Reconstructing spatially-varying multiplicative bias for Stage IV weak lensing galaxy surveys with a quadratic estimator

David Alonso; Joachim Harnois-D\'eraps; Konstantinos Tanidis; Lance Miller

arxiv: 2512.13679 · v1 · submitted 2025-12-15 · 🌌 astro-ph.CO

Reconstructing spatially-varying multiplicative bias for Stage IV weak lensing galaxy surveys with a quadratic estimator

Konstantinos Tanidis , David Alonso , Lance Miller , Joachim Harnois-D\'eraps This is my paper

Pith reviewed 2026-05-16 21:54 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords weak lensingmultiplicative biasquadratic estimatorEB mode couplingspatially varying biasshear systematicscosmic shearStage IV surveys

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The pith

A quadratic estimator reconstructs spatially varying multiplicative bias in weak lensing shear by isolating EB mode coupling.

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

The paper introduces a quadratic estimator to detect and map multiplicative bias in weak lensing shear measurements. This bias, which can vary across the sky due to instrumental effects, couples E and B modes in the observed field. The estimator combines these modes using inverse-variance weights to recover an unbiased map of the bias field to first order. For data volumes expected from large upcoming surveys, percent-level rms variations in the bias become detectable at high significance after stacking measurements from hundreds of sky patches. The reconstruction remains stable across different bias patterns and does not generate false signals from additive bias.

Core claim

The central claim is that a quadratic estimator constructed from the inverse-variance weighted product of E and B modes recovers the spatially dependent multiplicative bias m(θ) without bias to leading order. This follows because the position-dependent bias induces a coupling between the otherwise orthogonal E and B components of the shear field. When the estimator is applied to data and the resulting maps are stacked over many patches, the bias morphology is recovered with signal-to-noise that scales with the rms amplitude of the variations and the number of patches used.

What carries the argument

The quadratic estimator that isolates EB mode coupling generated by a spatially varying multiplicative bias m(θ), using inverse-variance weighting of the lensing modes to produce an unbiased reconstruction to first order.

If this is right

Percent-level rms variations in m-bias can be detected at 20 sigma significance after stacking 400 to 1000 patches.
The reconstruction shows no spurious response to additive bias.
The signal-to-noise ratio depends on the spatial scale and morphology of the bias pattern.
Results hold when intrinsic alignments or baryonic effects are present and across different cosmological models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The estimator could be run on real survey data to produce correction maps that are subtracted from shear catalogs before cosmological analysis.
Stacking strategy could be optimized by testing reconstruction fidelity on simulations with different patch sizes and overlap.
The first-order approximation implies that the method's accuracy at larger bias amplitudes should be checked directly in end-to-end simulations.

Load-bearing premise

The multiplicative bias produces EB mode coupling that can be isolated with inverse-variance weighting to give an unbiased reconstruction to first order, with higher-order terms and other systematics remaining subdominant after stacking.

What would settle it

Apply the estimator to simulated weak lensing maps that contain a known injected spatially varying m-bias pattern of 5 percent rms amplitude and verify whether the reconstructed map recovers the input pattern to within the predicted noise after stacking several hundred patches.

Figures

Figures reproduced from arXiv: 2512.13679 by David Alonso, Joachim Harnois-D\'eraps, Konstantinos Tanidis, Lance Miller.

**Figure 1.** Figure 1: Left: Galaxy redshift distribution 𝑝(𝑧) for a Euclid-like photometric sample (see Eq.20). Middle: A blob-like 𝑚−bias given by Eq.22. Right: A strip-like 𝑚−bias model given by Eq.23. Both 𝑚−bias patterns have an rms of 5% and are 20 × 20 pixel grids. with 𝜎𝑒 = 0.28 the intrinsic ellipticity variance for a Euclid-like galaxy sample (Euclid Collaboration: Mellier et al. 2025). Besides the contribution from gr… view at source ↗

**Figure 2.** Figure 2: The angular power spectra from an N-body (green) and a Gaussian (red) simulation at an area of 1 deg2 with their 1𝜎 uncertainties and the input theory prediction (dashed black). The high-ℓ suppression visible in the figure is due to finite-resolution effects. 3.5 N-body simulations Apart from using Gaussian realisations, we also test our quadratic estimator with N-body simulations from the cosmo-SLICS suit… view at source ↗

