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REVIEW 3 major objections 5 minor 68 references

Bias to CMB lensing from lensed foregrounds

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that extragalactic foregrounds in CMB temperature maps are lensed by the same large-scale structure that lenses the CMB, so standard lensing reconstructions pick up a correlated foreground-lensing signal that biases them…

desk verdict A genuinely new, carefully quantified systematic in CMB lensing, with a real order-unity uncertainty that the authors openly admit; worth refereeing, but the headline percent-level numbers should be read as estimates, not predictions. read the letter →

arxiv 1908.08057 v2 pith:HAP7ZXTW submitted 2019-08-21 astro-ph.CO astro-ph.GA

Nishant Mishra , Emmanuel Schaan This is my paper · ORCID
classification astro-ph.COastro-ph.GA
keywords CMBlensingextragalacticforegroundslensedforegroundbiasquadraticestimatorshearmagnificationcosmicinfraredbackgroundSunyaev-Zeldovicheffect
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

The paper establishes that extragalactic foregrounds in CMB temperature maps are themselves gravitationally lensed, and because this foreground lensing is correlated with CMB lensing, standard CMB lensing reconstruction measures a combination of both and is biased. For a next-generation stage-III 143 GHz temperature map with a 1.4 arcmin beam and 7 microK-arcmin white noise, the bias is about one percent of the CMB lensing auto-spectrum, comparable to the statistical uncertainty, and larger than the statistical error in cross-correlation with a deep galaxy sample. This matters because upcoming lensing measurements aim for percent-level precision, and the bias persists even for perfectly Gaussian foregrounds, so methods that only remove foreground non-Gaussianity do not cure it. The paper derives analytic bias formulas, confirms them in simulations, and proposes modified estimators that null the bias at some noise cost.

What carries the argument

The load-bearing object is the response function $R_L^f$, the ratio of a quadratic estimator's response to foreground lensing, built from the foreground power spectrum $C_\ell^f$, to its response to CMB lensing, built from $C_\ell^{\rm CMB}$. Multiplying this response by the correlation $C_L^{\kappa_f\kappa_{\rm CMB}}$ between foreground and CMB convergence gives the bias to the auto-spectrum, and by $C_L^{g\kappa_f}$ gives the bias in cross-correlation with galaxies. The foreground lensing convergence is defined through an effective kernel $W^{\kappa_f}(\chi,L)$, constructed from the redshift distribution of the foreground power spectrum $\mathrm{d}C_f/\mathrm{d}\chi_S$, evaluated at $\ell=3000$ in this paper. A secondary bias term, given by a four-dimensional integral, cancels exactly in the limit $L,L_0\ll\ell$ and dominates at high lensing multipoles.

What would settle it

Build a realistic foreground map split into redshift slices, lens each slice by its own convergence field correlated with the CMB convergence, add it to a lensed CMB map, apply the quadratic, shear, and magnification estimators, and compare the measured bias with this paper's single-effective-convergence prediction; a mismatch larger than the simulation error would falsify that approximation.

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Extended reading notes

Core claim

The central claim is that CMB lensing quadratic estimators applied to a temperature map containing lensed extragalactic foregrounds reconstruct a weighted sum of CMB lensing and foreground lensing, and that the correlation between the two convergences converts this response into a bias. The paper computes this lensed foreground bias for the standard quadratic estimator, the shear estimator, and the magnification estimator, finding percent-level biases for a single-frequency 143 GHz stage-III experiment. The foreground lensing convergence itself is 5% to 85% as large as the CMB lensing convergence depending on the component, but the estimators' response to foreground lensing is only about a percent, which sets the bias size. In the CMB lensing auto-spectrum the bias is comparable to the statistical uncertainty, while in cross-correlation with a deep galaxy sample it exceeds the statistical error and is thus highly significant. The bias formulas include primary, secondary, and four-point terms; the secondary term cancels at low lensing multipoles and dominates at multipoles of a few thousand.

