{"id":"9aa40fc7-6535-4473-973c-bfa01f1a1c47","arxiv_id":"2507.18789","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A maximum-likelihood quadratic estimator recovers the window-free E-mode galaxy-galaxy lensing power spectrum to a few percent in simulations and enables 3D P_gm(k) estimation via redshift-sliced lens samples.","lead":"This paper develops a quadratic estimator that measures the E- and B-mode angular power spectra of galaxy-galaxy weak lensing, correcting for survey window and mask effects. It validates the method on ray-tracing simulations and uses it to estimate the 3D galaxy-matter power spectrum P_gm(k) from multiple lens redshift slices.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central validation depends on an unpublished flat-sky spin-2 basis transformation (Section 4.1); if that transformation is wrong, the recovered C_gE and B-mode leakage results would not hold, so the central claim is not independently verified.","rationale":"The reader's weakest assumption identifies the unpublished flat-sky basis transformation as the key risk; my reading agrees. The central claim is the percent-level recovery of C_gE, and every element of the validation depends on the projected flat-sky maps. The paper gives no details of the spin-2 rotation, so the validation is not reproducible. The full-sky healpy comparison provides some assurance, but it is an end-to-end check and could mask a systematic rotation error if the same error also affects the interpretation. The edge-bin failures in Fig. 2 are a second, explicitly acknowledged limitation: the abstract's claim of accuracy 'in each multipole bin' is not literally true. However, the edge-bin issue is a known band-power binning artifact that can be mitigated by extending the multipole range, and it does not undermine the interior-bin results. The P_gm(k) application is promising but only qualitatively compared to the Dark Emulator, so it does not strengthen the central claim. Overall, the estimator derivation appears mathematically sound, including unbiasedness for arbitrary diagonal H and the Monte Carlo Fisher estimation, and the simulations demonstrate percent-level recovery in interior bins. The concern is therefore about verifiability and scope rather than a demonstrated internal inconsistency. The CONDITIONAL verdict already captures this; no verdict change is needed.","tokens_in":19846,"tokens_out":10496,"duration_ms":113042,"concrete_test":"Independently reimplement the flat-sky projection of the HEALPix shear maps using a documented spin-2 rotation, for example rotating each pixel's (gamma1, gamma2) by the angle between the local meridian and the flat-sky x-axis, using the same Nside=8192 maps and 10x10 degree patches, then rerun the Section 4.2 validation pipeline. If the resulting C_gE band powers change by more than the quoted few percent, or if the B-mode leakage rises above the quoted sub-percent level, the flat-sky transformation is the cause and the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.1 states that the flat-sky projection of the HEALPix shear maps uses 'the basis transformation of the shear field (Terawaki et al. in preparation)' and the NGP gridding method. Every subsequent quantity in the validation, including the FFT-based quadratic estimator products, the Fisher matrix, the recovered C_gE(l), and the E/B separation, is computed from these projected maps. The paper does not specify the rotation convention that converts the spin-2 shear components from the spherical pixel basis to the flat 2D basis, nor does it quantify the sensitivity of the results to that convention. If the rotation is incorrect, for example having the wrong sign or angle, the E and B modes in the flat maps would be mixed, biasing the headline few-percent recovery and the claimed sub-percent B-mode leakage. The comparison with the full-sky healpy measurement is a partial check of the end-to-end E-mode result, but it does not isolate the transformation, and the same pipeline would silently produce biased results on real data if the transformation were flawed. Additionally, Fig. 2 explicitly shows the first and last multipole bins are not within the few-percent accuracy, so the abstract's 'in each multipole bin' claim is overstated; this is acknowledged in the text but omitted from the abstract. The combination of an unverifiable linchpin transformation and the edge-bin discrepancy justifies the CONDITIONAL verdict.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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%.","tokens_in":20132,"tokens_out":29553,"duration_ms":291236,"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":[{"comment":"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.","section":"Section 4.1"},{"comment":"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.","section":"Abstract; Section 4.2, Fig. 2"},{"comment":"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.","section":"Section 5.1, Eqs. (5.1)-(5.3)"}],"minor_comments":[{"comment":"'Throughput this paper' should read 'Throughout this paper', and '19502 grids' should presumably read '1950² grids'.","section":"Section 4.1"},{"comment":"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.","section":"Abstract; Sections 5.2, 6"},{"comment":"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.","section":"Section 4.2, Fig. 3"},{"comment":"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.","section":"Section 4.2, Fig. 2"},{"comment":"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.","section":"Section 5.2, Fig. 5"},{"comment":"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.","section":"Section 5.1"},{"comment":"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.","section":"Sections 2-3"}],"recommendation":"major_revision","confidential_remarks":"The main roadblock is the unpublished '(Terawaki et al. in preparation)' citation in Section 4.1, which is by the lead author of this manuscript; as a referee I cannot verify the spin-2 basis transformation, and the paper's validation depends on it. This is fixable by moving the transformation into the paper or a companion that is available at revision time. The edge-bin discrepancy with the abstract is also fixable. The overall contribution is solid and the derivation is internally consistent, so I would not reject; I chose major revision because two claim-level issues (the unverifiable linchpin transformation and the abstract-vs-Fig. 2 discrepancy) need to be resolved before the results can be taken at face value. I do not treat the ray-tracing validation as circular, since the reference spectra are produced by an independent method (healpy on full-sky maps) with no fitted parameters."