**Figure 3.** Figure 3: The ’template’, ’data’ and ’peak’ SNR definitions as a function of the number of patches. We consider patches of 10 × 10 deg2 and a 5% rms blob-like 𝑚−bias. unknown constant, and n is the reconstruction noise component. The significance of the detection in this case may be estimated as SNRtemp = 𝛼¯ /𝜎𝛼, (28) where 𝛼¯ and 𝜎𝛼 are the best-fit value of 𝛼 and its statistical uncertainty, given by 𝛼¯ = t T𝐶 −1… view at source ↗

**Figure 4.** Figure 4: Top left: One input 𝛾1 component of the cosmic shear from Gaussian realisations. Top middle: Reconstruction of the 5% rms blob-like 𝑚−bias field at SNRtemp=10 with 600 realisations. Top right: Reconstruction of the same 𝑚−bias field at SNRtemp=20 with 2800 realisations. The grid area is 100 deg2 with 20 × 20 pixels. Similarly, the bottom panels show the 5% rms strip-like 𝑚−bias model reconstructed with 200… view at source ↗

**Figure 5.** Figure 5: Top left: One input 𝛾1 component of the cosmic shear from N-body realisations. Top right: Reconstruction of the 20% rms 𝑚−bias field at SNRtemp=81 with 50 simulations. The grid area is 100 deg2 , with 100 × 100 pixels. Bottom panels are for a 20% rms strip-like 𝑚−bias, reconstructed with SNRtemp=81, using also 50 simulations. number used above (see Sec. 3.2). The corresponding number of galaxies residing i… view at source ↗

**Figure 6.** Figure 6: Top left: One 𝛾1 component patch of the cosmic shear from N-body realisations. Top right: Reconstruction of the 𝑚−bias field with 5% amplitude at SNRtemp = 20 with 400 patches. Bottom panels same as top but for a strip-like 𝑚−bias model with 1000 patches. The reconstructed grid area is 0.68 × 0.68 deg2 with 21 × 21 pixels. Despite the inconsistent cosmology models between the simulations in the shear maps… view at source ↗

read the original abstract

We present a quadratic estimator that detects and reconstructs spatially-varying multiplicative ($m-$) bias in weak lensing shear measurements, by exploiting the $EB$ mode coupling that it generates. The method combines $E$ and $B$ modes with inverse-variance weights, to yield an unbiased reconstruction of $m(\boldsymbol{\theta})$ to first order. We study the ability of future Stage IV surveys to obtain an unbiased reconstruction of the $m$-bias in differing scenarios, considering differing bias morphologies, and characteristic scales, as well as differing metrics to quantify the signal-to-noise ratio of the reconstructed map. Considering an $m$ pattern repeating on $\sim 1^\circ\times1^\circ$ sky patches, as might be the case for an $m$ field caused by focal-plane systematics. With a Euclid-like redshift distribution, we find that $\sim5\%$ rms variations in $m$-bias may be detected at the 20$\sigma$ level, after stacking between $\sim400$ and $\sim1000$ patches (rising to between $\sim2800$ and $\sim7600$ for $1\%$ rms variations, data volumes that are becoming available with upcoming surveys), depending on the morphology of the $m$ pattern. We show that these results are robust against the cosmological model assumed in the reconstruction, as well as the presence of intrinsic alignments or baryonic effects, and that the method shows no spurious response to additive ($c-$) bias. These results demonstrate that percent-level, spatially-varying $m-$bias can be detected at high significance, enabling diagnosis and mitigation in the Stage IV weak lensing era.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a quadratic estimator that reconstructs spatially varying multiplicative bias from EB coupling in weak lensing, with simulations showing 20-sigma detection of 5% rms variations after stacking a few hundred patches.

read the letter

The core contribution is a quadratic estimator that combines E and B modes with inverse-variance weights to recover a map of m(theta) to first order from the coupling the bias itself produces. The simulations use Euclid-like redshift distributions and test patterns on roughly one-degree scales, reporting that 5% rms variations reach 20 sigma after stacking 400 to 1000 patches, with numbers rising for smaller amplitudes. They also check that the reconstruction stays robust when cosmology, intrinsic alignments, or baryons are varied, and that additive bias produces no spurious signal. These checks are straightforward and directly relevant to Stage IV pipelines. The approach avoids fitting a global parameter and instead builds the estimator from the physical mode coupling, which keeps the logic clean. The main soft spot is that the abstract and stress-test summary leave the full derivation and error budget implicit, so it is not yet clear how cleanly higher-order terms or survey-specific mask effects are controlled once the stacking is removed. The quoted S/N values come from the stacked maps, and real data may have additional position-dependent systematics that the current tests do not fully capture. This work is aimed at people building shear calibration pipelines for Euclid, LSST, or Roman. Anyone already running EB-mode analyses or focal-plane systematics studies will see immediate use for the estimator. I would send it to peer review; the method is timely, the tests are encouraging, and the central claim is worth a careful referee check on the math and simulation details.

