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Cosmic magnetic fields are observed to be coherent on large scales and couldhave a primordial origin. Non-Gaussian signals in the cosmic microwavebackground (CMB) are generated by primordial magnetic fields as the magneticstresses and temperature anisotropy they induce depend quadratically on themagnetic field. We compute the CMB scalar trispectrum on large angular scales,for nearly scale-invariant magnetic fields, sourced via the Sachs-Wolfe effect.The trispectra induced by magnetic energy density and by magnetic scalaranisotropic stress are found to have typical magnitudes of approximately$10^{-29}$ and $10^{-19}$, respectively. The scalar anisotropic stresstrispectrum is also calculated in the flat-sky approximation and yields asimilar result. Observational limits on CMB non-Gaussianity from the Planckmission data allow us to set upper limits of $B_0 \lesssim 0.6 $ nG on thepresent value of the primordial cosmic magnetic field. Considering theinflationary magnetic curvature mode in the trispectrum can further tighten themagnetic field upper limit to $B_0 \lesssim 0.05 $ nG. These sub-nanoGaussconstraints from the magnetic trispectrum are the most stringent limits so faron the strength of primordial magnetic fields, on megaparsec scales,significantly better than the limits obtained from the CMB bispectrum and theCMB power spectrum.

Primordial Magnetic Field Limits from CMB Trispectrum - Scalar Modes and Planck Constraints