Primordial Magnetic Field Limits from Cosmic Microwave Background
  Bispectrum of Magnetic Passive Scalar Modes
  Primordial magnetic fields lead to non-Gaussian signals in the cosmic
microwave background (CMB) even at the lowest order, as magnetic stresses and
the temperature anisotropy they induce depend quadratically on the magnetic
field. In contrast, CMB non-Gaussianity due to inflationary scalar
perturbations arises only as a higher order effect. Apart from a compensated
scalar mode, stochastic primordial magnetic fields also produce scalar
anisotropic stress that remains uncompensated till neutrino decoupling. This
gives rise to an adiabatic-like scalar perturbation mode that evolves passively
thereafter (called the passive mode). We compute the CMB reduced bispectrum
($b_{l_{_1}l_{_2}l_{_3}}$) induced by this passive mode, sourced via the
Sachs-Wolfe effect, on large angular scales. For any configuration of
bispectrum, taking a partial sum over mode-coupling terms, we find a typical
value of $l_1(l_1+1)l_3(l_3+1) b_{l_{_1}l_{_2}l_{_3}} \sim 6-9 \times
10^{-16}$, for a magnetic field of $B_0 \sim 3$ nG, assuming a nearly
scale-invariant magnetic spectrum . We also evaluate, in full, the bispectrum
for the squeezed collinear configuration over all angular mode-coupling terms
and find $l_1(l_1+1)l_3(l_3+1) b_{l_{_1}l_{_2}l_{_3}} \approx -1.4 \times
10^{-16}$. These values are more than $\sim 10^6$ times larger than the
previously calculated magnetic compensated scalar mode CMB bispectrum.
Observational limits on the bispectrum from WMAP7 data allow us to set upper
limits of $B_0 \sim 2$ nG on the present value of the cosmic magnetic field of
primordial origin. This is over 10 times more stringent than earlier limits on
$B_0$ based on the compensated mode bispectrum.
