Cosmic microwave background bispectrum of vector modes induced from primordial magnetic fields

We calculate the cosmic microwave background bispectrum of vector modes induced from primordial magnetic fields. We take into account the full angular dependence of the bispectrum and discuss the amplitude and also the shape of the bispectrum in $\mathrm{\ell }$ space. In the squeezed limit as $\ell_1 = \ell_2 \gg \ell_3$,weestimateatypicalvaluesofthenormalizedreducedbispectrumas$\ell_1(\ell_1 + 1)\ell_3(\ell_3+1)|b_{\ell_1\ell_2\ell_3}| \sim 2 \times 10^{-19}$,forthestrengthoftheprimordialmagneticfieldsmoothedon$1 {\rm Mpc}$scale$B_{1 \rm Mpc} = 4.7 \rm nG$ assuming a nearly scale-invariant spectrum of magnetic fields. We find that a constraint on the magnetic field strength will be roughly estimated as $B_{1\mathrm{M}\mathrm{p}\mathrm{c}}<10\mathrm{n}\mathrm{G}$ if PLANCK places a limit on the nonlinearity parameter of local-type configuration as $\mathrm{\mid }{f}_{\mathrm{N}\mathrm{L}}^{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}\mathrm{\mid }<5$.