify which black line is which; please label the curves directly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the forecast pipeline is self-contained, and the contested beta=-omega relation is an explicitly conditional hypothesis rather than a fitted or constructed output.

full rationale

The paper's forecasts are not derived from the quantity they claim to predict. Reconstruction responses and N^(0)/N^(1) biases are obtained from likelihood gradients, standard bispectrum kernels, and flat/curved-sky checks against full-sky simulations; the LSS-template signal-to-noise is computed from the theoretical omega-LSS bispectrum with no parameter fitted to the target. The central statement that observations can decide the controversy is supported by comparing two distinct scenarios (omega-only and beta=-omega) in Fig. 7. The beta=-omega relation is explicitly treated as a hypothesis: in Section 2 the paper says it does not want to take a stand and simply studies whether such a rotation, if present, could be detected, setting beta=-omega as claimed in Refs [18-20]. Those references overlap with the present authors, and the headline S/N values in Tables 1 and 4 are conditional on that exact relation; however, the paper provides the omega-only alternative (Table 3, SPT-3G-7y S/N=4.9) and does not claim to derive the relation. The lack of a partial-coherence calculation is a robustness limitation, not a circular reduction. Score 2 accounts for the self-citation dependence of the combined-signal scenario without treating a stated assumption as a circular derivation.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central forecasts rest on theory inputs and modeling choices that are not derived in this paper: the fiducial post-Born curl spectrum, the contested β=-ω relation, the delensing residuals, the neglect of N^1 after delensing, and the LSS tracer models. None of these are fitted to the forecast target; they are standard assumptions in the forecasting literature, but they set the scale of every quoted SNR.

free parameters (4)
  • Galaxy bias model b(z) = b(z) = 1 + 0.84 z
    Assumed LSST-like linear bias (Eq. B.3). Directly sets the galaxy window function W_g and the template correlation F_L.
  • Galaxy redshift distribution z0 = z0 = 0.311
    Assumed dN/dz shape for photometric galaxy sample (Eq. B.4).
  • CIB SED parameters = z_c=2, σ_z=2, T=34 K, ν'=353 GHz, β=α=2
    Fixed graybody SED and redshift kernel (Eqs. B.6-B.7) from Ref [57]; CIB contributes marginally to the template.
  • LSS shot noise levels = N_shot,g ≈ 2.1e-9; N_shot,I = 225.6e-12 MJy^2/sr
    Assumed shot noise floors (Eqs. B.5, B.7); they set the tracer inverse covariance used in Eq. (4.13).
assumptions (6)
  • domain assumption The post-Born lensing curl spectrum C_ωω_L is the correct signal spectrum for ΛCDM.
    Used throughout for SNR forecasts; computed from Eq. (2.3) with Planck 2018 parameters, not derived in this paper.
  • domain assumption β = -ω exactly, with C_ββ = C_ωω and C_βω = -C_ωω.
    Headline forecasts in Tables 1, 2, 4 and Fig. 7 depend on this contested relation from Refs [18-20].
  • domain assumption Iterative internal delensing achieves the assumed residual lensing B-mode power (95% removal at ℓ<500, 85-70% at higher ℓ).
    Section 3.2 and Fig. 4; based on analytic iteration calibrated to simulations [39,40], but not demonstrated for β in this paper.
  • domain assumption After delensing, the N^1 bias is negligible compared to N^0 and is neglected.
    Section 3.2, Fig. 3 shows N^1 reduced by an order of magnitude; the paper neglects it from then on.
  • domain assumption LSS tracers follow the models of Ref [21]: LSST-like galaxy clustering with b(z)=1+0.84z and shot noise 2.1e-9, and CIB with fixed SED parameters.
    Appendix B, Eqs. (B.3)-(B.7); these windows set the template correlation F_L.
  • domain assumption The Limber approximation and Halofit power spectrum accurately model the LSS spectra and ω-LSS bispectra.
    Appendix B, Eqs. (B.1), (B.10), (B.15)-(B.16); small-scale accuracy matters since the β=-ω signal peaks around L~2000.

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Pith. "Pith review of Detecting rotation from lensing in the CMB." pith.science (2026). https://pith.science/paper/DYOJI5ET

@misc{pith2026250104158,
  author       = {Pith},
  title        = {Pith review of: Detecting rotation from lensing in the CMB},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYOJI5ET}},
  note         = {Machine review of arXiv:2501.04158}
}
read the original abstract

An excellent estimate of the lensing signal is expected from the availability of deep and high-resolution polarization data in the near future. This is most important to allow for efficient delensing, needed to detect the primordial B-mode power and with it the famous tensor-to-scalar ratio. Here we discuss in a joint manner estimators of the rotation of polarization, of the second order lensing field rotation, and standard gradient lensing reconstruction. All are most efficient when able to probe the EB power created locally, have comparable reconstruction noise in this regime, and can benefit substantially from delensing. We discuss several ongoing and planned CMB experiments. We determine their noise for lensing field rotation and polarization rotation and discuss their prospects for measuring these effects. There is an on-going controversy on whether the lensing field rotation also rotates the polarization -- if so this will be observed at high significance soon with already on going observations of the South Pole Telescope, SPT-3G, in cross-correlation with tracers of large scale structure, as we show in this paper.

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