Generation of Large-Scale Magnetic Fields in Single-Field Inflation

TL;DR

This work analyzes how breaking conformal invariance during inflation through a time-dependent gauge-kinetic function $f(\phi)$ can generate large-scale magnetic fields with observationally allowed spectra. By solving the gauge-field dynamics under the ansatz $f\propto a^{\alpha}$ and connecting $f(\phi)$ to inflationary potentials, the authors uncover two main branches of viable magnetogenesis: $\alpha\approx -3$ (power-law inflation) and $\alpha\approx 2$ (small-field inflation). They derive constraints from CMB, BBN, RM, and dynamo requirements, and show that back-reaction imposes strong conditions on the reheating history, favoring prolonged reheating with a low inflation scale for the $\alpha\approx -3$ branch, while the $\alpha\approx 2$ branch can avoid back-reaction but is hard to realize in string-inspired models. Overall, consistent models exist only in restricted regions of parameter space, with large-field models remaining problematic for natural model-building. The work highlights how the interplay between the gauge coupling dynamics, inflationary dynamics, and reheating shapes the viability of inflationary magnetogenesis and points to specific avenues for constructing string- or supergravity-motivated realizations.

Abstract

We consider the generation of large-scale magnetic fields in slow-roll inflation. The inflaton field is described in a supergravity framework where the conformal invariance of the electromagnetic field is generically and naturally broken. For each class of inflationary scenarios, we determine the functional dependence of the gauge coupling that is consistent with the observations on the magnetic field strength at various astrophysical scales and, at the same time, avoid a back-reaction problem. Then, we study whether the required coupling functions can naturally emerge in well-motivated, possibly string-inspired, models. We argue that this is non trivial and can be realized only for a restricted class of scenarios. This includes power-law inflation where the inflaton field is interpreted as a modulus. However, this scenario seems to be consistent only if the energy scale of inflation is low and the reheating stage prolonged. Another reasonable possibility appears to be small field models since no back-reaction problem is present in this case but, unfortunately, the corresponding scenario cannot be justified in a stringy framework. Finally, large field models do not lead to sensible model building.

Figures (5)

• Figure 1: Amplitude of the magnetic power spectrum (solid line) at the end of inflation, as given by the function ${\cal F}(\alpha )$ defined in Eq. (\ref{['eq:defF']}), in terms of the index $\alpha$ characterizing the shape of the gauge coupling. An almost scale-invariant power spectrum, $\beta \simeq -2$, has been assumed such that $\gamma \equiv \alpha (1+\beta )\simeq -\alpha$. The divergence at $\alpha =-1/2$ signals the transition between the two branches of the spectrum. The amplitude of the electric power spectrum at the end of inflation is also displayed (dotted line). As shown in the following, the amplitude is given by the function ${\cal G}(\alpha )$ defined in Eq. (\ref{['eq:defG']}).
• Figure 2: Spectral index $n_{_{B}}$ (solid red line) of the magnetic power spectrum as a function of the index $\alpha$, assuming, as before, a background expansion closed to de Sitter, i.e.$\beta \simeq -2$. Scale invariance of the magnetic power spectrum corresponds to the values $\alpha =2$ and $\alpha =-3$. The dashed red line represents the quantity $\delta$ (see the text) as a function of $\alpha$. The corresponding quantities for the electric field are also displayed, namely the spectral index $n_{_{E}}$ (green dotted line) and the function $\iota$ (green dotted-dashed line), see the definition of $\iota$ in the text after Eq. (\ref{['eq:defG']}) and Eq. (\ref{['eq:psE']}). Scale-invariance of the magnetic power spectrum corresponds to a spectral index of the electric power spectrum given by $n_{_{E}}=-2$ for $\alpha =-3$ and $n_{_{E}}=2$ for $\alpha =2$. Scale invariance of the electric power spectrum is realized for $\alpha =-2$ or $\alpha =3$.
• Figure 3: Constraints on the amplitude of the present-day magnetic field at different scales. The bounds come from CMB measurements, Big Bang nucleosynthesis (BBN) and Faraday rotation (RM). The dynamo theoretical constraint is also plotted. The two solid blue lines represent the reddest spectra compatible with the CMB constraint at the Hubble scale in the case of a general model of inflation and in the case of large field models, respectively, while the dotted green line is the reddest spectrum compatible with the CMB constraint and the dynamo limit (valid in the case of a general model). Finally, the two dashed red lines are the bluest spectra (for a general model and for large field, respectively) compatible with the dynamo constraint.
• Figure 4: Constraints obtained from the WMAP3 data on the parameter $\ln R$ at $1\sigma$ and $2\sigma$, see Ref. Martin:2006rs. The solid lines represent the variation of $\ln R$ as a function of the energy density at the end of inflation for various reheating temperatures (or, equivalently, for various energy densities at the end of reheating). No observational constraint on $T_{\rm reh}$ is obtained in the case of large field models.
• Figure 5: Inflationary (solid line), magnetic (dashed line) and electric (dotted line) energy densities at the end of inflation in the case where $\alpha =-3$, $H_{\rm inf}/m_{{\mathrm{Pl}}} =10^{-22}$$n_{_{B}}=0$ and $n_{_{E}}=-2$. This situation corresponds to the limiting case treated in the text where there is no back-reaction problem, the electric energy density being always below the background energy density and where the dynamo constraint is still satisfied.