Parity-violating scalar trispectrum from helical primordial magnetic fields

Some recent observations of the cosmic microwave background (CMB) anisotropies and the large-scale structure of the Universe imply cosmic parity violation. Among possible parity-violating sources, helical primordial magnetic fields (PMFs) are of particular interest, as they inherently violate parity symmetry and can explain the observed magnetic fields, especially in void regions. PMFs, if generated in the early universe, can source curvature perturbations, which evolve into the present density fluctuations observed in CMB and galaxy surveys. Motivated by this, we study the imprint of helical PMFs on the trispectrum of the sourced primordial curvature perturbations, which is a leading-order scalar statistics sensitive to parity-violating signals. We derive full expressions for the trispectrum of the primordial curvature perturbations sourced by both the helical and non-helical PMFs and reduce them to computationally-feasible ones using a proper approximation. From numerical works, we confirm that parity-odd signals are efficiently enhanced and surpass parity-even ones in specific momentum and parameter spaces. Parity-violating signatures found in this paper are partially testable with observational implications reported so far. Assuming nearly scale-invariant PMF power spectra and the PMF strength of $B_{r}=4.7 \, {\rm nG}$, we obtain a rough upper bound on the helical-to-non-helical power ratio as $r_H\lesssim 4\times 10^{-4}$. Our findings highlight the primordial trispectrum as a promising probe of helical PMFs and provide a theoretical basis for future precise observations of higher-order statistics in the CMB anisotropies and the galaxy clustering.