Revisiting constraints on primordial vector modes and implications for sourced magnetic fields and observed $EB$ power spectrum

We revisit regular primordial vector modes sustained by the anisotropic stress of free-streaming neutrinos. We consider two classes of neutrino-sector initial conditions, the neutrino velocity isocurvature mode ($\nu\mathrm{VI}$) and the neutrino octupole mode ($\nu\mathrm{OCT}$). We update their observational constraints using current cosmological data, and examine the impact of including the BICEP/Keck 2018 $B$-mode polarization data. From an MCMC analysis, we obtain the 95\% C.L. upper bounds on the vector-to-scalar ratio as $r_\mathrm{v}<1.55\times10^{-4}$ and $r_\mathrm{v}<1.04\times10^{-2}$ for the $\nu\mathrm{VI}$ and $\nu\mathrm{OCT}$ modes at the vector pivot scale $k_{0} = 0.01\,{\rm Mpc}^{-1}$, respectively. We then study two consequences of these bounds. First, we estimate the magnetic fields inevitably generated in the pre-recombination plasma associated with the vector modes. We find that the magnetic-field amplitude at recombination with a coherent length of $1~{\rm Mpc}$ is bounded by $B\sim\mathcal{O}(10^{-23})\,{\rm G}$ and $B\sim\mathcal{O}(10^{-21})\,{\rm G}$ for the $\nu\mathrm{VI}$ and $\nu\mathrm{OCT}$ modes, respetively, which is too small to provide the seed of magnetic fields observed today. Second, assuming the helical vector mode, we compute the induced CMB $EB$ spectrum. We show that even a fully helical primordial vector mode cannot reproduce the currently observed $EB$ signal while remaining consistent with parity-even CMB constraints.