Metacalibration in Weak Lensing

• Metacalibration is a self-calibration framework for weak lensing that directly estimates shear biases by artificially shearing observed galaxy images.
• It employs a finite-difference method to measure the shear response, incorporating corrections for selection effects and correlated noise.
• This method enables high-precision shear recovery in major surveys like DES, LSST, Euclid, and Roman, reducing multiplicative biases to sub-percent levels.

Metacalibration is a self-calibration framework for bias correction in weak gravitational lensing shear measurements. It provides a methodology for directly estimating multiplicative and additive shear biases from the observed survey data by artificially shearing galaxy images, thus circumventing the need for calibration from external image simulations. Metacalibration has become a standard for shear measurement pipelines in modern cosmological surveys, enabling precision shear recovery that meets the stringent requirements of contemporary and upcoming experiments such as DES, LSST, Euclid, and Roman.

1. Principle and Formalism of Metacalibration

Metacalibration operates by generating artificially sheared versions of real survey images, allowing a direct numerical estimation of the response of measured galaxy shapes (ellipticities) to small, known shear perturbations. In the weak lensing regime, the lensed coordinate transformation is nearly linear, described by a mapping

$$A = \begin{pmatrix} 1+\kappa+\gamma_1 & \gamma_2 \ \gamma_2 & 1+\kappa-\gamma_1 \end{pmatrix},$$

with the observable being the reduced shear $g = \gamma/(1-\kappa)$. Measured ellipticities $e$ are modeled as

$\langle e \rangle = (1 + m)\langle g \rangle + c,$

where $m$ is the multiplicative bias and $c$ is the additive bias. Metacalibration aims to empirically determine the responsivity $R$, i.e., the derivative $\partial e / \partial g$, using finite-difference measurements: $R = \frac{e_+ - e_-}{2\Delta g},$ where $e_+$ and $e_-$ are the ellipticities from images sheared by $+\Delta g$ and $-\Delta g$, respectively.

To account for selection biases—where cuts on measured properties can introduce additional shear-dependent sample selection—metacalibration generalizes the response into two terms,

$R_{\text{total}} = R_\gamma + R_S,$

with $R_\gamma$ from shear perturbations and $R_S$ from selection response, measured by applying the selection on parameters extracted from sheared images. The ensemble shear is then estimated via

$$\langle \gamma \rangle \approx \langle R_{\text{total}}\rangle^{-1} \langle e\rangle.$$

This formalism is applicable to any shear estimation method whose estimator is a weighted average over galaxy shapes (Huff et al., 2017, Sheldon et al., 2017).

2. Implementation: Image Manipulations and Noise Corrections

The metacalibration procedure involves manipulating real galaxy images in Fourier space. For each image $I$ (the convolution of galaxy $G$ with PSF $P$), the forward model is: $$I'(\mathbf{x}|g) = \Gamma \star \left[ s_g(P^{-1}\star I)\right],$$ where $s_g$ is the shear operator and $\Gamma$ is a smoothing PSF to suppress noise amplification from deconvolution. Artificially sheared images are produced by deconvolving the original PSF, applying the shear, and reconvolving with the slightly larger $\Gamma$. For selection response correction, selection cuts are applied on both unsheared and sheared versions.

A critical practical aspect is noise: the sequence of deconvolution, shearing, and reconvolution couples noise between image pixels and can generate correlated noise that biases the shear response estimate by $\sim$5–10%. To correct this, an empirical fixnoise procedure adds a random noise realization, subjected to the same operations but with opposite shear, effectively cancelling noise correlations. This correction results in a modest $\sim$20% increase in the statistical noise of the measurement (Sheldon et al., 2017, Zhang et al., 2022).

Recent developments include deep-field metacalibration, which leverages deep survey fields to measure the shear response with substantially less noise penalty, reducing the precision degradation from $\sim$20% (standard metacalibration) to $\lesssim$5% (Zhang et al., 2022).

3. Extension to Blending and Detection: Metadetection

Blended images—where galaxies overlap due to projection and PSF convolution—create detection ambiguities that can be shear-dependent, introducing significant bias if not handled correctly. Standard metacalibration fails in heavily blended regimes because object detection (which identifies objects for shape measurement) changes as a function of shear.

Metadetection solves this by applying artificial shear to large image regions, rerunning the detection algorithm on these sheared images, and recalculating shape estimates. The averaged shear response over these multiple detection catalogs captures the shear dependence of object detection, thus eliminating otherwise large detection biases. This approach reduces multiplicative biases in blended fields to below the sub-percent threshold required for future surveys (Sheldon et al., 2019).

