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REVIEW 3 major objections 7 minor 1 cited by

Quadratic estimators for unwindowed power spectrum of galaxy-galaxy weak lensing and its application to $P_{\rm gm}(k)$ estimation

T0 review · 3 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A quadratic estimator for galaxy-galaxy weak lensing recovers the E-mode power spectrum to within a few percent in simulations and keeps E-to-B leakage below one percent.

desk verdict A solid, well-validated estimator whose main caveats are an unpublished spin-2 projection step and an overstated abstract; deserves peer review. read the letter →

arxiv 2507.18789 v1 pith:UOJYSJIT submitted 2025-07-24 astro-ph.CO

Taisei Terawaki , Masahiro Takada , Takanori Taniguchi This is my paper · ORCID
classification astro-ph.CO
keywords galaxy-galaxyweaklensingquadraticestimatorE/B-modedecompositionflat-skyapproximationsurveywindowcorrectionFFTpowerspectrumestimationgalaxy-matterredshiftslicing
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 develops a quadratic estimator for the E- and B-mode angular power spectra of galaxy-galaxy weak lensing, $C_{\mathrm{g}E}(\ell)$ and $C_{\mathrm{g}B}(\ell)$, that is designed to be free of survey-window distortions such as masks and survey geometry. The estimator accounts for the spin-2 nature of shear, uses FFTs on pixellated maps under the flat-sky approximation, and deconvolves window effects through an inverse Fisher matrix. Validated on ray-tracing simulations with halo catalogs as lens galaxies, it recovers the input E-mode power spectrum to within a few percent in each multipole bin up to $\ell \sim 3000$ and keeps E-mode leakage into the B-mode below one percent. The same estimator is then used to estimate the 3D galaxy-matter power spectrum $P_{\mathrm{gm}}(k)$ by dividing lens galaxies into redshift slices, with an inverse-variance weighting that reduces statistical errors by roughly 20 to 30 percent.

What carries the argument

The central object is the quadratic estimator of Eqs. (3.13)-(3.15), a maximum-likelihood-derived band-power estimator whose data term is a quadratic product of the weighted lens density field and the two shear components, and whose normalization is the inverse Fisher matrix $F^{-1}$ computed by Monte Carlo from ancillary Gaussian fields. The Fisher matrix carries the window information: its off-diagonal entries encode mask-induced couplings between multipole bins, and its E/B off-diagonal blocks encode E-to-B leakage; inverting it deconvolves both effects. The argument is carried by the flat-sky FFT implementation of Eq. (3.20), which reduces the naive $O(N_{\mathrm{grid}}^2)$ computation to $O(N_{\mathrm{grid}} \log N_{\mathrm{grid}})$, making the estimator practical on maps with roughly a million pixels.

What would settle it

Recompute the spherical-to-flat shear projection with an independent implementation and rerun the 972-realization validation; the central claim fails if the recovered $C_{\mathrm{g}E}(\ell)$ band powers shift by more than the quoted few-percent accuracy or if the E-to-B leakage rises above the percent level. A simpler check is to run the estimator on a mock map with no masks and trivial geometry, where the window correction should be inactive, and compare against the known input spectrum.

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Extended reading notes

Core claim

The central claim is that Eqs. (3.13)-(3.15) give a closed-form, window-free quadratic estimator for galaxy-galaxy weak lensing band powers, and that when applied to simulated lens halos and source shears it recovers the underlying E-mode power spectrum $C_{\mathrm{g}E}(\ell)$ to within a few percent fractional error in each multipole bin over $200 \le \ell \le 3000$, while suppressing E-mode leakage into the B-mode below one percent. The spin-2 structure of the E/B decomposition is built into the Fisher matrix, whose inverse corrects for the survey window and for the E/B mixing induced by masks and survey geometry; the window-convolved spectrum deviates significantly from the underlying spectrum, whereas the estimator does not. For the $P_{\mathrm{gm}}(k)$ application, dividing lenses into five redshift slices and combining their $C_{\mathrm{g}E}$ estimates through Eq. (5.2) gives a halo-matter cross spectrum consistent with the emulator-based model prediction, and the inverse-variance weight of Eq. (5.3) reduces statistical errors by 20 to 30 percent relative to no weighting.

Load-bearing premise

The validation rests on an unpublished coordinate transformation that converts the simulated shear field from spherical map pixels to flat two-dimensional grids; if that transformation is wrong, the recovered band powers would be biased in a way the paper's tests do not expose.

