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 $-\frac{1}{4}I(\varphi )F_{\mu \nu }{\tilde{F}}^{\mu \nu }$, where $I(\varphi )$ is a pseudoscalar function of a non-trivial background field $\varphi$. 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 $I(k,\eta )=g\mathrm{/}(-k\eta {)}^{\beta }$ during inflation, while it is negligibly small subsequently. Here, $g$ is a positive dimensionless constant, $k$ the wavenumber, $\eta$ the conformal time, and $\beta$ a real positive number. We find that only when $\beta =1$ and $0.1 \lesssim g \lesssim 2$ astrophysically interesting fields can be produced as excitation of the vacuum, and that they are maximally helical.