Accurate Galaxy Cluster Shear and Mass Calibration for LSST with AnaCal

Technical Summary: Accurate Galaxy Cluster Shear and Mass Calibration for LSST with AnaCal

Problem Statement
Galaxy cluster cosmology relies on establishing an accurate relationship between cluster observables and cluster masses, a process known as mass calibration. Weak gravitational lensing is the standard tool for this calibration because it probes total mass (including dark matter) without relying on dynamical or thermal equilibrium assumptions. However, a significant challenge arises in the cluster environment: the shear signal near cluster centers can reach the non-linear regime (e.g., $|g| \sim 0.15$), where many shear estimators rely on perturbative assumptions that require explicit validation. Furthermore, cluster lensing introduces unique complexities not present in standard cosmic shear measurements, including:

• High Shear Regime: Tangential shear in inner regions ($\sim 0.2$ Mpc) is several times larger than values typically used in perturbation-based calibration tests.
• Inhomogeneous Deflection: The cluster lensing field shifts the apparent positions of background galaxies, altering their overlap and blending configurations, which impacts detection and shape measurement.
• Radial Dependence: The convergence field varies radially, potentially driving a radially decreasing mean shear response.
• Higher-Order Effects: Near the cluster center, higher-order distortions (e.g., flexion) and anisotropic light from intra-cluster halos may introduce spurious signals.

While the LSST Dark Energy Science Collaboration (DESC) has selected AnaCal and metadetection as fiducial shear estimation methods, their performance had not been rigorously tested under the specific high-shear and inhomogeneous conditions characteristic of cluster lenses.

Methodology
The authors utilize image simulations representative of the 10-year LSST data to test the AnaCal shear estimator. The methodology involves:

1. Simulation Framework: Using descwl-shear-sims and xlens, the team generated realistic cluster-lensing image simulations. They modeled 9 halo masses ranging from $10^{14} M_\odot$ to $10^{15} M_\odot$ as NFW halos at a lens redshift of $z_l = 0.25$, with source galaxies at $z_s = 1.0$.

   • Simulations included 2178 realizations per mass, covering $\approx 1700$ deg$^2$ total.
   • To isolate shear-estimation effects, the simulations excluded cluster member galaxies, intra-cluster light, and redshift calibration errors, focusing strictly on the impact of the high-shear, inhomogeneous deflection field.
   • A constant circular Moffat PSF ($FWHM = 0.8''$) was used to represent expected Rubin Observatory image quality.

2. AnaCal Implementation: The study employs the AnaCal framework, which analytically derives the shear response of the full measurement pipeline (observables, detection, and selection) using auto-differentiation and a shapelet basis.

   • Observables: Images are projected onto detection kernels and polar shapelet basis functions.
   • Response Calculation: The shear response is calculated by differentiating observables with respect to controlled shear perturbations.
   • Noise Correction: A "renoising" scheme is applied to correct for noise bias by adding synthetic noise realizations to images.
   • Weighting: Selection and detection weights are applied based on signal-to-noise ratio (SNR) and galaxy size, designed to be invariant under 90-degree rotations.

3. Analysis and Fitting:

   • Shear Estimation: Tangential and cross ellipticities are calculated, and the response matrix is rotated into the tangential frame. Crucially, the authors use the shear response in each angular bin separately rather than averaging across all bins to avoid bias from the radial dependence of the convergence field.
   • Mass Fitting: A forward-modeling approach fits the observed tangential reduced shear profile ($g_T$) to NFW models using a likelihood function that incorporates an analytical covariance matrix derived from 10-year LSST forecasts.
   • Bias Quantification: The shear multiplicative bias ($m$) is fitted to the form $m = a + b(g_{true})^2$, testing for linear and third-order non-linear biases.

Key Results

• Shear Recovery: AnaCal recovers the input shear with minimal bias even at mildly high shear levels ($|g| \sim 0.15$).
• Radial Response Trend: The study confirms a radially decreasing mean shear response, driven by the radial dependence of the convergence field. The authors demonstrate that if this effect is unmodeled (i.e., if a global average response is used), it biases shear inference.
• Third-Order Bias: A positive shear-estimation bias is detected at the third order in the reduced shear near the cluster center.
• Mass Calibration Accuracy: Despite the presence of high-shear regimes and third-order biases, the impact on the final mass estimate is negligible. For cluster lens masses in the range $[10^{14} M_\odot, 10^{15} M_\odot]$ (adopting a scale cut of $\sim 0.2$ Mpc at $z=0.25$), the resulting mean cluster-mass bias is $0.24 \pm 0.26\%$ under ideal settings. This small bias is attributed to the fact that only a small fraction of galaxies lie in the high-shear regime, and those measurements are downweighted by the covariance matrix.

Significance and Claims
The paper claims that these results demonstrate AnaCal is a robust tool for accurate cluster mass calibration in the LSST era. Specifically:

• It validates that perturbation-based shear estimation (via AnaCal) remains accurate even in the high-shear regime typical of cluster centers, provided the radial dependence of the response is properly modeled.
• It confirms that the non-linearities and high-shear effects do not significantly degrade the precision of mass estimates for cosmologically relevant cluster samples.
• The work provides a critical validation step for the LSST DESC, ensuring that the selected shear estimation methods can meet the stringent requirements for cluster cosmology without introducing significant systematic errors.

The authors explicitly note that their simulations did not include effects from cluster members, intra-cluster light, or higher-order distortions like flexion, isolating the performance of the shear estimator against the primary challenge of high shear and inhomogeneous deflection fields.