Editorial extensions

If this is right

  • If the estimator performs on real data as it does on the simulations, galaxy-galaxy lensing power spectra can be compared directly with theory without modeling the survey window, removing a known source of systematic error.
  • Because E-to-B leakage is suppressed below the percent level, the measured $C_{\mathrm{g}B}(\ell)$ can serve as a clean null test for systematics such as shear calibration errors, PSF residuals, or intrinsic alignments.
  • The $P_{\mathrm{gm}}(k)$ estimator converts window-free $C_{\mathrm{g}E}(\ell)$ measurements from multiple lens redshift slices into a direct, redshift-space-distortion-free estimate of the 3D galaxy-matter power spectrum, which can be combined with the galaxy auto-spectrum $P_{\mathrm{gg}}(k)$ in full-shape cosmological analyses.
  • The inverse-variance weighting of lens slices yields up to about 20 to 30 percent smaller statistical errors, which at fixed observing time is equivalent to a substantially larger survey area.

Reading between the lines

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

  • A natural next test, not performed in the paper, is to replace the unpublished spherical-to-flat shear projection with an independent implementation and rerun the validation; agreement would settle the least-supported link in the chain.
  • The same maximum-likelihood structure could be extended to cosmic shear auto-spectra and to shear bispectra with full $C^{-1}$ weighting, which would make the estimator optimal in the sample-variance-limited regime as well as in the noise-limited regime.
  • For a real survey, the number of Monte Carlo ancillary realizations needed should be checked per dataset; the paper shows sub-percent convergence with 300 realizations for its specific maps, but a different mask geometry or shape-noise level could require more.
  • Applied to current wide-area imaging surveys, the method would deliver harmonic-space galaxy-galaxy lensing spectra whose scale cuts could be chosen independently of real-space analyses, potentially changing combined-probe cosmological constraints.
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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 / 7 minor

Summary. This paper develops a quadratic estimator, in the maximum-likelihood tradition of Refs. [27-31], for the E- and B-mode angular power spectra of galaxy-galaxy weak lensing, C_gE(ℓ) and C_gB(ℓ), under the flat-sky approximation. The estimator is closed-form (no auto-spectra are needed), corrects for survey windows and masks, and is implemented with FFTs, with the Fisher matrix computed from Monte Carlo ancillary fields (Eqs. 3.13-3.20). The method is validated on 972 patches extracted from the Takahashi et al. full-sky ray-tracing simulations, with halos as lenses and shear at z_s=1.03, 15% masks, and 20 ℓ-bins over 200≤ℓ≤3000: the recovered C_hE agrees with the full-sky healpy measurement to within a few percent for the interior bins, with B-mode leakage below the percent level; the first and last bins are inaccurate, as the text acknowledges. The paper also applies the estimator to estimate the 3D galaxy-matter power spectrum P_gm(k) by splitting the lens sample into redshift slices, and derives an inverse-variance weight claimed to reduce errors by up to 20-30%.

Significance. If the claims hold, this is a useful and practical contribution: it provides a window-free harmonic-space route to galaxy-galaxy lensing power spectra with E/B separation and O(N log N) cost, which is timely for ongoing and upcoming surveys, and the P_gm(k) application gives a direct path from projected galaxy-galaxy lensing measurements to a 3D statistic. The paper's strengths include a largely self-contained derivation of the estimator and its FFT implementation, the explicit N_MC convergence test (Fig. 3), a validation on a large number of independent simulated patches, a non-circular reference measurement (the full-sky healpy spectra are computed by an independent method, with no parameters fitted), and an external comparison with Dark Emulator for P_gm. The two caveats that prevent full endorsement are the unpublished flat-sky spin-2 basis transformation used at the start of the validation pipeline (Section 4.1) and the inconsistency between the abstract's 'in each multipole bin' claim and the acknowledged failure of the first and last bins (Section 4.2, Fig. 2). Both are fixable within the scope of a revision.