Load-bearing premise

The calculation assumes each foreground component is lensed as a whole by a single effective convergence field built from its adopted redshift distribution, and it assumes those foreground redshift distributions and power spectra are accurate; if they are not, the percent-level bias estimate changes.

Editorial extensions

If this is right

  • For a stage-III single-frequency temperature experiment, the lensed foreground bias must be included in the error budget because it is comparable to the statistical uncertainty on the CMB lensing auto-spectrum.
  • In cross-correlation with a deep galaxy sample, the bias is larger than the statistical error, so ignoring it would produce a biased measurement of the galaxy-CMB lensing correlation.
  • The shear and magnification estimators have opposite-sign responses to foreground lensing, so comparing them provides a null test for the presence of lensed foreground bias.
  • Mitigation methods that rely on foreground non-Gaussianity, such as standard bias hardening or the shear estimator alone, do not remove this bias; reducing the foreground level in the map, scale cuts, or the lensed-foreground bias-hardened estimators are needed.
  • At lensing multipoles of a few thousand the secondary bias dominates, so small-scale lensing measurements from temperature need special treatment.

Reading between the lines

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

  • The bias depends directly on foreground redshift distributions, so measuring those distributions, for example by cross-correlating foreground maps with galaxy positions, could turn the predicted bias into a correctable template; the paper's statement that a 10% theory error is acceptable sets the required calibration.
  • The single-effective-convergence approximation could be checked with redshift-sliced foreground mocks; if slice-by-slice lensing matters, the bias formulas would need the full $\mathrm{d}C_f/\mathrm{d}\chi_S$ integral rather than a single $\ell=3000$ evaluation.
  • The same correlated-lensing logic should also bias CMB lensing cross-correlations with the foreground fields themselves, such as CIB or tSZ tracers, since those tracers share the foreground lensing convergence and would enter through the same response formulas.
  • The opposite signs of the shear and magnification responses suggest an internal consistency test on real data: the difference of the two estimators should flip sign where lensed foregrounds dominate, independently of any assumed foreground model.
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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

3 major / 5 minor

Summary. This paper quantifies a new systematic for CMB lensing reconstruction: the lensing of extragalactic foregrounds by the same large-scale structure that lenses the CMB. The authors derive analytic expressions for the response of the standard quadratic, shear, and magnification estimators to foreground lensing (Eq. 14), the resulting primary, secondary, and 4-point biases in the lensing auto-spectrum (Eqs. 19-22), and the cross-correlation bias (Eq. 17). Using foreground power spectra and redshift distributions from the literature, a Simons Observatory-like single-frequency 143 GHz configuration, and the LSST gold sample, they find percent-level biases comparable to the statistical errors in the auto-spectrum and larger than the expected cross-correlation errors. They validate the analytic expressions against 8060-map Gaussian simulations, provide a public code repository, and propose bias-hardening estimators.

Significance. If the amplitude estimate is reliable, this is an important first quantification of a systematic that has been noted but not computed, and it is directly relevant to AdvACT, SPT-3G, Simons Observatory, and CMB-S4. The analytic derivation is careful, the exact cancellation of the secondary bias at low L is a clean result, and the extensive simulation cross-checks plus public code are clear strengths. The main limitation is that the bias amplitude inherits the uncertainty in the approximate effective foreground source distribution, which the paper itself flags as needing quantification; consequently the quantitative headline is not yet fully robust.

