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Helical Magnetic Fields from Inflation

We analyze the generation of seed magnetic fields during de Sitter inflation considering a non-invariant conformal term in the electromagnetic Lagrangian of the form $-\frac14 I(ϕ) F_{μν} \widetilde{F}^{μν}$, where $I(ϕ)$ is a pseudoscalar function of a non-trivial background field $ϕ$. In particular, we consider a toy model, that could be realized owing to the coupling between the photon and either a (tachyonic) massive pseudoscalar field and a massless pseudoscalar field non-minimally coupled to gravity, where $I$ follows a simple power-law behavior $I(k,η) = g/(-kη)^β$ during inflation, while it is negligibly small subsequently. Here, $g$ is a positive dimensionless constant, $k$ the wavenumber, $η$ the conformal time, and $β$ a real positive number. We find that only when $β= 1$ and $0.1 \lesssim g \lesssim 2$ astrophysically interesting fields can be produced as excitation of the vacuum, and that they are maximally helical.