Analytical weak-lensing shear response of galaxy model fitting

Abstract

Galaxy model fitting is widely employed to estimate properties such as galaxy shape, size, and color. Understanding how the outputs of galaxy model fitting respond to weak-lensing shear distortions is crucial for accurate shear estimation and mitigating shear-related systematics in weak lensing image analyses. In this paper, we investigate how the fitted parameters - specifically flux, size, and shape - respond to weak-lensing shear distortions within the AnaCal framework. To achieve this, we introduce quintuple numbers, a novel algebraic system inspired by dual numbers from automatic differentiation. Quintuple numbers enable the propagation of shear response information throughout the entire model-fitting process by linking analytical pixel shear responses to those of the fitted parameters. We integrate quintuple numbers into the AnaCal framework to derive the shear responses of shapes estimated with model fitting and validate the pipeline using image simulations that include realistic blending. Our results demonstrate that the multiplicative bias remains below 0.003 for ground-based, oversampled images.

Figures (4)

• Figure 1: We show the noiseless image $f_p(\mathbf{x})$, the smoothed image $f_h(\mathbf{x})$, and the filtered images $(G_1(\mathbf{x})\,, G_2(\mathbf{x})\,, J_1(\mathbf{x})\,, J_2(\mathbf{x}))$, which are key components of the pixel-level shear response Anacal_Li2023. The images $G_1$ and $G_2$ represent the response of the smoothed image $f_h(\mathbf{x})$ to small shear distortions along the $\gamma_1$ and $\gamma_2$ directions, respectively. In contrast, $J_1$ and $J_2$ quantify how this shear response changes with shifts in the reference point about which the shear distortion is applied. The original image has a seeing size (FWHM) of $0\hbox{$.\!\!^{\prime\prime}$}80$, and the FWHM of the Gaussian smoothing kernel is $0\hbox{$.\!\!^{\prime\prime}$}85$.
• Figure 2: The left and middle panels display the multiplicative ($m_1$) and additive ($c_1$) shear estimation biases, respectively, for galaxies with varying Sersic indices. The right panel shows the relative flux bias, defined as the ratio of measured to true flux minus one. Despite using a Gaussian model to fit galaxies with non-Gaussian morphologies---leading to a 15% flux bias---the shear estimation biases remain below the 0.2% level relative to the input shear distortion ($\gamma_1 = 0.02$).
• Figure 3: We present a 2 deg$^2$ image simulation used to evaluate the accuracy of our shear estimator. Details of the simulation setup are provided in Section \ref{['sec:sim_sim']}. The red ellipses indicate the 2$\sigma$ contours from Gaussian fits to the detected sources.
• Figure 4: The multiplicative (upper panel) and additive bias (lower panel) as a function of magnitude cut. The black lines show the result for fixed-kernel shape estimator (FPFS) and the blue lines are for model-fitting code introduced in Section \ref{['subsec:method_fit']}. The shaded region shows the requirement on the control of multiplicative bias in LSST Dark Energy Science Collaboration LSSTRequirement2018.