major comments (3)
  1. [Sec. III B (Eq. 11) and Sec. V] The exact effective foreground source distribution Wf(χS,L) defined in Eq. (11) is estimator-weighted and L-dependent, but the numerical results replace it with Wf ∝ dCf(ℓ=3000)/dχS, dropping the F/N_L weighting and all L dependence. Since the bias in Eq. (19) is linear in the resulting C^{κf κCMB}_L, an order-unity error in the effective source redshift distribution would change the percent-level claim and the 'highly significant' LSST cross-correlation conclusion. The paper itself states in Sec. V that quantifying the uncertainty in the foreground source distributions is needed, but no such quantification is provided. Please evaluate Eq. (11) exactly for representative L values, or propagate explicit uncertainties from the adopted source models, and report how the bias changes.
  2. [Sec. IV and Sec. V] The single-convergence approximation, where each foreground map is lensed as a whole by one convergence field κf, is stated in Sec. IV and in the Conclusions to be 'sufficient' and believed to be comparable to the redshift-distribution uncertainty, but no test is given. Because each redshift slice of a foreground is lensed by a different convergence, and because the estimator weights in Eq. (11) depend on the source distribution, this assumption directly affects the same C^{κf κCMB}_L that sets the headline amplitude. Please test the approximation, for example by splitting a foreground into two or more redshift slices and recomputing the bias.
  3. [App. B and Figs. 4, 6] The simulation validation uses foreground convergence maps κf generated from the same approximate Wf used in the analytic calculation (App. B, Eqs. B1-B3). The agreement between simulation and analytic curves therefore validates the algebra of the response and bias formulae, but it does not validate the Wf approximation itself. The paper should make this explicit and, more importantly, the approximate input should be tested against exact or observationally calibrated source distributions.
minor comments (5)
  1. [Sec. III B] The sentence 'For the CMB S3 experiment we consider, and assuming lmax T = 3500, most of the CMB lensing signal-to-noise comes from temperature multipoles ℓ ~ 3000' is given without support; a quantitative statement of the signal-to-noise contribution would strengthen the justification for evaluating dCf/dχ at ℓ=3000.
  2. [App. C (Eq. C10)] The second line of Eq. (C10) contains an undefined weight w'_{L,ℓ}; the primary-bias term should presumably involve (Σ_ℓ w_{L,ℓ} R^f_{L,ℓ})(Σ_ℓ w_{L,ℓ}) or simply (Σ_ℓ w_{L,ℓ} R^f_{L,ℓ}) since the estimator has unit CMB response. Please clarify or correct.
  3. [Figs. 5 and 6 captions] The grey shaded areas are described as 'statistical uncertainty on the amplitude' with fsky and Lmax,κ specified; stating clearly whether this includes cosmic variance of the CMB lensing field and of the tracer, and how the bandpower errors are combined, would help the reader.
  4. [Abstract and Sec. V] The term 'lensed foreground bias-hardening' is used in the abstract and in Sec. V; the hyphenation makes it read as if 'lensed foreground' modifies 'bias-hardening'. Consider rewording as 'hardening against the lensed foreground bias'.
  5. [Sec. V] The statement 'This bias is thus marginally significant in auto-correlation, and highly significant in cross-correlation' would benefit from repeating the quantitative numbers (e.g., percent-level bias versus 0.6% cross-correlation uncertainty and 1% auto-spectrum uncertainty) so that the conclusion does not depend on the figures alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lensed-foreground bias is computed from independent foreground spectra and source distributions, with internal simulation checks; the overlap with Schaan, Ferraro, Spergel (2018) is not load-bearing.

full rationale

The paper's central derivation is not circular. The lensed-foreground bias is computed from (i) foreground power spectra and redshift distributions taken from independent literature (Dunkley et al. 2013 for the spectra; the CIB halo model, Hill & Pajer tSZ model, Shaw et al. kSZ model, and NVSS-based radio source distributions), (ii) the standard quadratic estimator response formalism, whose analytic evaluation is checked against 8060 first-order lensing simulations in App. B, and (iii) the nonlinear matter power spectrum. No parameter is fitted to the target bias. The only overlapping-author citation, Schaan, Ferraro, and Spergel (2018), supplies the response and effective-kernel formalism, but the present paper independently computes the analytic integrals and validates them with simulations (Eqs. B1-B3 and Figs. 4, 6-8), so the citation is not load-bearing in the circularity sense. The approximation Wf ∝ dCf/dχ at ℓ=3000 and the single-lens-plane simplification are acknowledged accuracy limitations (Secs. III B, IV, V), not circular reductions: the predicted bias does not reduce to its input foreground spectra by construction. The paper's own statements that the foreground source distributions are uncertain and that the bias should be treated as a reasonable value rather than exact further support reading this as an uncertainty caveat, not a circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central bias calculation rests on foreground power spectra (Dunkley et al. 2013), foreground redshift distributions (from several models), and the approximation of a single effective lensing convergence per foreground. These are external inputs or acknowledged simplifications, not derived in this paper. No free parameters are fitted; no new entities are introduced.

