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 Mpc } / 4.7 { nG } ) ^ { 6 } depending on the energy scale of the magnetic field production from 10 ^ { 14 } GeV to 10 ^ { 3 } GeV .
Here , B _ { 1 { Mpc } } is the strength of the primordial magnetic field smoothed on 1 { 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 { Mpc } } < 2.6 - 4.4 { nG } .