4. Performance, Validation, and Accuracy

Metacalibration has been extensively validated on both synthetic and real images. Testing with simulated datasets such as GREAT3, BDK (parametric bulge+disk+knots models), and real COSMOS galaxies demonstrates that, when both shear response and selection response corrections are applied, residual multiplicative biases are reduced from levels of several percent (pre-calibration) to less than a part in a thousand ($m \lesssim 0.1\%$), even in the presence of significant selection effects and PSF aberrations (Huff et al., 2017, Sheldon et al., 2017). Table 1 summarizes the observed correction levels:

Context	Pre-metacalibration $m$	Post-metacalibration $m$
Simulations (BDK, S/N cut 10)	$> 0.005$	$\sim 0.00003$
Real galaxy simulations (COSMOS)	$>0.005$	$\sim 0.00003$
DES Y1 shapes (Zuntz et al., 2017)	$-$	$\sigma_m = 0.013$

Similarly, additive biases due to PSF anisotropy are reduced with reconvolution to a symmetrized target PSF, while fixes for correlated noise maintain the total error budget within survey requirements.

5. Influence on Survey Analyses and Systematic Error Budgets

Metacalibration's internal (data-driven) calibration substantially reduces the dependency on external, high-fidelity image simulations, which are often limited by their inability to capture the full range of galaxy morphology and imaging artifacts present in real data. Its flexibility and applicability to any shape estimation method based on weighted averages of galaxy shapes make it the dominant calibration approach in recent major lensing surveys.

In DES Y1 and Y3, metacalibration enabled shape catalogs with $\sim$35M–100M sources over 1,500–4,000 deg$^2$, achieving multiplicative bias control at the $\lesssim$1\% level and passing stringent null tests (PSF modeling diagnostics, $B$-mode tests, and galaxy property correlation checks). The self-calibration approach is essential to ensure the systematic error does not exceed the statistical error in upcoming large-scale surveys (Zuntz et al., 2017, Gatti et al., 2020).

Moreover, the method's ability to directly estimate the calibration uncertainty from the data yields a lower marginalization penalty on cosmological parameters, compared to simulation-reliant codes such as IM3SHAPE, which have larger $\sigma_m$ and greater sensitivity to model assumptions.

6. Limitations and Ongoing Developments

A number of limitations and areas of ongoing development remain:

• For undersampled, space-based imaging (Euclid, Roman), the metacalibration deconvolution–shearing–reconvolution cycle may introduce aliasing biases. These can be minimized by using wider, non-adaptive Gaussian weight functions in moment measurements, trading bias reduction for increased variance (Kannawadi et al., 2020, Yamamoto et al., 2022).
• In cases with severe blending, metacalibration requires extension to metadetection to account for detection biases.
• Implementing metadetection (and even standard metacalibration) at survey scale involves challenges with PSF spatial variation, world coordinate system consistency, and selection propagation, which are the focus of current algorithmic work.
• Analytical and renoising techniques have been developed to correct for noise bias without computing high-order derivatives, providing more robust calibration and millisecond-level computational performance per galaxy for large survey data streams (Li et al., 12 Aug 2024).

7. Impact on Cosmological Inference

Metacalibration underpins the accuracy of cosmic shear, galaxy-galaxy lensing, and higher-order weak lensing statistics in current cosmological analyses. The technique's self-calibration rigor ensures that multiplicative and additive calibration biases are not dominant contributors to the error budgets on parameters such as $S_8$, $\sigma_8$, and $\Omega_m$. Furthermore, as the statistical precision of surveys increases with effective source density and area, the ability to control residual systematic biases using direct calibration on real images is critical to future dark energy constraints (Zuntz et al., 2017, Gatti et al., 2020, Prat et al., 2021, Guinot et al., 2022).

Metacalibration's role in enabling accurate propagation of shear response corrections into map-based and harmonic-space inference pipelines has also become central, as shown in the context of power spectrum analyses (Kitching et al., 2023).

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In summary, metacalibration is a statistically principled, data-driven bias calibration methodology for weak lensing shear, combining empirical measurement of shear response functions with explicit correction for selection biases and instrumental effects. Its adoption across major surveys reflects its robustness and indispensable contribution to the reliability of cosmic shear, galaxy-galaxy lensing, and the broader cosmic acceleration science case.