assumptions (5)
  • standard math Born approximation and a single source plane for CMB lensing.
    Used in Eq. (1) for the CMB lensing convergence; standard in CMB lensing.
  • standard math Limber and flat-sky approximations for power spectra.
    Used in Eqs. (3) and (13) to compute lensing power spectra; standard and well justified at the multipoles of interest.
  • domain assumption Effective foreground source distribution Wf(chiS,L) is approximated as proportional to dCf(l=3000)/dchiS, ignoring L-dependence.
    Eqs. (10)-(11) define a scale-dependent source distribution; the paper fixes it at l=3000, affecting the foreground lensing kernel and bias amplitude.
  • domain assumption Each foreground map is lensed by a single convergence field kappa_f.
    Stated in Sec. IV; ignores that each redshift slice is lensed separately. Acknowledged as a simplification.
  • domain assumption Foreground power spectra and redshift distributions from the literature are sufficiently accurate.
    Sec. III adopts models from Dunkley et al. 2013 and various redshift distribution papers; the bias scales with these inputs.

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Cite this review

Pith. "Pith review of Bias to CMB lensing from lensed foregrounds." pith.science (2026). https://pith.science/paper/HAP7ZXTW

@misc{pith2026190808057,
  author       = {Pith},
  title        = {Pith review of: Bias to CMB lensing from lensed foregrounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HAP7ZXTW}},
  note         = {Machine review of arXiv:1908.08057}
}
read the original abstract

Extragalactic foregrounds are known to constitute a limiting systematic in temperature-based CMB lensing with AdvACT, SPT-3G, Simons Observatory and CMB S4. Furthermore, since these foregrounds are emitted at cosmological distances, they are also themselves lensed. The correlation between this foreground lensing and CMB lensing causes an additional bias in CMB lensing estimators. In this paper, we quantify for the first time this "lensed foreground bias" for the standard CMB lensing quadratic estimator, the CMB shear and the CMB magnification estimators, in the case of Simons Observatory and in the absence of multi-frequency component separation. This percent-level bias is highly significant in cross-correlation of CMB lensing with LSST galaxies, and comparable to the statistical uncertainty in CMB lensing auto-spectrum. We discuss various mitigation strategies, and show that "lensed foreground bias-hardening" methods can reduce this bias at some cost in signal-to-noise. The code used to generate our theory curves is publicly available at https://github.com/EmmanuelSchaan/LensedForegroundBias .

Figures

Figures reproduced from arXiv: 1908.08057 by the authors.

Figure 1
Figure 1. FIG. 1: Extragalactic foreground power spectra at 143GHz from [52], compared to the lensed CMB and the detector noise [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Extragalactic foregrounds are themselves lensed, because they are emitted at cosmological distances. The exact [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Since foreground sources lie at lower redshift than the CMB, the lensing they experience is less than that of the CMB, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The lensing responses [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The relative bias from lensed foregrounds on the cross correlation of CMB lensing from Simons Observatory and [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The primary relative bias from lensed foregrounds on the power spectrum of CMB lensing from Simons Observatory [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The secondary relative bias from lensed foregrounds on the power spectrum of CMB lensing is suppressed at low lensing [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The total (primary + secondary) relative bias from lensed foregrounds on the power spectrum of CMB lensing from [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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Reviewed August 14, 2026 · model on record in the stance chip above.