major comments (3)
  1. [Section 4.1] The entire validation of the headline claims (the few-percent recovery of C_gE and the sub-percent E-to-B leakage) is carried out on maps that are projected to flat 2D coordinates using a spin-2 basis transformation cited only as '(Terawaki et al. in preparation)'. The manuscript does not specify either the map projection or the per-pixel rotation that converts the shear components from the HEALPix spherical basis to the flat basis. This is load-bearing: an error in the rotation convention would mix E and B modes in a position- and scale-dependent way, and the existing end-to-end check against the full-sky healpy spectrum does not isolate this step, since a small rotation error would still leave the recovered E-mode close to the true one when the true B-mode is subdominant. Please include the explicit transformation (or a complete companion reference) and add a sensitivity test, e.g., perturbing the rotation angle and verifying that the few-percent recovery and the sub-percent B-mode leakage claimed in Fig. 2 are stable.
  2. [Abstract; Section 4.2, Fig. 2] The abstract states that the estimator recovers C_gE(ℓ) 'to within a few percent in fractional error ... in each multipole bin over the wide range of multipoles (up to ℓ∼3000 studied in this paper)'. However, Section 4.2 states that the first and last ℓ bins are 'not accurate' because the Fisher matrix cannot correct for cross-correlations with multipoles outside the fitted range, and the middle panel of Fig. 2 shows these bins lying outside the few-percent band. The headline claim is therefore contradicted by the paper's own figure. Please either restrict the claim to the interior bins or extend the binning (e.g., with guard bins beyond ℓ=200 and 3000) so that the edge bins can be deconvolved and discarded, and adjust the abstract and Section 4.2 accordingly.
  3. [Section 5.1, Eqs. (5.1)-(5.3)] The derivation of the inverse-variance weight in Eq. (5.3) is not shown, and the displayed expression does not obviously follow from the one-sentence justification given in the text. If Eq. (5.3) is as printed, with \bar n_{2D}(z_L)^2 in the numerator, the stated basis (σ²(Δg) ∝ 1/[S_L \bar n_{2D}(z_L)]) would support w_L ∝ \bar n_{2D}(z_L) rather than the square; in either case, the conversion factors Σcrit(χ̄_L) χ̄_L² in Eqs. (5.1)-(5.2) should enter the variance of the estimated P_gm and hence the weight, and no χ̄_L dependence appears in Eq. (5.3) unless f_AL is intended to absorb it. Since the claimed 20-30% error reduction (Section 5.2, Fig. 5, right panel) rests on this weight, please provide the explicit derivation, including the shape-noise and C_gE² terms in the variance of Ĉ_gE, and check the claimed optimality against the simulated error bars.
minor comments (7)
  1. [Section 4.1] 'Throughput this paper' should read 'Throughout this paper', and '19502 grids' should presumably read '1950² grids'.
  2. [Abstract; Sections 5.2, 6] The weight improvement is quoted as 'up to ~20%' in the abstract, 'up to 20-30%' in Section 5.2 and Fig. 5, and 'about 20-30%' in Section 6; these numbers should be harmonized.
  3. [Section 4.2, Fig. 3] The statement that a ~20% reduction in statistical errors at ℓ∼3000 'is equivalent to increasing the survey area by 40%' appears to be incorrect under the usual σ ∝ A^{-1/2} scaling, which would give roughly a 56% area increase for σ_w/σ_uw ≃ 0.8; please check this equivalence.
  4. [Section 4.2, Fig. 2] The claim that the E-to-B leakage is 'below the percent level' should be quantified (e.g., the maximum of |C_hB|/C_hE over multipole bins and its uncertainty), since the plotted ratios with error bars do not by themselves allow the reader to verify the sub-percent statement.
  5. [Section 5.2, Fig. 5] The Dark Emulator prediction is evaluated at a single effective redshift z=0.28 for a lens slice spanning 0.15<z<0.35; please specify how the effective redshift, the halo mass threshold, and the redshift evolution of the halo bias enter the model curve.
  6. [Section 5.1] The statement that lens-slice correlations are negligible for slice widths thicker than ∼30 h^{-1}Mpc is asserted without support; a brief justification or reference would be helpful.
  7. [Sections 2-3] The flat-sky Fourier conventions (sign convention of the transform, definition of the two-dimensional ℓ, and the normalization of C(ℓ) in Eq. 2.5) are only implicit; a short paragraph defining them would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quadratic estimator is validated against independent full-sky healpy measurements and an external emulator; no fitted parameter or input spectrum is repackaged as a prediction.

full rationale

The derivation chain is self-contained. The estimator in Eqs. (3.13)-(3.15) is obtained from the quadratic maximum-likelihood form (Eqs. (3.7)-(3.9)); the final band-power estimates are F^{-1}q built from quadratic products of the weighted maps (Eqs. (3.14), (3.18)), with no band-power guess or fitted astrophysical parameter entering the prediction. The validation in Fig. 2 compares these outputs to C_hE(l) measured from the same light-cone simulations by the independent, full-sky healpy route ('the true band powers measured from the original full-sky simulations using the healpy code'), so the agreement is not enforced by construction. The Fisher matrix used for window and E/B deconvolution is estimated from 300 realizations of ancillary Gaussian fields (Eqs. (3.17), (3.20)), not from the lensing data, and its convergence is checked in Fig. 3; the E/B leakage is therefore an output of Monte Carlo matrix inversion rather than a fitted residual. The P_gm(k) application follows by direct Limber inversion (Eqs. (2.6), (5.1)) with analytic inverse-variance slice weights (Eq. (5.3)), and is checked against the Dark Emulator prediction, a separate emulator rather than a quantity fitted by this paper. Two caveats do not amount to circularity: Section 4.1 delegates the flat-sky spin-2 basis transformation to '(Terawaki et al. in preparation)', an unpublished self-citation whose correctness is load-bearing for the end-to-end validation; and the first/last multipole bins are explicitly acknowledged as inaccurate in Section 4.2. These are verification and completeness risks, not cases where an equation or fitted parameter is repurposed as a prediction.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard statistical assumptions and a flat-sky projection that is only documented in an unpublished companion paper. No new physical entities are introduced.