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. This is a solid methods paper, not a breakthrough. The quadratic estimator for CgE and CgB is derived cleanly, and the validation is honest: 972 simulated patches, masks, shape noise, and recovery at the few-percent level for interior multipole bins. The B-mode leakage is below a percent. The Pgm(k) stacking estimator with optimal weights across lens slices is genuinely new in this context, and the 20-30% error reduction is useful. I would send it to a serious referee.\n\nWhere it's soft. First, the flat-sky projection of the HEALPix shear field relies on 'Terawaki et al. in preparation' (Sec. 4.1). That is the key step for all the validation, and the rotation convention is not specified. The full-sky healpy comparison is an end-to-end check that likely catches a gross error, but it does not isolate this step. If the companion paper is not available, the method is not reproducible as-is. That is a legitimate referee request, not a reason to reject. Second, the abstract says the recovery works 'in each multipole bin', but Fig. 2 and the text say the first and last bins are inaccurate. That overstatement should be fixed. Third, the Pgm(k) comparison to the Dark Emulator is qualitative and same-group; there is no independent truth test. It is enough to show the pipeline runs, but not a precision validation. Fourth, Appendix A confirms that with the diagonal H used in practice, the estimator is the pseudo-C_l method. So the novelty over Hikage & Oguri (2016) is real but incremental: explicit E/B mixing and Monte Carlo Fisher deconvolution, not a new estimator family.\n\nOverall, the central claim holds up for interior multipoles, conditional on the unpublished transformation. This paper deserves peer review. I would ask the authors to specify or reference the transformation, correct the abstract, and quantify the Pgm validation. No desk reject.","headline":"A solid, well-validated estimator whose main caveats are an unpublished spin-2 projection step and an overstated abstract; deserves peer review.","tokens_in":20675,"tokens_out":4232,"would_cite":true,"duration_ms":41882,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["galaxy-galaxy weak lensing","quadratic estimator","E/B-mode decomposition","flat-sky approximation","survey window correction","FFT power spectrum estimation","galaxy-matter power spectrum","redshift slicing"],"falsifier":"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.","tokens_in":19631,"feed_emoji":"🔭","tokens_out":10500,"duration_ms":90463,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quadratic estimator recovers lensing power spectra to a few percent","feed_subtitle":"Window-free E-mode galaxy-galaxy lensing spectra match simulations; B-mode leakage stays below one percent.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the maximum-likelihood quadratic estimator formulation from which Eqs. (3.13)-(3.15) are derived.","marker":"[27]"},{"why":"Supplies the window-free quadratic estimator and FFT implementation for galaxy clustering that the paper extends to the spin-2 galaxy-galaxy lensing case.","marker":"[30]"},{"why":"Provides the pseudo-spectrum analysis of galaxy-galaxy lensing that the estimator is compared with and shown equivalent to in the diagonal-weight limit.","marker":"[15]"},{"why":"Provides the full-sky ray-tracing simulations and halo catalogs used for all validation and for the $P_{\\mathrm{gm}}(k)$ tests.","marker":"[32]"},{"why":"Supplies the emulator-based halo-matter power spectrum prediction used as the theory curve for the $P_{\\mathrm{gm}}(k)$ validation.","marker":"[48]"},{"why":"Supplies the inverse-variance weighting logic used to derive the lens-slice weight in Eq. (5.3).","marker":"[44]"},{"why":"Supplies the shape-noise and source-density parameters used to set the simulated intrinsic ellipticity amplitude.","marker":"[21]"}],"fun_headline_variants":["Lensing estimator recovers E-mode spectrum to few percent","B-mode leakage kept below 1% in lensing spectra","Few-percent lensing E-mode spectra, <1% B-mode leakage","Quadratic estimator yields few-percent E-mode lensing spectra","Window-free lensing spectra match simulations to few percent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Lensing estimator recovers E-mode spectrum to few percent","B-mode leakage kept below 1% in lensing spectra","Few-percent lensing E-mode spectra, <1% B-mode leakage","Quadratic estimator yields few-percent E-mode lensing spectra","Window-free lensing spectra match simulations to few percent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000879,"raw_usage":{"total_tokens":3870,"prompt_tokens":1084,"completion_tokens":2786,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":700,"completion_tokens_details":{"reasoning_tokens":2700}},"tokens_in":700,"tokens_out":2786,"duration_ms":23178,"temperature":1.0,"reasoning_tokens":2700,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:08:34.886233+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the maximum-likelihood quadratic estimator formulation from which Eqs. (3.13)-(3.15) are derived."},{"cited_title":"Philcox,Cosmology Without Windows: Quadratic Estimators for the Galaxy Power Spectrum, Physical Review D103 (2021) 103504","cited_arxiv_id":null,"evidence_quote":"Supplies the window-free quadratic estimator and FFT implementation for galaxy clustering that the paper extends to the spin-2 galaxy-galaxy lensing case."},{"cited_title":"A pseudo-spectrum analysis of galaxy-galaxy lensing","cited_arxiv_id":"1603.07818","evidence_quote":"Provides the pseudo-spectrum analysis of galaxy-galaxy lensing that the estimator is compared with and shown equivalent to in the diagonal-weight limit."},{"cited_title":"Takahashi, T","cited_arxiv_id":null,"evidence_quote":"Provides the full-sky ray-tracing simulations and halo catalogs used for all validation and for the $P_{\\mathrm{gm}}(k)$ tests."},{"cited_title":"Nishimichi, M","cited_arxiv_id":null,"evidence_quote":"Supplies the emulator-based halo-matter power spectrum prediction used as the theory curve for the $P_{\\mathrm{gm}}(k)$ validation."},{"cited_title":"Hikage, M","cited_arxiv_id":null,"evidence_quote":"Supplies the shape-noise and source-density parameters used to set the simulated intrinsic ellipticity amplitude."}],"review_version":2}