Abstract
We propose a novel scenario to generate primordial magnetic fields during inflation induced by an oscillating coupling of the electromagnetic field to the inflaton. This resonant mechanism has two key advantages over previous proposals. First of all, it generates a narrow band of magnetic fields at any required wavelength, thereby allaying the usual problem of a strongly blue spectrum and its associated backreaction. Secondly, it avoids the need for a strong coupling as the coupling is oscillating rather than growing or decaying exponentially. Despite these major advantages, we find that the backreaction is still far too large during inflation if the generated magnetic fields are required to have a strength of \U0001d4aa(10−15 Gauss) today on observationally interesting scales. We provide a more general no-go argument, proving that this problem will apply to any model in which the magnetic fields are generated on subhorizon scales and freeze after horizon crossing.
Highlights
Background evolutionBefore we turn to a careful examination of the solution (2.6) and its consequences, we need to understand how a Mathieu equation can arise in the context of inflationary models
The simplest way to obtain the correct time evolution is to solve for the background dynamics, which returns the functional form for φ(η), and engineer a coupling functional which translates this into the desired oscillations
We have made a novel proposal for the generation of primordial magnetic fields induced by an oscillating axial interaction between the inflaton and the EM field
Summary
We will focus on one specific case, the Mathieu equation, bearing in mind that any periodic coupling term would exhibit similar properties, and share qualitatively our results Such an oscillating term with a constant, or slowly time-varying frequency, automatically selects and amplifies only a very small range of Fourier modes, leaving all other modes almost unperturbed. If a mode A is resonantly amplified through the coupling to the inflaton, its amplitude roughly grows like |A| ∝ eμωη, where μ is the k-dependent characteristic exponent or Floquet index, and ω is the frequency of the driving term in the Mathieu equation (see appendix B for details) For this frequency will be kept constant. This will be alleviated when we allow for a time-dependent frequency which will enlarge the range of modes that undergo resonant amplification
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