Three-Dimensional Parity Statistics for Galaxy Images in Phase Space

We construct a three-dimensional parity statistics of two-dimensional galaxy images, employing Wigner function shapelets (WFSs) in phase space Sp(4, R) associated with the image plane (configuration space) and visibility plane (Fourier space). The WFS can represent galaxy morphology in phase space through the Wigner function, allowing configuration-space and Fourier-space information to be treated simultaneously as different coordinate representations within symplectic transforms. Using the WFS coefficients, we distinguish their connected even–odd covariance from the Gaussian-exchange-subtracted connected four-point cumulant of the underlying image-mode amplitudes. The covariance-level quantity underlies the directly observable parity estimators, whereas the cumulant-level quantity defines parity-odd invariants that act as generalized trispectra in internal shape space. We then introduce local pseudoscalar observables associated with individual galaxy images and elevate them into three-dimensional parity estimators defined on the cosmological light cone. Practical observational effects, including point-spread functions, noise contamination, and finite mode truncation, are discussed at the estimator level. To demonstrate the behavior of the proposed observables, we formulate the linear response of the parity-violating signal. We find that an optimal matched filter, constructed from the null covariance of galaxy shapes in the absence of parity-odd signals, provides the maximum signal-to-noise ratio. This indicates that the proposed framework offers a viable route toward probing parity-odd morphology covariance and, after Gaussian exchange subtraction, non-Gaussian parity structure directly from image data.This is a preprint of a manuscript submitted to Physical Review D. It has not yet been accepted for publication.