Cosmic microwave background bispectrum of tensor passive modes induced
  from primordial magnetic fields
  If the seed magnetic fields exist in the early Universe, tensor components of
their anisotropic stresses are not compensated prior to neutrino decoupling and
the tensor metric perturbations generated from them survive passively.
Consequently, due to the decay of these metric perturbations after
recombination, the so-called integrated Sachs-Wolfe effect, the large-scale
fluctuations of CMB radiation are significantly boosted. This kind of CMB
anisotropy is called the "tensor passive mode." Because these fluctuations
deviate largely from the Gaussian statistics due to the quadratic dependence on
the strength of the Gaussian magnetic field, not only the power spectrum but
also the higher-order correlations have reasonable signals. With these motives,
we compute the CMB bispectrum induced by this mode. When the magnetic spectrum
obeys a nearly scale-invariant shape, we obtain an estimation of a typical
value of the normalized reduced bispectrum as $\ell_1(\ell_1 +
1)\ell_3(\ell_3+1)|b_{\ell_1\ell_2\ell_3}| \sim (130-6) \times 10^{-16} (B_{1
\rm Mpc} / 4.7 {\rm nG})^6$ depending on the energy scale of the magnetic field
production from $10^{14}$GeV to $10^3$GeV. Here, $B_{1 {\rm Mpc}}$ is the
strength of the primordial magnetic field smoothed on $1 {\rm Mpc}$. From the
above estimation and the current observational constraint on the primordial
non-Gaussianity, we get a rough constraint on the magnetic field strength as
$B_{1 {\rm Mpc}} < 2.6 - 4.4 {\rm nG}$.
