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We investigate bounds on the strength of the primordial magnetic field (PMF)from the cosmic microwave background (CMB) bispectra of the intensity(temperature) modes induced from the auto- and cross-correlated bispectra ofthe scalar and tensor components of the PMF anisotropic stress. At first, weconstruct a general formula for the CMB intensity and polarization bispectrafrom PMFs composed of any type of perturbation. Then we derive an approximateexpression which traces the exact shape of the CMB bispectrum in order toreduce the computation time with respect to a large number of the multipoleconfigurations, and also show that the non-Gaussian structure coming from PMFsis classified as the local-type configuration. Computing the signal-to-noiseratio on the basis of the approximate formula with the information of theinstrumental noises and resolutions, we find expected upper bounds on themagnetic field strength, when the magnetic spectrum is nearly scale invariant($n_B = -2.9$), smoothed on $1 {\rm Mpc}$ scale at 95% confidence level fromthe WMAP and PLANCK experiments as $B_{1 \rm Mpc} < 4.0 - 6.7 {\rm nG}$ and$3.8 - 6.5 {\rm nG}$, respectively, depending on the energy scale of themagnetic field production from $10^{14} {\rm GeV}$ to $10^3 {\rm GeV}$. Our newconsequences imply slight overestimations by the previous rough discussions.

Optimal limits on primordial magnetic fields from CMB temperature bispectrum of passive modes