Helical Magnetic Fields from Inflation

TL;DR

The paper investigates primordial seed magnetic field generation during inflation by coupling electromagnetism to a pseudoscalar via ${\cal L} \ni -\tfrac{1}{4} I F_{\mu\nu}\tilde{F}^{\mu\nu}$ with $I(\phi)$ peaking at large scales. A toy model with $I_k(\eta)=\dfrac{g}{(-k\eta)^{{\beta}}}$ shows that significant, maximally helical fields arise only for $\beta=1$ and $0.1\lesssim g\lesssim 2$, with the field strength on cosmological scales potentially exceeding seed thresholds; backreaction on inflation is negligible for plausible reheating temperatures. The generated magnetic helicity is nonzero mainly for $\beta=1$, yielding left-handed, nearly maximal helicity $H_{\lambda}^{\rm today} \sim - (B_{\lambda,-}^{\rm today})^2 \lambda$, albeit below current CMB-detectable levels. The results are supported by two particle-physics realizations, linking the required I-term to pseudoscalar couplings and showing parameter regions that realize the needed $\beta$ and $g$. Overall, the work provides a concrete mechanism for inflationary, helical seed fields capable of seeding galactic magnetism while maintaining negligible backreaction.

Abstract

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.

Figures (1)

• Figure 1: Actual magnetic field in the case $\beta = 1$ as a function of $g$ at the comoving scales $\lambda = 10\hbox{kpc}$ (thin lines) and $\lambda = 1\hbox{Mpc}$ (thick lines), for a reheating temperature $T_1 = 10^{-2} m_{\rm Pl}$. Dashed and continuous lines refer to the cases $\alpha = 1$ (positive helicity states) and $\alpha = 2$ (negative helicity states), respectively. The horizontal dotted lines refer to the minimum seed fields required for dynamo amplification, $B \simeq 10^{-33} \hbox{G}$, and to directly explain galactic magnetism, $B \simeq 10^{-14} \hbox{G}$.