Primordial Magnetic Field Generated in Natural Infation

TL;DR

This paper investigates the generation of primordial magnetic fields during Natural Inflation using the gauge invariant ${f^2}FF$ model, revealing limitations in achieving scale invariance and issues with backreaction, thus challenging the model's viability.

Contribution

It provides a detailed analysis of primordial magnetic field generation in Natural Inflation, highlighting constraints and the narrow parameter space for viable magnetic field production.

Findings

Scale invariance cannot be achieved under standard slow roll conditions.

Backreaction problem constrains the model parameters.

Electric field energy can be lower than magnetic field energy within a narrow parameter range.

Abstract

We study the simple gauge invariant model ${f^2}FF$ as a way to generate primordial magnetic fields (PMF) in Natural Inflation (NI). We compute both magnetic and electric spectra generated by the ${f^2}FF$ model in NI for different values of model parameters and find that both de Sitter and power law expansion lead to the same results at sufficiently large number of e-foldings. We also find that the necessary scale invariance property of the PMF cannot be obtained in NI in first order of slow roll limits under the constraint of inflationary potential, $V\left( 0 \right) \simeq 0$. Furthermore, if this constraint is relaxed to achieve scale invariance, then the model suffers from the backreaction problem for the co-moving wave number, $k \lesssim 8.0\times 10^{-7} \rm{Mpc^{-1}}$ and Hubble parameter, $H_i \gtrsim 1.25\times 10^{-3} \rm{M_{\rm{Pl}}}$. The former can be considered as a lower bound of $k$ and the later as an upper bound of $H_i$ for a model which is free from the backreaction problem. Further, we show that there is a narrow range of the height of the potential $\Lambda $ around ${\Lambda _{\min }} \approx 0.00874{M_{{\rm{Pl}}}}$ and of $k$ around ${k_{\min }} \sim 0.0173{\rm{Mp}}{{\rm{c}}^{ - 1}}$, at which the energy of the electric field can fall below the energy of the magnetic field. The range of $k$ lies within some observable scales. However, the relatively short range of $k$ presents a challenge to the viability of this model.