Referee Report

0 major / 2 minor

Summary. The paper presents a quadratic estimator that reconstructs spatially varying multiplicative bias m(θ) in weak lensing shear catalogs by isolating the EB-mode coupling induced by m-bias. The estimator combines E and B modes via inverse-variance weighting to produce an unbiased reconstruction to first order. For Euclid-like redshift distributions, the work claims that 5% rms m-bias variations repeating on ~1° patches can be recovered at ~20σ significance after stacking 400–1000 patches (scaling to 2800–7600 patches for 1% rms), with results shown to be robust against cosmology, intrinsic alignments, baryonic effects, and additive c-bias.

Significance. If the central claim holds, the method supplies a concrete, data-driven diagnostic for a leading systematic in Stage IV weak lensing analyses. The reported stacking thresholds and robustness tests indicate that percent-level spatially varying m-bias can be detected and potentially mitigated with existing and forthcoming survey volumes, directly supporting the accuracy of cosmological parameter inference.

minor comments (2)

[Abstract and §3] The abstract and results section quote S/N values and stacking numbers but do not tabulate the precise inverse-variance weights or the explicit first-order expansion that demonstrates unbiasedness; adding these expressions would strengthen reproducibility.
[Results] Figure captions and text should clarify whether the quoted rms values refer to the input m-field or the reconstructed map, and whether the reported detection thresholds include cosmic variance or only shape noise.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary and recommendation for minor revision. We are encouraged by the recognition that the quadratic estimator provides a practical, data-driven approach to diagnosing a key systematic for Stage IV weak lensing analyses.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The quadratic estimator is constructed directly from the known EB-mode coupling induced by spatially varying multiplicative bias, a standard effect in weak lensing shear. The unbiased reconstruction to first order follows mathematically from the inverse-variance weighting of E and B modes, without reducing to a fitted parameter or self-referential definition. Detection significance arises from stacking independent patches rather than any internal fit. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the derivation; robustness checks against cosmology, IA, baryons, and c-bias are external to the core estimator. The central claim remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the first-order EB coupling induced by multiplicative bias and the validity of inverse-variance weighting after stacking; no free parameters are explicitly fitted in the abstract, and no new entities are introduced.

axioms (1)

domain assumption Multiplicative bias m(θ) generates EB mode coupling to first order in weak lensing shear
This is the physical basis invoked for the quadratic estimator construction.

pith-pipeline@v0.9.0 · 5616 in / 1290 out tokens · 36098 ms · 2026-05-16T21:54:48.480492+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

A., MacCrann N., Troxel M., Fang X., 2019, Physical Review D, 100 Bridle S., King L., 2007, New Journal of Physics, 9, 444–444 Chisari N

Albrecht A., et al., 2006, arXiv e-prints, pp astro–ph/0609591 Alonso D., Sanchez J., Slosar A., 2019, Monthly Notices of the Royal Astro- nomical Society, 484, 4127–4151 BerlfeinF.,MandelbaumR.,DodelsonS.,SchaferC.,2024,MonthlyNotices of the Royal Astronomical Society, 531, 4954–4973 Blazek J. A., MacCrann N., Troxel M., Fang X., 2019, Physical Review D,...

work page doi:10.1007/978-3-540-30310-7_3 2006

[1] [1]

A., MacCrann N., Troxel M., Fang X., 2019, Physical Review D, 100 Bridle S., King L., 2007, New Journal of Physics, 9, 444–444 Chisari N

Albrecht A., et al., 2006, arXiv e-prints, pp astro–ph/0609591 Alonso D., Sanchez J., Slosar A., 2019, Monthly Notices of the Royal Astro- nomical Society, 484, 4127–4151 BerlfeinF.,MandelbaumR.,DodelsonS.,SchaferC.,2024,MonthlyNotices of the Royal Astronomical Society, 531, 4954–4973 Blazek J. A., MacCrann N., Troxel M., Fang X., 2019, Physical Review D,...

work page doi:10.1007/978-3-540-30310-7_3 2006