Can slow roll inflation induce relevant helical magnetic fields?

TL;DR

This work analyzes the generation of helical magnetic fields during single-field slow-roll inflation via a parity-violating coupling $f(\phi)F\tilde{F}$. It derives the perturbative regime bound $|f_N|<1$, solves the EM mode equations under slow-roll for common coupling forms, and shows that the resulting magnetic spectrum is always blue with $B^2(k) \propto k$; helicity drives an inverse cascade in the radiation era, but even with this transfer, typical high-scale inflation yields seed fields ($\sim 10^{-40}$ G at $0.1$ Mpc) too weak for galactic dynamos unless the inflation scale is very low ($T_* \lesssim 10^4$ GeV) and the coupling is large. The study finds deviations from slow-roll do not qualitatively alter the spectrum within observational bounds, reinforcing the challenge of achieving cosmologically relevant seeds in this setup. Overall, the results constrain viable magnetogenesis scenarios during inflation and highlight the importance of post-inflationary evolution in determining present-day field strengths.

Abstract

We study the generation of helical magnetic fields during single field inflation induced by an axial coupling of the electromagnetic field to the inflaton. During slow roll inflation, we find that such a coupling always leads to a blue spectrum with $B^2(k) \propto k$, as long as the theory is treated perturbatively. The magnetic energy density at the end of inflation is found to be typically too small to backreact on the background dynamics of the inflaton. We also show that a short deviation from slow roll does not result in strong modifications to the shape of the spectrum. We calculate the evolution of the correlation length and the field amplitude during the inverse cascade and viscous damping of the helical magnetic field in the radiation era after inflation. We conclude that except for low scale inflation with very strong coupling, the magnetic fields generated by such an axial coupling in single field slow roll inflation with perturbative coupling to the inflaton are too weak to provide the seeds for the observed fields in galaxies and clusters.

Figures (6)

• Figure 1: The evolution of the two helicity modes ($+$ solid red, $-$ dashed blue) for wavenumber $k=10/{\rm Mpc}$ is shown as a function of the number of e-foldings, $N$, during inflation. Here we consider an axial coupling function $f\propto\varphi^p$, as discussed in detail in Sec. \ref{['sec:powcoup']}. Both modes feel the axial coupling only around horizon crossing, at $N_{\rm cross}\simeq 10$, while the evolution ceases and the modes saturate quickly after crossing. Here inflation ends at $N\simeq 60$.
• Figure 2: The symmetric (solid, red) and antisymmetric (dashed, blue) power spectra of the magnetic fields, $P_{S,A}/k$, are plotted as a function of the effective coupling constant $|{f_N}|$. For vanishing coupling, the vacuum solution, $P_S/k=1$ and $P_A/k=0$ is recovered, while the larger the coupling, the smaller the difference between $P_S$ and $P_A$. As discussed in the main text, for the theory to be perturbative we must require $|{f_N}|<1$ and therefore, the amplification of the magnetic fields is small.
• Figure 3: The coupling term $|{f_N}|$ at Hubble crossing of the wave number $k$ is plotted for the power law coupling with $p=2$, $f_0=1/5$. The horizontal dotted blue line represents the constant value of ${f_N}=4/5$ in the slow roll approximation. The dashed black line indicates the behaviour of the coupling term for the potential in Eq. (\ref{['e:pot-step']}) with $\beta=0$, while the solid red line is the same for $\beta \neq 0$. For the later case, we have used the best fit values of the parameters of the potential as in Ref. Hazra:2010ve. The bump in the coupling term for $\beta \neq 0$ arises due to the short period of deviation from slow roll.
• Figure 4: The relative deviation of the magnetic field power spectrum from an exact $k$-spectrum is plotted as a function of the wave number $k$ for the power law coupling with $p=2$, $f_0=1/5$. The dashed black line indicates the numerical solution for slow roll inflation ($\beta=0$) while the solid red line is the spectrum resulting from a deviation from slow roll. The horizontal dotted blue line is the magnetic field spectrum from the slow roll approximation, ${f_N}=4/5$. The deviation from this approximated spectrum is always small. Only the scales which exit the Hubble radius around the time when a deviation from slow roll or equivalently a bump in the coupling term occurs are affected more strongly.
• Figure 5: The Reynolds number $\mathrm{Re}(T)$ (black solid) and the comoving correlation scale $k_c(T)/k_{\rm fin}$ (red dashed) are shown in log scale as a function of $\log(T_*/T)$ with $T_*=10^{14}\,{\rm GeV}$. Turbulence and with it the inverse cascade terminate at $T_{\rm fin}\simeq 1\,{\rm GeV}$ and $k_{\rm fin}=k_c(t_{\rm fin}) \simeq (10^{-12}\,{\rm Mpc})^{-1}$.