free parameters (1)
  • N_MC (number of ancillary field realizations) = 300 (convergence test up to 1000)
    The Fisher matrix is estimated from N_MC=300 Gaussian ancillary realizations; the paper shows convergence to <1% for N_MC>300, so the central claim of percent-level recovery depends on this hand-chosen number being large enough.
assumptions (7)
  • domain assumption The observed fields follow a Gaussian likelihood, so the quadratic estimator is unbiased in ensemble average.
    Section 3.1; non-Gaussianity would affect the covariance but the estimator's expectation value depends only on two-point functions.
  • ad hoc to paper Flat-sky approximation and the unpublished basis transformation of the shear field from HEALPix to flat coordinates.
    Section 4.1; the validation depends on a companion paper in preparation, which is not publicly available.
  • standard math Limber approximation for projecting 3D power spectra to angular power spectra.
    Eq. (2.6); standard approximation in cosmic shear analyses.
  • domain assumption The survey window function is exactly known and binary (W=0 or 1) on each grid.
    Section 4.1; real surveys have partial coverage and uncertain masks, but the method assumes exact knowledge of the window.
  • domain assumption Shape noise is Gaussian with known per-grid dispersion and the lens shot noise follows the random catalog prescription.
    Section 4.1; deviations could bias the error weighting and the covariance, but the estimator's mean remains unbiased.
  • domain assumption The power spectrum is constant within each multipole bin (20 logarithmically spaced bins over 200<=l<=3000).
    Section 3.1; this causes the stated edge-bin inaccuracy because the Fisher matrix cannot correct for correlations with multipoles outside the bin range.
  • domain assumption For P_gm(k), lens galaxies have secure spectroscopic redshifts and statistical isotropy holds.
    Section 5.1; enables redshift slicing and interpretation of the measured modes as k_perp.

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Cite this review

Pith. "Pith review of Quadratic estimators for unwindowed power spectrum of galaxy-galaxy weak lensing and its application to $P_{\rm gm}(k)$ estimation." pith.science (2026). https://pith.science/paper/UOJYSJIT

@misc{pith2026250718789,
  author       = {Pith},
  title        = {Pith review of: Quadratic estimators for unwindowed power spectrum of galaxy-galaxy weak lensing and its application to $P_\rm gm(k)$ estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UOJYSJIT}},
  note         = {Machine review of arXiv:2507.18789}
}
abstract

Galaxy-galaxy weak lensing provides a powerful means of measuring the average matter distribution around lens galaxies -- i.e., the galaxy bias relation. Properly accounting for the spin-2 nature of weak lensing distortions, we develop a quadratic estimator for measuring the $E$- and $B$-mode angular power spectra from galaxy-galaxy weak lensing, correcting for survey window effects arising from, for example, survey geometry and bright star masks. The estimator can be implemented efficiently by adopting FFTs on pixelized maps of the lens galaxy distribution and source galaxy ellipticities, under the flat-sky approximation. Using simulated weak lensing fields and halo catalogs in the light-cone ray-tracing simulations, we show that the estimator can recover the underlying $E$-mode power spectrum, $C_{{\rm g}E}(\ell)$, to within a few percent in fractional error, while minimizing the leakage of $E$-mode into the $B$-mode power spectrum, in each multipole bin over the wide range of multipoles (up to $\ell \sim 3000$ studied in this paper). We then discuss that the estimator can be used to estimate the 3D galaxy-matter power spectrum, $P_{\rm gm}(k)$, by dividing lens galaxies into multiple redshift slices. We also derive an optimal weighting for each lens redshift slice in the shot noise-limited regime for the estimation of $P_{\rm gm}(k)$, which reduces the statistical errors by up to $\sim$20\% compared to the case without weighting.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An optimal quadratic estimator for window-free cosmic shear power spectra

    astro-ph.CO 2026-07 conditional novelty 5.0 of 10

    A window-free quadratic estimator that is statistically optimal when the true covariance is known improves cosmic shear E-mode precision by 5–15% at ℓ≲500 and suppresses E→B leakage.

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