Constraint on the primordial vector mode and its magnetic field
  generation from seven-year Wilkinson Microwave Anisotropy Probe Observations
  A primordial vector mode and its associated magnetic field generation are
investigated. Firstly, we put an observational constraint on the amount of the
primordial vector mode from the seven-year WMAP data. The constraint is found
as $r_v \lesssim -\frac{r}{40}+0.012$, where $r_v$ and $r$ are the amounts of
vector and tensor perturbation amplitudes with respect to the scalar one,
respectively. Secondly, we calculate the spectrum of magnetic fields inevitably
created from the primordial vector mode, given the constraint on $r_v$. It is
found that the maximum amount of magnetic fields generated from the vector mode
is given by $B\lesssim 10^{-22}{\rm G} (\frac{r_v}{0.012})^{1/2}
(\frac{k}{0.002})^{(n_v+1)/2}$ with $n_v$ being a spectral index of the vector
mode. We find a non-trivial cancellation of the magnetic field generation in
the radiation dominated era, which creates a characteristic cut off in the
magnetic field spectrum around $k\approx 1.0$ Mpc$^{-1}$.
