Primordial Spectrum of Gauge Fields from Inflation

TL;DR

This work addresses the origin of primordial gauge fields during inflation and their potential to seed galactic magnetic fields. It introduces a natural mechanism in which conformal invariance is broken by the backreaction of a charged scalar in scalar electrodynamics, producing an effectively massive gauge field and an almost scale-invariant spectrum with $B_ extell \propto \ell^{-1}$. The authors extend the mechanism to the Standard Model Z field via Higgs backreaction and trace how the amplified spectrum can be transferred to the hypercharge and, after electroweak symmetry breaking, to the photon, yielding seed fields of order $10^{-29}$ G on 100 pc and potentially larger through preheating. They compute the magnetic-field spectrum and show that, across plausible inflation scales, these seeds can satisfy galactic dynamo requirements, with preheating offering an additional pathway to stronger seeds.

Abstract

We show that conformal invariance of gauge fields is naturally broken in inflation, having as a consequence amplification of gauge fields. The resulting spectrum of the field strength is approximately B_L ~ L^(-1), where L is the relevant coherence scale. One realisation of our scenario is scalar electrodynamics with a scalar whose mass is large enough to evade observational constraints - the obvious candidates being supersymmetric partners of the standard-model fermions. Our mechanism also leads naturally to amplification of the standard-model Z-boson field due to its coupling to the electroweak Higgs field. At preheating, the spectrum of the Z field is transferred to the hypercharge field, which remains frozen in the plasma and is converted into a magnetic field at the electroweak phase transition. With a reasonable model of field evolution one obtains a magnetic field strength of the order of $10^{-29}$ Gauss on a scale of 100 pc, the size of the largest turbulent eddy in a virialised galaxy. Resonant amplification in preheating can lead to primordial fields as large as $10^{-24}$ Gauss, consistent with the seed field required for the galactic dynamo mechanism.

Figures (3)

• Figure 1: Evolution of (the imaginary part of) the gauge-field amplitude ${\rm Im}\,[{\cal A}_{\vec{k}}]$ (red solid curve) and its momentum ${\rm Im}\,[\partial_\tau{\cal A}_{\vec{k}}]$ (blue dotted curve) for $k/H_{\rm I}=0.05$ near the inflation-radiation transition, which occurs at $\tau=\tau_{\rm I} =\pm H_{\rm I}^{-1}$. As a consequence of broken conformal invariance, toward the end of inflation the amplitude decreases and the derivative increases, leading to an enhanced amplitude in the radiation era. The normalisation is chosen so that both functions would have amplitude equal to unity in the conformally evolving case, and we use $\nu=0.2$ for the purpose of illustration.
• Figure 2: Projection of the primordial spectrum of the standard-model $Z$ field first onto the hypercharge field $Y$ (at the inflation-radiation transition) and then onto the photon field $A$ (at the electroweak phase transition).
• Figure 3: Magnetic-field spectra and relevant seed-field bounds. In green ( dash-dot-dot-dot) we show the vacuum spectrum $B_\ell\propto \ell^{-2}$ obtained from preheating, assuming an amplification factor of $10^5$. At the comoving scale $\ell_c\sim 10$ kpc, $B_{\ell_c}\sim 10^{-50}$ G. In red ( dots and solid) the spectrum $B_\ell\propto \ell^{-3/2+\nu}$ from inflation in our mechanism is shown, with and without preheating amplification. For this spectrum, $B_{\ell_c}\sim 10^{-29}$ G and $B_{\ell_c}\sim 10^{-34}$ G, respectively. We also show ( blue dash-dots) the spectrum enhanced by helical turbulence (at $\ell_c\sim 10$ kpc an enhancement of about 20 is obtained). This is to be compared with the dynamo bounds rescaled by a factor $5\times 10^3$ (see main text) $B_{\rm seed} \hbox{$\stackrel{>}{\sim}$} 2\times 10^{-27}$ G for a universe with critical matter density, and $B_{\rm seed} \hbox{$\stackrel{>}{\sim}$} 2\times 10^{-34}$ G for a flat, low-density universe.