Improving Precision in Kinematic Weak Lensing with MIRoRS: Model-Independent Restoration of Reflection Symmetries

Abstract

We present a novel, model-independent technique for fitting the cross-component of weak lensing shear, $γ_\times$, along a line of sight by combining kinematic and photometric measurements of a single lensed galaxy. Rather than relying on parametric models, we fit for the shear parameter that best transforms the velocity field to restore its underlying symmetries, while also incorporating photometric data for the change in position angle due to shear. We first validate our technique with idealized mock data, exploring the method's response to variations in shear, position angle, inclination, and noise. On this idealized mock data, our combined kinematic and photometric model demonstrates superior performance compared to traditional parametric or kinematic-only approaches. We also explore the effects of asymmetric warps and show that rotation direction can impart a small bias on the fit of $γ_\times$. Subsequently, we apply our method to a dataset of 358 halos from the Illustris TNG simulations, achieving a notable reduction in the uncertainty of $γ_\times$ to 0.039, marking a substantial improvement over previous analysis of the dataset with a parametric model. Finally, we introduce an outlier rejection method based on Moran's $I$ test for spatial autocorrelation. Identifying and filtering out halos with spatially correlated residuals reduces the overall uncertainty to 0.028. Our results underscore the efficacy of combining kinematic and photometric data for weak lensing studies, providing a more precise and targeted measurement of shear along an individual line of sight.

Figures (16)

• Figure 1: Velocity field of a nonsheared galaxy (top panel) and galaxy with $\gamma_{\times} =0.12$ shear applied (bottom panel) from a lens along the $y=x$ line from the source. The sheared galaxy displays a notable loss of symmetry in its kinematic field as well as an apparent rotation in its morphology.
• Figure 2: Pre- and post-lensing kinematic and photometric axes corresponding to an elliptical isophote overlaid on the sheared velocity field. Prior to lensing, both the kinematic and photometric axes are aligned (gray). After lensing, the isophote will remain elliptical with orthogonal photometric axes (black). For small values of $\gamma_{+}$, the transformation of the isophote can be well modeled as an ellipse rotated by an angle $\theta$ as shown in \ref{['eq:photo_axes']}. The kinematic axes (red), on the other hand, do not remain orthogonal. The discrepancy between the kinematic and photometric major and minor axes is given by \ref{['eq:delta_maj']} and \ref{['eq:delta_min']} respectively.
• Figure 3: Illustration of the MIRoRS algorithm. Starting in the detector frame (light background), data (green) is first masked around a central pixel. To create a model, the raw data is translated to the detector origin and inverse shear is applied, bringing the image into the source plane (dark background). A clockwise rotation orients the galaxy axes with the coordinate axes. Two symmetry transformations are done independently; a reflection about the major axis (red) and an inversion about the minor (blue) using \ref{['eq:v_minor_shift']}. Rotation and shear are then reapplied to bring the images back to the source plane where they are translated back to the original position. The two separate images are combined into a single model (purple) and masked with the same mask as applied to the data. The difference between the model and the data produces the residual (orange). Here, the data has an applied shear, $\gamma_{\times}=0.15$ and position angle $\phi=30^\circ$ while the model transformations use $\gamma_{\times}=0.12$ and $\phi=29^\circ$ so a nonzero residual is produced. Note, the parameter values and mask sizing were chosen to provide qualitative insight and are not necessarily indicative of typical values.
• Figure 4: Fisher information density for each fitted parameter of an ideal model ($\gamma_\times=\phi=x_c=y_c=v_c=0$), computed independently for the major-axis reflection, the minor-axis reflection, and the $180^\circ$ rotation operations. Each panel shows the spatial distribution of $(\partial v / \partial \theta)^2 / \sigma^2$ for a single parameter. The two reflection operators exhibit identical sensitivity to the shear parameter $\gamma_{\times}$, whereas the rotation operation provides no information on shear or position angle. The major-axis reflection constrains $y_c$ while the minor-axis reflection is sensitive to $x_c$ and $v_c$.
• Figure 5: Correlation matrices for the fitted parameters $(\gamma_\times,\ \phi,\ x_c,\ y_c,\ v_c)$ derived from the Fisher–matrix covariance for the three symmetry operations. The top row corresponds to the ideal model used in \ref{['fig:fisher']}, where the expected parameter couplings appear cleanly: both reflections correlate $\gamma_\times$ with $\phi$, the minor-axis reflection couples $x_c$ and $v_c$, and the rotation operation leaves all parameters independent. The bottom row shows results for a model with nonzero parameters ($\gamma_\times=0.01$, $\phi=5^\circ$, $x_c=y_c=1$, $v_c=15\ \mathrm{km\,s^{-1}}$), which introduce additional cross-correlations, particularly for the minor-axis reflection.