Primordial Magnetic Field Limits from CMB Trispectrum - Scalar Modes and
  Planck Constraints
  Cosmic magnetic fields are observed to be coherent on large scales and could
have a primordial origin. Non-Gaussian signals in the cosmic microwave
background (CMB) are generated by primordial magnetic fields as the magnetic
stresses and temperature anisotropy they induce depend quadratically on the
magnetic 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 scalar
anisotropic stress are found to have typical magnitudes of approximately
$10^{-29}$ and $10^{-19}$, respectively. The scalar anisotropic stress
trispectrum is also calculated in the flat-sky approximation and yields a
similar result. Observational limits on CMB non-Gaussianity from the Planck
mission data allow us to set upper limits of $B_0 \lesssim 0.6 $ nG on the
present value of the primordial cosmic magnetic field. Considering the
inflationary magnetic curvature mode in the trispectrum can further tighten the
magnetic field upper limit to $B_0 \lesssim 0.05 $ nG. These sub-nanoGauss
constraints from the magnetic trispectrum are the most stringent limits so far
on the strength of primordial magnetic fields, on megaparsec scales,
significantly better than the limits obtained from the CMB bispectrum and the
CMB power spectrum.
