Constraint on the primordial vector mode and its magnetic field generation from seven-year Wilkinson Microwave Anisotropy Probe observations

Abstract

It is often stated that the cosmological vector mode has only a decaying mode and gives negligible effects on cosmological observations. The vector mode, however, has a growing mode if there exists anisotropic stress in energy distributions in the universe. In this paper, we consider a primordial vector mode sustained by free-streaming neutrinos and its associated magnetic field generation. First, we put an observational constraint on the amount of the vector mode from the 7-year WMAP data. The constraint is found as ${r}_{v}\ensuremath{\lesssim}\ensuremath{-}\frac{r}{40}+0.012$, where ${r}_{v}$ and $r$ are the amounts of vector and tensor perturbation amplitudes with respect to the scalar one, respectively. Second, 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 $B\ensuremath{\lesssim}{10}^{\ensuremath{-}22}\text{ }\text{ }\mathrm{G}(\frac{{r}_{v}}{0.012}{)}^{1/2}(\frac{k}{0.002}{)}^{({n}_{v}+1)/2}$, with ${n}_{v}$ being a spectral index of the vector mode. We find a nontrivial cancellation of the magnetic field generation in the radiation-dominated era, which creates a characteristic cutoff in the magnetic field spectrum around $k\ensuremath{\approx}1.0\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$.